Integral de 1/(3sinxcosx+1) dx

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                                                         ___________    |        ___     ___   /       ___          |                                        ___________    |        ___     ___   /       ___          |                                                   |        ___     ___   /       ___          |                                                       |        ___     ___   /       ___          |                                                       |        ___     ___   /       ___          |                                                       |        ___     ___   /       ___          |                                                       |        ___     ___   /       ___          |                                                       |        ___     ___   /       ___          |                                                       |        ___     ___   /       ___          |                                                       |        ___     ___   /       ___          |                                            ___________    |        ___     ___   /       ___          |                                        ___________    |        ___     ___   /       ___          |                                           ___________    |        ___     ___   /       ___          |                                        ___________    |        ___     ___   /       ___          |                                        ___________    |        ___     ___   /       ___          |                                        ___________    |        ___     ___   /       ___          |         
  /                                                     /       ___     |  3   \/ 5    \/ 6 *\/  3 - \/ 5        /x\|                                       /       ___     |  3   \/ 5    \/ 6 *\/  3 - \/ 5        /x\|                                            ___    |  3   \/ 5    \/ 6 *\/  3 - \/ 5        /x\|                                                ___    |  3   \/ 5    \/ 6 *\/  3 - \/ 5        /x\|                                               ____    |  3   \/ 5    \/ 6 *\/  3 + \/ 5        /x\|                                               ____    |  3   \/ 5    \/ 6 *\/  3 + \/ 5        /x\|                                               ____    |  3   \/ 5    \/ 6 *\/  3 - \/ 5        /x\|                                               ____    |  3   \/ 5    \/ 6 *\/  3 - \/ 5        /x\|                                                ___    |  3   \/ 5    \/ 6 *\/  3 + \/ 5        /x\|                                                ___    |  3   \/ 5    \/ 6 *\/  3 + \/ 5        /x\|                                           /       ___     |  3   \/ 5    \/ 6 *\/  3 + \/ 5        /x\|                                       /       ___     |  3   \/ 5    \/ 6 *\/  3 + \/ 5        /x\|                                    ___   /       ___     |  3   \/ 5    \/ 6 *\/  3 + \/ 5        /x\|                                 ___   /       ___     |  3   \/ 5    \/ 6 *\/  3 + \/ 5        /x\|                                 ___   /       ___     |  3   \/ 5    \/ 6 *\/  3 - \/ 5        /x\|                                 ___   /       ___     |  3   \/ 5    \/ 6 *\/  3 - \/ 5        /x\|         
 |                                          721827843*\/  3 + \/ 5  *log|- - + ----- + -------------------- + tan|-||                           721827843*\/  3 + \/ 5  *log|- - + ----- - -------------------- + tan|-||                                270196287*\/ 6 *log|- - + ----- + -------------------- + tan|-||                                    270196287*\/ 6 *log|- - + ----- - -------------------- + tan|-||                                   120835453*\/ 30 *log|- - - ----- + -------------------- + tan|-||                                   120835453*\/ 30 *log|- - - ----- - -------------------- + tan|-||                                   120835453*\/ 30 *log|- - + ----- + -------------------- + tan|-||                                   120835453*\/ 30 *log|- - + ----- - -------------------- + tan|-||                                    270196287*\/ 6 *log|- - - ----- + -------------------- + tan|-||                                    270196287*\/ 6 *log|- - - ----- - -------------------- + tan|-||                               721827843*\/  3 + \/ 5  *log|- - - ----- + -------------------- + tan|-||                           721827843*\/  3 + \/ 5  *log|- - - ----- - -------------------- + tan|-||                        322811225*\/ 5 *\/  3 + \/ 5  *log|- - - ----- + -------------------- + tan|-||                     322811225*\/ 5 *\/  3 + \/ 5  *log|- - - ----- - -------------------- + tan|-||                     322811225*\/ 5 *\/  3 + \/ 5  *log|- - + ----- + -------------------- + tan|-||                     322811225*\/ 5 *\/  3 + \/ 5  *log|- - + ----- - -------------------- + tan|-||         
 |          1                                                           \  2     2              2                \2//                                                       \  2     2              2                \2//                                                   \  2     2              2                \2//                                                       \  2     2              2                \2//                                                       \  2     2              2                \2//                                                       \  2     2              2                \2//                                                       \  2     2              2                \2//                                                       \  2     2              2                \2//                                                       \  2     2              2                \2//                                                       \  2     2              2                \2//                                                           \  2     2              2                \2//                                                       \  2     2              2                \2//                                                          \  2     2              2                \2//                                                       \  2     2              2                \2//                                                       \  2     2              2                \2//                                                       \  2     2              2                \2//         
 | ------------------- dx = C - ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------
 | 3*sin(x)*cos(x) + 1                          ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________                   ___________                                                           ___________
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/                               - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5     - 1614056125*\/  3 + \/ 5   - 604177265*\/ 6  + 270196287*\/ 30  + 721827843*\/ 5 *\/  3 + \/ 5  

