\(\int \frac {1}{(\cos (3 x)+\sin (2 x))^6} \, dx\) [6]

Optimal result

Integrand size = 11, antiderivative size = 1 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^6} \, dx=0 \] Output:

0
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 6.13 (sec) , antiderivative size = 547, normalized size of antiderivative = 547.00 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^6} \, dx =\text {Too large to display} \] Input:

Integrate[(Cos[3*x] + Sin[2*x])^(-6),x]
 

Output:

(-1792*Sqrt[2/(5*(5 - Sqrt[5]))]*(6566 + 2937*Sqrt[5])*ArcTanh[(Sec[x/2]*( 
4*Cos[x/2] - Sin[x/2] - Sqrt[5]*Sin[x/2]))/Sqrt[2*(5 - Sqrt[5])]])/15625 - 
 (1792*Sqrt[2/(5*(5 + Sqrt[5]))]*(-6566 + 2937*Sqrt[5])*ArcTanh[(Sec[x/2]* 
(4*Cos[x/2] - Sin[x/2] + Sqrt[5]*Sin[x/2]))/Sqrt[2*(5 + Sqrt[5])]])/15625 
+ 1/(40*(Cos[x/2] - Sin[x/2])^4) + 379/(240*(Cos[x/2] - Sin[x/2])^2) + Sin 
[x/2]/(20*(Cos[x/2] - Sin[x/2])^5) + (379*Sin[x/2])/(120*(Cos[x/2] - Sin[x 
/2])^3) + (6001*Sin[x/2])/(30*(Cos[x/2] - Sin[x/2])) + Sin[x/2]/(312500*(C 
os[x/2] + Sin[x/2])^5) - 1/(625000*(Cos[x/2] + Sin[x/2])^4) + (139*Sin[x/2 
])/(1875000*(Cos[x/2] + Sin[x/2])^3) - 139/(3750000*(Cos[x/2] + Sin[x/2])^ 
2) + (31*Sin[x/2])/(18750*(Cos[x/2] + Sin[x/2])) + (128*(4*Cos[x] + 7*Sin[ 
2*x]))/(625*(-1 + 2*Cos[2*x] + 2*Sin[x])^5) - (96*(139*Cos[x] + 192*Sin[2* 
x]))/(3125*(-1 + 2*Cos[2*x] + 2*Sin[x])^4) - (2624*(3446*Cos[x] + 4473*Sin 
[2*x]))/(234375*(-1 + 2*Cos[2*x] + 2*Sin[x])^2) + (512*(18289*Cos[x] + 244 
25*Sin[2*x]))/(78125*(-1 + 2*Cos[2*x] + 2*Sin[x])) + (32*(20019*Cos[x] + 2 
6527*Sin[2*x]))/(46875*(-1 + 2*Cos[2*x] + 2*Sin[x])^3)
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 34.64 (sec) , antiderivative size = 6697, normalized size of antiderivative = 6697.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {3042, 4830, 2462, 7239, 2036, 7293, 7239, 2036, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (3 x))^6} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (3 x))^6}dx\)

\(\Big \downarrow \) 4830

\(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{17}}{\left (-\tan ^6\left (\frac {x}{2}\right )-4 \tan ^5\left (\frac {x}{2}\right )+15 \tan ^4\left (\frac {x}{2}\right )-15 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 2462

\(\displaystyle 2 \int \left (-\frac {384 \left (20803 \tan ^2\left (\frac {x}{2}\right )+132116 \tan \left (\frac {x}{2}\right )-114597\right )}{78125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {413}{4 \left (\tan \left (\frac {x}{2}\right )-1\right )^2}+\frac {283}{312500 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {1024 \left (11284 \tan ^3\left (\frac {x}{2}\right )-117793 \tan ^2\left (\frac {x}{2}\right )+913340 \tan \left (\frac {x}{2}\right )-8373233\right )}{3125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {79}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )^3}-\frac {31}{125000 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {32768 \left (2337536 \tan ^3\left (\frac {x}{2}\right )-10776721 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )-410962073\right )}{3125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {83}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )^4}+\frac {7}{25000 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {262144 \left (76868836 \tan ^3\left (\frac {x}{2}\right )-259826893 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )-5110635645\right )}{625 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {1}{\left (\tan \left (\frac {x}{2}\right )-1\right )^5}-\frac {1}{15625 \left (\tan \left (\frac {x}{2}\right )+1\right )^5}-\frac {4194304 \left (432157152 \tan ^3\left (\frac {x}{2}\right )-1176722883 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )-9670732787\right )}{125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^5}+\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^6}+\frac {1}{31250 \left (\tan \left (\frac {x}{2}\right )+1\right )^6}-\frac {67108864 \left (768284400 \tan ^3\left (\frac {x}{2}\right )-1777619297 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )[phone]\right )}{25 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^6}\right )d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{17}}{\left (1-\tan \left (\frac {x}{2}\right )\right )^6 \left (\tan \left (\frac {x}{2}\right )+1\right )^6 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 2036

