Integrand size = 11, antiderivative size = 154 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^5} \, dx=-\frac {1063 \text {arctanh}(\cos (x))}{262144}+\frac {3603 \text {arctanh}\left (\sqrt {2} \cos (x)\right )}{16384 \sqrt {2}}-\frac {27761 \sec (x)}{262144}-\frac {13349 \sec ^3(x)}{786432}-\frac {6143 \sec ^5(x)}{1310720}-\frac {635 \sec ^7(x)}{458752}-\frac {1477 \sec ^9(x)}{4718592}+\frac {59 \sec ^9(x) \sec (2 x)}{524288}+\frac {159 \sec ^9(x) \sec ^2(2 x)}{262144}+\frac {149 \sec ^9(x) \sec ^3(2 x)}{393216}+\frac {45 \sec ^9(x) \sec ^4(2 x)}{131072}-\frac {37 \csc ^2(x) \sec ^9(x) \sec ^4(2 x)}{262144}-\frac {\csc ^4(x) \sec ^9(x) \sec ^4(2 x)}{131072} \] Output:
-1063/262144*arctanh(cos(x))+3603/32768*arctanh(cos(x)*2^(1/2))*2^(1/2)-27 761/262144*sec(x)-13349/786432*sec(x)^3-6143/1310720*sec(x)^5-635/458752*s ec(x)^7-1477/4718592*sec(x)^9+59/524288*sec(x)^9*sec(2*x)+159/262144*sec(x )^9*sec(2*x)^2+149/393216*sec(x)^9*sec(2*x)^3+45/131072*sec(x)^9*sec(2*x)^ 4-37/262144*csc(x)^2*sec(x)^9*sec(2*x)^4-1/131072*csc(x)^4*sec(x)^9*sec(2* x)^4
Result contains complex when optimal does not.
Time = 6.26 (sec) , antiderivative size = 1603, normalized size of antiderivative = 10.41 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^5} \, dx =\text {Too large to display} \] Input:
Integrate[(Sin[3*x] + Sin[5*x])^(-5),x]
Output:
(-102055729*Cos[x]^10*(-Sin[x] + Sin[3*x])^5)/(1290240*(Sin[3*x] + Sin[5*x ])^5) - ((3603/32 + (3603*I)/32)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)* Sec[x/2]*(Cos[x/2] + Sin[x/2])]*Cos[x]^10*(-Sin[x] + Sin[3*x])^5)/(Sin[3*x ] + Sin[5*x])^5 + ((3603/32 + (3603*I)/32)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)* (-1)^(3/4)*Sec[x/2]*(Cos[x/2] - Sin[x/2])]*Cos[x]^10*(-Sin[x] + Sin[3*x])^ 5)/(Sin[3*x] + Sin[5*x])^5 - (63*Cos[x]^10*Csc[x/2]^2*(-Sin[x] + Sin[3*x]) ^5)/(1024*(Sin[3*x] + Sin[5*x])^5) - (Cos[x]^10*Csc[x/2]^4*(-Sin[x] + Sin[ 3*x])^5)/(2048*(Sin[3*x] + Sin[5*x])^5) - (1063*Cos[x]^10*Log[Cos[x/2]]*(- Sin[x] + Sin[3*x])^5)/(256*(Sin[3*x] + Sin[5*x])^5) + (1063*Cos[x]^10*Log[ Sin[x/2]]*(-Sin[x] + Sin[3*x])^5)/(256*(Sin[3*x] + Sin[5*x])^5) + (63*Cos[ x]^10*Sec[x/2]^2*(-Sin[x] + Sin[3*x])^5)/(1024*(Sin[3*x] + Sin[5*x])^5) + (Cos[x]^10*Sec[x/2]^4*(-Sin[x] + Sin[3*x])^5)/(2048*(Sin[3*x] + Sin[5*x])^ 5) - (Cos[x]^10*(-Sin[x] + Sin[3*x])^5)/(9216*(Cos[x/2] - Sin[x/2])^8*(Sin [3*x] + Sin[5*x])^5) - (503*Cos[x]^10*(-Sin[x] + Sin[3*x])^5)/(129024*(Cos [x/2] - Sin[x/2])^6*(Sin[3*x] + Sin[5*x])^5) - (35551*Cos[x]^10*(-Sin[x] + Sin[3*x])^5)/(430080*(Cos[x/2] - Sin[x/2])^4*(Sin[3*x] + Sin[5*x])^5) - ( 3876529*Cos[x]^10*(-Sin[x] + Sin[3*x])^5)/(2580480*(Cos[x/2] - Sin[x/2])^2 *(Sin[3*x] + Sin[5*x])^5) - (Cos[x]^10*Sin[x/2]*(-Sin[x] + Sin[3*x])^5)/(4 608*(Cos[x/2] - Sin[x/2])^9*(Sin[3*x] + Sin[5*x])^5) - (503*Cos[x]^10*Sin[ x/2]*(-Sin[x] + Sin[3*x])^5)/(64512*(Cos[x/2] - Sin[x/2])^7*(Sin[3*x] +...