$$\int \frac{1}{3 \sin{\left(x \right)} \cos{\left(x \right)} + 1}\, dx = C - \frac{322811225 \sqrt{5} \sqrt{\sqrt{5} + 3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} - \frac{\sqrt{5}}{2} + \frac{\sqrt{6} \sqrt{\sqrt{5} + 3}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} - \frac{120835453 \sqrt{30} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} - \frac{\sqrt{5}}{2} + \frac{\sqrt{6} \sqrt{\sqrt{5} + 3}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} + \frac{270196287 \sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} - \frac{\sqrt{5}}{2} + \frac{\sqrt{6} \sqrt{\sqrt{5} + 3}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} + \frac{721827843 \sqrt{\sqrt{5} + 3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} - \frac{\sqrt{5}}{2} + \frac{\sqrt{6} \sqrt{\sqrt{5} + 3}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} - \frac{721827843 \sqrt{\sqrt{5} + 3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} - \frac{\sqrt{6} \sqrt{3 - \sqrt{5}}}{2} + \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} - \frac{270196287 \sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} - \frac{\sqrt{6} \sqrt{3 - \sqrt{5}}}{2} + \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} + \frac{120835453 \sqrt{30} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} - \frac{\sqrt{6} \sqrt{3 - \sqrt{5}}}{2} + \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} + \frac{322811225 \sqrt{5} \sqrt{\sqrt{5} + 3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} - \frac{\sqrt{6} \sqrt{3 - \sqrt{5}}}{2} + \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} - \frac{721827843 \sqrt{\sqrt{5} + 3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} + \frac{\sqrt{6} \sqrt{3 - \sqrt{5}}}{2} + \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} - \frac{270196287 \sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} + \frac{\sqrt{6} \sqrt{3 - \sqrt{5}}}{2} + \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} + \frac{120835453 \sqrt{30} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} + \frac{\sqrt{6} \sqrt{3 - \sqrt{5}}}{2} + \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} + \frac{322811225 \sqrt{5} \sqrt{\sqrt{5} + 3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} + \frac{\sqrt{6} \sqrt{3 - \sqrt{5}}}{2} + \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} - \frac{322811225 \sqrt{5} \sqrt{\sqrt{5} + 3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{6} \sqrt{\sqrt{5} + 3}}{2} - \frac{3}{2} - \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} - \frac{120835453 \sqrt{30} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{6} \sqrt{\sqrt{5} + 3}}{2} - \frac{3}{2} - \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} + \frac{270196287 \sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{6} \sqrt{\sqrt{5} + 3}}{2} - \frac{3}{2} - \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}} + \frac{721827843 \sqrt{\sqrt{5} + 3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{6} \sqrt{\sqrt{5} + 3}}{2} - \frac{3}{2} - \frac{\sqrt{5}}{2} \right)}}{- 1614056125 \sqrt{\sqrt{5} + 3} - 604177265 \sqrt{6} + 270196287 \sqrt{30} + 721827843 \sqrt{5} \sqrt{\sqrt{5} + 3}}$$

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