\(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{17}}{\left (1-\tan ^2\left (\frac {x}{2}\right )\right )^6 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (-\frac {384 \left (20803 \tan ^2\left (\frac {x}{2}\right )+132116 \tan \left (\frac {x}{2}\right )-114597\right )}{78125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {413}{4 \left (\tan \left (\frac {x}{2}\right )-1\right )^2}+\frac {283}{312500 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {1024 \left (11284 \tan ^3\left (\frac {x}{2}\right )-117793 \tan ^2\left (\frac {x}{2}\right )+913340 \tan \left (\frac {x}{2}\right )-8373233\right )}{3125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {79}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )^3}-\frac {31}{125000 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {32768 \left (2337536 \tan ^3\left (\frac {x}{2}\right )-10776721 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )-410962073\right )}{3125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {83}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )^4}+\frac {7}{25000 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {262144 \left (76868836 \tan ^3\left (\frac {x}{2}\right )-259826893 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )-5110635645\right )}{625 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {1}{\left (\tan \left (\frac {x}{2}\right )-1\right )^5}-\frac {1}{15625 \left (\tan \left (\frac {x}{2}\right )+1\right )^5}-\frac {4194304 \left (432157152 \tan ^3\left (\frac {x}{2}\right )-1176722883 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )-9670732787\right )}{125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^5}+\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^6}+\frac {1}{31250 \left (\tan \left (\frac {x}{2}\right )+1\right )^6}-\frac {67108864 \left (768284400 \tan ^3\left (\frac {x}{2}\right )-1777619297 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )[phone]\right )}{25 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^6}\right )d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{17}}{\left (1-\tan \left (\frac {x}{2}\right )\right )^6 \left (\tan \left (\frac {x}{2}\right )+1\right )^6 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 2036

\(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{17}}{\left (1-\tan ^2\left (\frac {x}{2}\right )\right )^6 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (-\frac {384 \left (20803 \tan ^2\left (\frac {x}{2}\right )+132116 \tan \left (\frac {x}{2}\right )-114597\right )}{78125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {413}{4 \left (\tan \left (\frac {x}{2}\right )-1\right )^2}+\frac {283}{312500 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {1024 \left (11284 \tan ^3\left (\frac {x}{2}\right )-117793 \tan ^2\left (\frac {x}{2}\right )+913340 \tan \left (\frac {x}{2}\right )-8373233\right )}{3125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {79}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )^3}-\frac {31}{125000 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {32768 \left (2337536 \tan ^3\left (\frac {x}{2}\right )-10776721 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )-410962073\right )}{3125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {83}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )^4}+\frac {7}{25000 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {262144 \left (76868836 \tan ^3\left (\frac {x}{2}\right )-259826893 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )-5110635645\right )}{625 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {1}{\left (\tan \left (\frac {x}{2}\right )-1\right )^5}-\frac {1}{15625 \left (\tan \left (\frac {x}{2}\right )+1\right )^5}-\frac {4194304 \left (432157152 \tan ^3\left (\frac {x}{2}\right )-1176722883 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )-9670732787\right )}{125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^5}+\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^6}+\frac {1}{31250 \left (\tan \left (\frac {x}{2}\right )+1\right )^6}-\frac {67108864 \left (768284400 \tan ^3\left (\frac {x}{2}\right )-1777619297 \tan ^2\left (\frac {x}{2}\right )[phone] \tan \left (\frac {x}{2}\right )[phone]\right )}{25 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^6}\right )d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {64 \sqrt {\frac {1}{5} \left (8695085546875825+1259018405558642 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625}+\frac {128 \sqrt {\frac {2}{5} \left (141850517196265+61058226939259 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625}-\frac {512 \left (287077635668047+37671805033778 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{78125 \sqrt {5-2 \sqrt {5}}}+\frac {128 \left (227459771388189+29848451614037 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625 \sqrt {5-2 \sqrt {5}}}+\frac {256 \left (5489320781629+720324020529 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{78125 \sqrt {5-2 \sqrt {5}}}+\frac {256 \left (16506895269+2155237343 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{78125 \sqrt {5-2 \sqrt {5}}}+\frac {896 \left (2807548135034-1249009950747 \sqrt {5}\right ) \sqrt {\frac {1}{5} \left (491045+219602 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625}-\frac {448 \left (5615096852059-2498020161765 \sqrt {5}\right ) \sqrt {\frac {1}{5} \left (491045+219602 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625}+\frac {32256 \sqrt {\frac {1}{5} \left (214045+93962 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{78125}-\frac {256 \left (230908136503-105038638193 \sqrt {5}\right ) \sqrt {\frac {1}{5} \left (27365+12238 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625}+\frac {512 \left (19506957513983-8892542155366 \sqrt {5}\right ) \sqrt {\frac {1}{5} \left (27365+12238 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625}-\frac {256 \left (38783007150929-17680045788591 \sqrt {5}\right ) \sqrt {\frac {1}{5} \left (27365+12238 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625}+\frac {128 \left (36557138261-32065231708 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625 \sqrt {5 \left (5-2 \sqrt {5}\right )}}+\frac {128 \left (2344698376661-2047707259012 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{3125 \sqrt {5 \left (5-2 \sqrt {5}\right )}}+\frac {64 \left (478188055400771-416754786287400 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{3125 \sqrt {5 \left (5-2 \sqrt {5}\right )}}-\frac {128 \left (1207230227055423-1052157602365700 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625 \sqrt {5 \left (5-2 \sqrt {5}\right )}}-\frac {384 \left (4684724605-694973687 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{78125 \left (5-2 \sqrt {5}\right )^{3/2}}-\frac {384 \left (2146986122415-271127200652 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{78125 \left (5-2 \sqrt {5}\right )^{3/2}}-\frac {192 \left (100122572902055-12191554480147 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{15625 \left (5-2 \sqrt {5}\right )^{3/2}}+\frac {384 \left (252458103049835-30750708350822 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{78125 \left (5-2 \sqrt {5}\right )^{3/2}}+\frac {192 \sqrt {\frac {2}{5} \left (1283236265-546155621 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{78125}+\frac {128 \left (1207230227055423+1052157602365700 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{15625 \sqrt {5 \left (5+2 \sqrt {5}\right )}}-\frac {64 \left (478188055400771+416754786287400 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{3125 \sqrt {5 \left (5+2 \sqrt {5}\right )}}+\frac {384 \left (252458103049835+30750708350822 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{78125 \left (5+2 \sqrt {5}\right )^{3/2}}-\frac {192 \left (100122572902055+12191554480147 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{15625 \left (5+2 \sqrt {5}\right )^{3/2}}+\frac {256 \left (10834214641097+3422915573747 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{3125 \left (5+2 \sqrt {5}\right )^{5/2}}-\frac {128 \left (2344698376661+2047707259012 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{3125 \sqrt {5 \left (5+2 \sqrt {5}\right )}}-\frac {512 \left (5448795748864+1721873203251 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{3125 \left (5+2 \sqrt {5}\right )^{5/2}}-\frac {448 \left (3095282642645+1259907104707 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{3125 \left (5+2 \sqrt {5}\right )^{7/2}}+\frac {896 \left (1547641167700+629953483667 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{3125 \left (5+2 \sqrt {5}\right )^{7/2}}-\frac {384 \left (2146986122415+271127200652 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{78125 \left (5+2 \sqrt {5}\right )^{3/2}}-\frac {128 \left (36557138261+32065231708 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{15625 \sqrt {5 \left (5+2 \sqrt {5}\right )}}+\frac {256 \left (63376917959+20830860117 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{3125 \left (5+2 \sqrt {5}\right )^{5/2}}-\frac {384 \left (4684724605+694973687 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{78125 \left (5+2 \sqrt {5}\right )^{3/2}}-\frac {192 \sqrt {\frac {2}{5} \left (1283236265+546155621 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{78125}-\frac {64 \left (70200244+15804901 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{15625 \sqrt {5+2 \sqrt {5}}}+\frac {128 \left (23286995+8604389 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{15625 \left (5+2 \sqrt {5}\right )^{3/2}}+\frac {256 \left (16506895269-2155237343 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{78125 \sqrt {5+2 \sqrt {5}}}+\frac {256 \left (5489320781629-720324020529 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{78125 \sqrt {5+2 \sqrt {5}}}+\frac {128 \left (227459771388189-29848451614037 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{15625 \sqrt {5+2 \sqrt {5}}}-\frac {512 \left (287077635668047-37671805033778 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{78125 \sqrt {5+2 \sqrt {5}}}+\frac {32256 \sqrt {\frac {1}{5} \left (214045-93962 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{78125}+\frac {413}{4 \left (1-\tan \left (\frac {x}{2}\right )\right )}-\frac {283}{312500 \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {384 \left (190956686348191+70232532869112 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {192 \left (75739463941761+27857474680675 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {256 \left (29049620294933+12530618400665 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {512 \left (14601697674756+6298324166197 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {448 \left (4743334946099+2105727304083 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {896 \left (2371667471554+1052863652181 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {384 \left (1604731721111+587667248314 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {256 \left (153775059291+66029930783 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {384 \left (3294777231+1178916155 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {32256 \left (149+28 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {128 \left (5+2 \sqrt {5}\right ) \left (4 \left (1058670128002455-385796246167204 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-10752202292671511 \sqrt {5}+22674642150788265\right )}{390625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {64 \left (5+2 \sqrt {5}\right ) \left (2 \left (838814340800575-305677284706193 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-4259505595015581 \sqrt {5}+8982612562234025\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {128 \left (5+2 \sqrt {5}\right ) \left (2 \left (20243363702855-7377021460613 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+3 \left (72483429866485-34375018383377 \sqrt {5}\right )\right )}{390625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {128 \left (5+2 \sqrt {5}\right ) \left (2 \left (60982102915-22237603823 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+3 \left (219934835325-104313926017 \sqrt {5}\right )\right )}{390625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {64 \left (5+2 \sqrt {5}\right ) \left (2 \left (23286995-8604389 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+5 \left (52094303-24659491 \sqrt {5}\right )\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {32256 \left (149-28 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {384 \left (3294777231-1178916155 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {256 \left (153775059291-66029930783 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {384 \left (1604731721111-587667248314 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {896 \left (2371667471554-1052863652181 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {448 \left (4743334946099-2105727304083 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {512 \left (14601697674756-6298324166197 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {256 \left (29049620294933-12530618400665 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {192 \left (75739463941761-27857474680675 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {384 \left (190956686348191-70232532869112 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {64 \left (5-2 \sqrt {5}\right ) \left (2 \left (23286995+8604389 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+5 \left (52094303+24659491 \sqrt {5}\right )\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {128 \left (5-2 \sqrt {5}\right ) \left (2 \left (60982102915+22237603823 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+3 \left (219934835325+104313926017 \sqrt {5}\right )\right )}{390625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {128 \left (5-2 \sqrt {5}\right ) \left (2 \left (20243363702855+7377021460613 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+3 \left (72483429866485+34375018383377 \sqrt {5}\right )\right )}{390625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {64 \left (5-2 \sqrt {5}\right ) \left (2 \left (838814340800575+305677284706193 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+4259505595015581 \sqrt {5}+8982612562234025\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {128 \left (5-2 \sqrt {5}\right ) \left (4 \left (1058670128002455+385796246167204 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+10752202292671511 \sqrt {5}+22674642150788265\right )}{390625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {79}{16 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {31}{250000 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {512 \left (19941917468015+4553851413459 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {1024 \left (10025246711810+2288225481473 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {896 \left (2659401689665+1041966628217 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {1792 \left (1329700835960+520983317797 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {512 \left (108575988625+22599535333 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {21504 \left (465-158 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {256 \left (2+\sqrt {5}\right ) \left (\left (658669747853780-313959519751479 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-1849406048571739 \sqrt {5}+4212659920524695\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {128 \left (2+\sqrt {5}\right ) \left (\left (261202918204845-124505681862349 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+2 \left (835236766137515-366677887621217 \sqrt {5}\right )\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {256 \left (2+\sqrt {5}\right ) \left (\left (5649608248090-2689240523719 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+13 \left (2799380718185-1229201796273 \sqrt {5}\right )\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {256 \left (2+\sqrt {5}\right ) \left (\left (12844317645-6074671979 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+8 \left (10517615395-4623412868 \sqrt {5}\right )\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {21504 \left (465+158 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {512 \left (108575988625-22599535333 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {1792 \left (1329700835960-520983317797 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {896 \left (2659401689665-1041966628217 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {1024 \left (10025246711810-2288225481473 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {512 \left (19941917468015-4553851413459 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {256 \left (2-\sqrt {5}\right ) \left (\left (12844317645+6074671979 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+8 \left (10517615395+4623412868 \sqrt {5}\right )\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {256 \left (2-\sqrt {5}\right ) \left (\left (5649608248090+2689240523719 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+13 \left (2799380718185+1229201796273 \sqrt {5}\right )\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {128 \left (2-\sqrt {5}\right ) \left (\left (261202918204845+124505681862349 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+2 \left (835236766137515+366677887621217 \sqrt {5}\right )\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {256 \left (2-\sqrt {5}\right ) \left (\left (658669747853780+313959519751479 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1849406048571739 \sqrt {5}+4212659920524695\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {83}{24 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {7}{75000 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {86016 \left (781-344 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {7168 \left (287734200366-10897016587 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {3584 \left (575468433231-21794047649 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {1024 \left (5+2 \sqrt {5}\right ) \left (2 \left (88400228942955-38783007150929 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-487718616082673 \sqrt {5}+1084383745917085\right )}{234375 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {1024 \left (5+2 \sqrt {5}\right ) \left (4 \left (44462710776830-19506957513983 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-490667197301287 \sqrt {5}+1090942266635645\right )}{234375 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {1024 \left (5+2 \sqrt {5}\right ) \left (2 \left (525193190965-230908136503 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-2948581903981 \sqrt {5}[phone]\right )}{234375 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {3584 \left (575468433231+21794047649 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {7168 \left (287734200366+10897016587 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{46875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {86016 \left (781+344 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {1024 \left (5-2 \sqrt {5}\right ) \left (2 \left (525193190965+230908136503 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )[phone] \sqrt {5}[phone]\right )}{234375 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {1024 \left (5-2 \sqrt {5}\right ) \left (4 \left (44462710776830+19506957513983 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )[phone] \sqrt {5}+1090942266635645\right )}{234375 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {1024 \left (5-2 \sqrt {5}\right ) \left (2 \left (88400228942955+38783007150929 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )[phone] \sqrt {5}+1084383745917085\right )}{234375 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {1}{4 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{62500 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {73728 \left (7345-3282 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {1024 \left (2+\sqrt {5}\right ) \left (\left (12490100808825-5615096852059 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+4 \left (19782738172590-8832717227141 \sqrt {5}\right )\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {2048 \left (2+\sqrt {5}\right ) \left (\left (6245049753735-2807548135034 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-17665434236969 \sqrt {5}[phone]\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {73728 \left (7345+3282 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {2048 \left (2-\sqrt {5}\right ) \left (\left (6245049753735+2807548135034 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )[phone] \sqrt {5}[phone]\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {1024 \left (2-\sqrt {5}\right ) \left (\left (12490100808825+5615096852059 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+4 \left (19782738172590+8832717227141 \sqrt {5}\right )\right )}{15625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {1}{10 \left (1-\tan \left (\frac {x}{2}\right )\right )^5}-\frac {1}{156250 \left (\tan \left (\frac {x}{2}\right )+1\right )^5}-\frac {16384 \left (5+2 \sqrt {5}\right ) \left (4 \left (131745-58918 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-37327 \sqrt {5}+83465\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^5}-\frac {16384 \left (5-2 \sqrt {5}\right ) \left (4 \left (131745+58918 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+37327 \sqrt {5}+83465\right )}{78125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^5}\right )\)

Input:

Int[(Cos[3*x] + Sin[2*x])^(-6),x]
 

Output:

2*((192*Sqrt[(2*(1283236265 - 546155621*Sqrt[5]))/5]*ArcTanh[(1 - Sqrt[5] 
+ Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/78125 + (384*(252458103049835 - 30750708 
350822*Sqrt[5])*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/(78 
125*(5 - 2*Sqrt[5])^(3/2)) - (192*(100122572902055 - 12191554480147*Sqrt[5 
])*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/(15625*(5 - 2*Sq 
rt[5])^(3/2)) - (384*(2146986122415 - 271127200652*Sqrt[5])*ArcTanh[(1 - S 
qrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/(78125*(5 - 2*Sqrt[5])^(3/2)) - ( 
384*(4684724605 - 694973687*Sqrt[5])*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt 
[5 - 2*Sqrt[5]]])/(78125*(5 - 2*Sqrt[5])^(3/2)) - (128*(1207230227055423 - 
 1052157602365700*Sqrt[5])*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqr 
t[5]]])/(15625*Sqrt[5*(5 - 2*Sqrt[5])]) + (64*(478188055400771 - 416754786 
287400*Sqrt[5])*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/(31 
25*Sqrt[5*(5 - 2*Sqrt[5])]) + (128*(2344698376661 - 2047707259012*Sqrt[5]) 
*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/(3125*Sqrt[5*(5 - 
2*Sqrt[5])]) + (128*(36557138261 - 32065231708*Sqrt[5])*ArcTanh[(1 - Sqrt[ 
5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/(15625*Sqrt[5*(5 - 2*Sqrt[5])]) - (25 
6*(38783007150929 - 17680045788591*Sqrt[5])*Sqrt[(27365 + 12238*Sqrt[5])/5 
]*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/15625 + (512*(195 
06957513983 - 8892542155366*Sqrt[5])*Sqrt[(27365 + 12238*Sqrt[5])/5]*ArcTa 
nh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/15625 - (256*(2309081...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2036

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p 
_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2 
*x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && E 
qQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && Gt 
Q[a2, 0]))
 

rule 2462

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u*Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ 
[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 0 
] && RationalFunctionQ[u, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4830

Int[(cos[(n_.)*((c_.) + (d_.)*(x_))]*(b_.) + (a_.)*sin[(m_.)*((c_.) + (d_.) 
*(x_))])^(p_), x_Symbol] :> Simp[2/d   Subst[Int[Simplify[TrigExpand[a*Sin[ 
2*m*ArcTan[x]] + b*Cos[2*n*ArcTan[x]]]]^p/(1 + x^2), x], x, Tan[(1/2)*(c + 
d*x)]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && Intege 
rQ[n]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.59 (sec) , antiderivative size = 374, normalized size of antiderivative = 374.00

\[-\frac {256 \left (-\frac {167530}{3}-\frac {3544418 \tan \left (\frac {x}{2}\right )}{3}-6401814 \tan \left (\frac {x}{2}\right )^{2}[phone] \tan \left (\frac {x}{2}\right )^{3}[phone] \tan \left (\frac {x}{2}\right )^{4}-\frac {580831840 \tan \left (\frac {x}{2}\right )^{5}}{3}-\frac {7305842584 \tan \left (\frac {x}{2}\right )^{6}}{3}+\frac {11662085264 \tan \left (\frac {x}{2}\right )^{7}}{3}[phone] \tan \left (\frac {x}{2}\right )^{8}-\frac {113167452436 \tan \left (\frac {x}{2}\right )^{9}}{3}[phone] \tan \left (\frac {x}{2}\right )^{10}-\frac {39613206364 \tan \left (\frac {x}{2}\right )^{11}}{3}-\frac {11145371096 \tan \left (\frac {x}{2}\right )^{12}}{3}+\frac {8320893136 \tan \left (\frac {x}{2}\right )^{13}}{3}+\frac {477319784 \tan \left (\frac {x}{2}\right )^{14}}{3}-225437760 \tan \left (\frac {x}{2}\right )^{15}-\frac {54674698 \tan \left (\frac {x}{2}\right )^{16}}{3}+\frac {23176198 \tan \left (\frac {x}{2}\right )^{17}}{3}+1375914 \tan \left (\frac {x}{2}\right )^{18}+64606 \tan \left (\frac {x}{2}\right )^{19}\right )}{78125 \left (\tan \left (\frac {x}{2}\right )^{4}+4 \tan \left (\frac {x}{2}\right )^{3}-14 \tan \left (\frac {x}{2}\right )^{2}+4 \tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {1792 \left (-\frac {3629}{2}-\frac {8119 \sqrt {5}}{10}\right ) \operatorname {arctanh}\left (\frac {2 \tan \left (\frac {x}{2}\right )-2 \sqrt {5}+2}{2 \sqrt {5-2 \sqrt {5}}}\right )}{15625 \sqrt {5-2 \sqrt {5}}}+\frac {1792 \left (\frac {3629}{2}-\frac {8119 \sqrt {5}}{10}\right ) \operatorname {arctanh}\left (\frac {2 \tan \left (\frac {x}{2}\right )+2+2 \sqrt {5}}{2 \sqrt {5+2 \sqrt {5}}}\right )}{15625 \sqrt {5+2 \sqrt {5}}}-\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {1}{5 \left (\tan \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {83}{12 \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {79}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {413}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )}-\frac {1}{78125 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {1}{31250 \left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {7}{37500 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {31}{125000 \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {283}{156250 \left (\tan \left (\frac {x}{2}\right )+1\right )}\]

Input:

int(1/(cos(3*x)+sin(2*x))^6,x)
 

Output:

-256/78125*(-167530/3-3544418/3*tan(1/2*x)-6401814*tan(1/2*x)^2+17962534*t 
an(1/2*x)^3+194410616*tan(1/2*x)^4-580831840/3*tan(1/2*x)^5-7305842584/3*t 
an(1/2*x)^6+11662085264/3*tan(1/2*x)^7+11244480332*tan(1/2*x)^8-1131674524 
36/3*tan(1/2*x)^9+39264676220*tan(1/2*x)^10-39613206364/3*tan(1/2*x)^11-11 
145371096/3*tan(1/2*x)^12+8320893136/3*tan(1/2*x)^13+477319784/3*tan(1/2*x 
)^14-225437760*tan(1/2*x)^15-54674698/3*tan(1/2*x)^16+23176198/3*tan(1/2*x 
)^17+1375914*tan(1/2*x)^18+64606*tan(1/2*x)^19)/(tan(1/2*x)^4+4*tan(1/2*x) 
^3-14*tan(1/2*x)^2+4*tan(1/2*x)+1)^5-1792/15625*(-3629/2-8119/10*5^(1/2))/ 
(5-2*5^(1/2))^(1/2)*arctanh(1/2*(2*tan(1/2*x)-2*5^(1/2)+2)/(5-2*5^(1/2))^( 
1/2))+1792/15625*(3629/2-8119/10*5^(1/2))/(5+2*5^(1/2))^(1/2)*arctanh(1/2* 
(2*tan(1/2*x)+2+2*5^(1/2))/(5+2*5^(1/2))^(1/2))-1/2/(tan(1/2*x)-1)^4-1/5/( 
tan(1/2*x)-1)^5-83/12/(tan(1/2*x)-1)^3-79/8/(tan(1/2*x)-1)^2-413/2/(tan(1/ 
2*x)-1)-1/78125/(tan(1/2*x)+1)^5+1/31250/(tan(1/2*x)+1)^4-7/37500/(tan(1/2 
*x)+1)^3+31/125000/(tan(1/2*x)+1)^2-283/156250/(tan(1/2*x)+1)
 

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.23 (sec) , antiderivative size = 640, normalized size of antiderivative = 640.00 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^6} \, dx=\text {Too large to display} \] Input:

integrate(1/(cos(3*x)+sin(2*x))^6,x, algorithm="fricas")
 

Output:

-1/468750*(135222722560*cos(x)^14 - 450050539520*cos(x)^12 + 564945510400* 
cos(x)^10 - 317707407360*cos(x)^8 + 67470466880*cos(x)^6 + 28000000*cos(x) 
^4 - 1344*sqrt(2)*(1024*cos(x)^15 - 6400*cos(x)^13 + 14400*cos(x)^11 - 152 
80*cos(x)^9 + 7820*cos(x)^7 - 1563*cos(x)^5 + 2*(1280*cos(x)^13 - 4480*cos 
(x)^11 + 5936*cos(x)^9 - 3512*cos(x)^7 + 781*cos(x)^5)*sin(x))*sqrt(279085 
621*sqrt(5) + 624054425)*log(-448*sqrt(2)*sqrt(279085621*sqrt(5) + 6240544 
25)*(9503*sqrt(5) - 21251)*cos(x) + 15670144*(sqrt(5) + 1)*sin(x) - 626805 
76) + 1344*sqrt(2)*(1024*cos(x)^15 - 6400*cos(x)^13 + 14400*cos(x)^11 - 15 
280*cos(x)^9 + 7820*cos(x)^7 - 1563*cos(x)^5 + 2*(1280*cos(x)^13 - 4480*co 
s(x)^11 + 5936*cos(x)^9 - 3512*cos(x)^7 + 781*cos(x)^5)*sin(x))*sqrt(27908 
5621*sqrt(5) + 624054425)*log(-448*sqrt(2)*sqrt(279085621*sqrt(5) + 624054 
425)*(9503*sqrt(5) - 21251)*cos(x) - 15670144*(sqrt(5) + 1)*sin(x) + 62680 
576) - 3*(1024*cos(x)^15 - 6400*cos(x)^13 + 14400*cos(x)^11 - 15280*cos(x) 
^9 + 7820*cos(x)^7 - 1563*cos(x)^5 + 2*(1280*cos(x)^13 - 4480*cos(x)^11 + 
5936*cos(x)^9 - 3512*cos(x)^7 + 781*cos(x)^5)*sin(x))*sqrt(-11202720095436 
8*sqrt(5) + 250500438630400)*log(-(9503*sqrt(5) + 21251)*sqrt(-11202720095 
4368*sqrt(5) + 250500438630400)*cos(x) + 15670144*(sqrt(5) - 1)*sin(x) + 6 
2680576) + 3*(1024*cos(x)^15 - 6400*cos(x)^13 + 14400*cos(x)^11 - 15280*co 
s(x)^9 + 7820*cos(x)^7 - 1563*cos(x)^5 + 2*(1280*cos(x)^13 - 4480*cos(x)^1 
1 + 5936*cos(x)^9 - 3512*cos(x)^7 + 781*cos(x)^5)*sin(x))*sqrt(-1120272...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^6} \, dx=\text {Timed out} \] Input:

integrate(1/(cos(3*x)+sin(2*x))**6,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^6} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(1/(cos(3*x)+sin(2*x))^6,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.12 (sec) , antiderivative size = 311, normalized size of antiderivative = 311.00 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^6} \, dx =\text {Too large to display} \] Input:

integrate(1/(cos(3*x)+sin(2*x))^6,x, algorithm="giac")
 

Output:

-2/46875*(9801627*tan(1/2*x)^29 + 207539160*tan(1/2*x)^28 + 1086782010*tan 
(1/2*x)^27 - 4091973852*tan(1/2*x)^26 - 39877891959*tan(1/2*x)^25 + 480575 
05836*tan(1/2*x)^24 + 611846565540*tan(1/2*x)^23 - 822349312680*tan(1/2*x) 
^22 - 4527294198541*tan(1/2*x)^21 + 9882693980368*tan(1/2*x)^20 + 85457576 
88614*tan(1/2*x)^19 - 36646833294420*tan(1/2*x)^18 + 7064910503625*tan(1/2 
*x)^17 + 60480105024596*tan(1/2*x)^16 - 43130634999688*tan(1/2*x)^15 - 462 
65973899504*tan(1/2*x)^14 + 58402675048825*tan(1/2*x)^13 + 9742317629080*t 
an(1/2*x)^12 - 35972981583386*tan(1/2*x)^11 + 7234969571388*tan(1/2*x)^10 
+ 9848510973859*tan(1/2*x)^9 - 4181395911980*tan(1/2*x)^8 - 850989600860*t 
an(1/2*x)^7 + 567880320536*tan(1/2*x)^6 + 50680356857*tan(1/2*x)^5 - 36634 
647232*tan(1/2*x)^4 - 3982452230*tan(1/2*x)^3 + 976976500*tan(1/2*x)^2 + 1 
90845707*tan(1/2*x) + 9052204)/(tan(1/2*x)^6 + 4*tan(1/2*x)^5 - 15*tan(1/2 
*x)^4 + 15*tan(1/2*x)^2 - 4*tan(1/2*x) - 1)^5 - 0.0179538142054400*log(tan 
(1/2*x) + 6.31375151468000) + 0.0179538142049280*log(tan(1/2*x) + 0.158384 
440325000) + 286.503025128960*log(tan(1/2*x) - 0.509525449494000) - 286.50 
3025128960*log(tan(1/2*x) - 1.96261050551000)
 

Mupad [B] (verification not implemented)

Time = 22.46 (sec) , antiderivative size = 802, normalized size of antiderivative = 802.00 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^6} \, dx=\text {Too large to display} \] Input:

int(1/(cos(3*x) + sin(2*x))^6,x)
 

Output:

((381691414*tan(x/2))/46875 + (15631624*tan(x/2)^2)/375 - (1592980892*tan( 
x/2)^3)/9375 - (73269294464*tan(x/2)^4)/46875 + (101360713714*tan(x/2)^5)/ 
46875 + (1135760641072*tan(x/2)^6)/46875 - (340395840344*tan(x/2)^7)/9375 
- (1672558364792*tan(x/2)^8)/9375 + (19697021947718*tan(x/2)^9)/46875 + (4 
823313047592*tan(x/2)^10)/15625 - (71945963166772*tan(x/2)^11)/46875 + (38 
96927051632*tan(x/2)^12)/9375 + (4672214003906*tan(x/2)^13)/1875 - (925319 
47799008*tan(x/2)^14)/46875 - (86261269999376*tan(x/2)^15)/46875 + (120960 
210049192*tan(x/2)^16)/46875 + (37679522686*tan(x/2)^17)/125 - (4886244439 
256*tan(x/2)^18)/3125 + (17091515377228*tan(x/2)^19)/46875 + (197653879607 
36*tan(x/2)^20)/46875 - (9054588397082*tan(x/2)^21)/46875 - (109646575024* 
tan(x/2)^22)/3125 + (81579542072*tan(x/2)^23)/3125 + (32038337224*tan(x/2) 
^24)/15625 - (26585261306*tan(x/2)^25)/15625 - (2727982568*tan(x/2)^26)/15 
625 + (144904268*tan(x/2)^27)/3125 + (27671888*tan(x/2)^28)/3125 + (653441 
8*tan(x/2)^29)/15625 + 18104408/46875)/(20*tan(x/2) + 85*tan(x/2)^2 - 560* 
tan(x/2)^3 - 3595*tan(x/2)^4 + 10004*tan(x/2)^5 + 57425*tan(x/2)^6 - 16160 
0*tan(x/2)^7 - 371075*tan(x/2)^8 + 1588100*tan(x/2)^9 - 292495*tan(x/2)^10 
 - 4918000*tan(x/2)^11 + 5124625*tan(x/2)^12 + 5671300*tan(x/2)^13 - 11792 
275*tan(x/2)^14 + 11792275*tan(x/2)^16 - 5671300*tan(x/2)^17 - 5124625*tan 
(x/2)^18 + 4918000*tan(x/2)^19 + 292495*tan(x/2)^20 - 1588100*tan(x/2)^21 
+ 371075*tan(x/2)^22 + 161600*tan(x/2)^23 - 57425*tan(x/2)^24 - 10004*t...
 

Reduce [F]

\[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^6} \, dx=\int \frac {1}{\cos \left (3 x \right )^{6}+6 \cos \left (3 x \right )^{5} \sin \left (2 x \right )+15 \cos \left (3 x \right )^{4} \sin \left (2 x \right )^{2}+20 \cos \left (3 x \right )^{3} \sin \left (2 x \right )^{3}+15 \cos \left (3 x \right )^{2} \sin \left (2 x \right )^{4}+6 \cos \left (3 x \right ) \sin \left (2 x \right )^{5}+\sin \left (2 x \right )^{6}}d x \] Input:

int(1/(cos(3*x)+sin(2*x))^6,x)
 

Output:

int(1/(cos(3*x)**6 + 6*cos(3*x)**5*sin(2*x) + 15*cos(3*x)**4*sin(2*x)**2 + 
 20*cos(3*x)**3*sin(2*x)**3 + 15*cos(3*x)**2*sin(2*x)**4 + 6*cos(3*x)*sin( 
2*x)**5 + sin(2*x)**6),x)