Time = 0.61 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.53, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.273, Rules used = {3042, 4824, 27, 374, 27, 441, 441, 441, 27, 441, 27, 441, 27, 445, 27, 445, 27, 445, 27, 445, 27, 445, 25, 397, 219}
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.
| \(\displaystyle \int \frac {1}{(\sin (3 x)+\sin (5 x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (3 x)+\sin (5 x))^5}dx\) |
| \(\Big \downarrow \) 4824 |
| \(\displaystyle -\int -\frac {\sec ^{10}(x)}{32768 \left (1-2 \cos ^2(x)\right )^5 \left (1-\cos ^2(x)\right )^3}d\cos (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right )^5 \left (1-\cos ^2(x)\right )^3}d\cos (x)}{32768}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {\frac {1}{8} \int \frac {2 \left (13-21 \cos ^2(x)\right ) \sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^3}d\cos (x)+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \int \frac {\left (13-21 \cos ^2(x)\right ) \sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^3}d\cos (x)+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \int \frac {\left (123-95 \cos ^2(x)\right ) \sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^3}d\cos (x)+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (1851-2567 \cos ^2(x)\right ) \sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^3}d\cos (x)+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {3 \left (4639-5675 \cos ^2(x)\right ) \sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^3}d\cos (x)+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \int \frac {\left (4639-5675 \cos ^2(x)\right ) \sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^3}d\cos (x)+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}-\frac {1}{4} \int -\frac {8 \left (3485-3367 \cos ^2(x)\right ) \sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}d\cos (x)\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \int \frac {\left (3485-3367 \cos ^2(x)\right ) \sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}d\cos (x)+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (-\frac {1}{2} \int -\frac {4 \left (649 \cos ^2(x)+1477\right ) \sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \int \frac {\left (649 \cos ^2(x)+1477\right ) \sec ^{10}(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (-\frac {1}{9} \int -\frac {18 \left (2540-1477 \cos ^2(x)\right ) \sec ^8(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {1477}{9} \sec ^9(x)\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \int \frac {\left (2540-1477 \cos ^2(x)\right ) \sec ^8(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \left (-\frac {1}{7} \int -\frac {7 \left (6143-5080 \cos ^2(x)\right ) \sec ^6(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {2540}{7} \sec ^7(x)\right )-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \left (\int \frac {\left (6143-5080 \cos ^2(x)\right ) \sec ^6(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {2540 \sec ^7(x)}{7}\right )-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \left (-\frac {1}{5} \int -\frac {5 \left (13349-12286 \cos ^2(x)\right ) \sec ^4(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {2540}{7} \sec ^7(x)-\frac {6143 \sec ^5(x)}{5}\right )-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \left (\int \frac {\left (13349-12286 \cos ^2(x)\right ) \sec ^4(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {2540}{7} \sec ^7(x)-\frac {6143 \sec ^5(x)}{5}\right )-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \left (-\frac {1}{3} \int -\frac {3 \left (27761-26698 \cos ^2(x)\right ) \sec ^2(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {2540}{7} \sec ^7(x)-\frac {6143 \sec ^5(x)}{5}-\frac {13349 \sec ^3(x)}{3}\right )-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \left (\int \frac {\left (27761-26698 \cos ^2(x)\right ) \sec ^2(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {2540}{7} \sec ^7(x)-\frac {6143 \sec ^5(x)}{5}-\frac {13349 \sec ^3(x)}{3}\right )-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \left (-\int -\frac {56585-55522 \cos ^2(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {2540}{7} \sec ^7(x)-\frac {6143 \sec ^5(x)}{5}-\frac {13349 \sec ^3(x)}{3}-27761 \sec (x)\right )-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \left (\int \frac {56585-55522 \cos ^2(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {2540}{7} \sec ^7(x)-\frac {6143 \sec ^5(x)}{5}-\frac {13349 \sec ^3(x)}{3}-27761 \sec (x)\right )-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \left (57648 \int \frac {1}{1-2 \cos ^2(x)}d\cos (x)-1063 \int \frac {1}{1-\cos ^2(x)}d\cos (x)-\frac {2540}{7} \sec ^7(x)-\frac {6143 \sec ^5(x)}{5}-\frac {13349 \sec ^3(x)}{3}-27761 \sec (x)\right )-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} \left (2 \left (2 \left (2 \left (-1063 \text {arctanh}(\cos (x))+28824 \sqrt {2} \text {arctanh}\left (\sqrt {2} \cos (x)\right )-\frac {2540}{7} \sec ^7(x)-\frac {6143 \sec ^5(x)}{5}-\frac {13349 \sec ^3(x)}{3}-27761 \sec (x)\right )-\frac {1477 \sec ^9(x)}{9}\right )-\frac {59 \sec ^9(x)}{1-\cos ^2(x)}\right )+\frac {259 \sec ^9(x)}{\left (1-\cos ^2(x)\right )^2}\right )+\frac {1135 \sec ^9(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}\right )+\frac {151 \sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}\right )+\frac {5 \sec ^9(x)}{6 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}\right )+\frac {\sec ^9(x)}{4 \left (1-2 \cos ^2(x)\right )^4 \left (1-\cos ^2(x)\right )^2}}{32768}\) |
Input:
Int[(Sin[3*x] + Sin[5*x])^(-5),x]
Output:
(Sec[x]^9/(4*(1 - 2*Cos[x]^2)^4*(1 - Cos[x]^2)^2) + ((5*Sec[x]^9)/(6*(1 - 2*Cos[x]^2)^3*(1 - Cos[x]^2)^2) + ((151*Sec[x]^9)/(4*(1 - 2*Cos[x]^2)^2*(1 - Cos[x]^2)^2) + ((1135*Sec[x]^9)/(2*(1 - 2*Cos[x]^2)*(1 - Cos[x]^2)^2) + (3*((259*Sec[x]^9)/(1 - Cos[x]^2)^2 + 2*((-59*Sec[x]^9)/(1 - Cos[x]^2) + 2*((-1477*Sec[x]^9)/9 + 2*(-1063*ArcTanh[Cos[x]] + 28824*Sqrt[2]*ArcTanh[S qrt[2]*Cos[x]] - 27761*Sec[x] - (13349*Sec[x]^3)/3 - (6143*Sec[x]^5)/5 - ( 2540*Sec[x]^7)/7)))))/2)/4)/6)/4)/32768
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si mp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 )^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[((a_.)*sin[(m_.)*((c_.) + (d_.)*(x_))] + (b_.)*sin[(n_.)*((c_.) + (d_.) *(x_))])^(p_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Simplify[TrigExpand[a* Sin[m*ArcCos[x]] + b*Sin[n*ArcCos[x]]]]^p/Sqrt[1 - x^2], x], x, Cos[c + d*x ]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[(p - 1)/2, 0] && IntegerQ[(m - 1)/ 2] && IntegerQ[(n - 1)/2]
Time = 139.91 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.01
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(\frac {1}{524288 \left (1+\cos \left (x \right )\right )^{2}}+\frac {63}{524288 \left (1+\cos \left (x \right )\right )}-\frac {1063 \ln \left (1+\cos \left (x \right )\right )}{524288}-\frac {\frac {1043 \cos \left (x \right )^{7}}{256}-\frac {10193 \cos \left (x \right )^{5}}{1536}+\frac {11183 \cos \left (x \right )^{3}}{3072}-\frac {1389 \cos \left (x \right )}{2048}}{8 \left (2 \cos \left (x \right )^{2}-1\right )^{4}}+\frac {3603 \,\operatorname {arctanh}\left (\sqrt {2}\, \cos \left (x \right )\right ) \sqrt {2}}{32768}-\frac {1}{294912 \cos \left (x \right )^{9}}-\frac {13}{229376 \cos \left (x \right )^{7}}-\frac {3}{5120 \cos \left (x \right )^{5}}-\frac {265}{49152 \cos \left (x \right )^{3}}-\frac {2435}{32768 \cos \left (x \right )}-\frac {1}{524288 \left (\cos \left (x \right )-1\right )^{2}}+\frac {63}{524288 \left (\cos \left (x \right )-1\right )}+\frac {1063 \ln \left (\cos \left (x \right )-1\right )}{524288}\) | \(125\) |
| risch | \(-\frac {8744715 \,{\mathrm e}^{41 i x}+40585440 \,{\mathrm e}^{39 i x}+69873594 \,{\mathrm e}^{37 i x}+47900640 \,{\mathrm e}^{35 i x}-34247585 \,{\mathrm e}^{33 i x}-171621760 \,{\mathrm e}^{31 i x}-274841576 \,{\mathrm e}^{29 i x}-189119360 \,{\mathrm e}^{27 i x}+43279510 \,{\mathrm e}^{25 i x}+277989440 \,{\mathrm e}^{23 i x}+404201564 \,{\mathrm e}^{21 i x}+277989440 \,{\mathrm e}^{19 i x}+43279510 \,{\mathrm e}^{17 i x}-189119360 \,{\mathrm e}^{15 i x}-274841576 \,{\mathrm e}^{13 i x}-171621760 \,{\mathrm e}^{11 i x}-34247585 \,{\mathrm e}^{9 i x}+47900640 \,{\mathrm e}^{7 i x}+69873594 \,{\mathrm e}^{5 i x}+40585440 \,{\mathrm e}^{3 i x}+8744715 \,{\mathrm e}^{i x}}{41287680 \left ({\mathrm e}^{2 i x}+1\right )^{9} \left ({\mathrm e}^{6 i x}-{\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}-1\right )^{4}}-\frac {1063 \ln \left ({\mathrm e}^{i x}+1\right )}{262144}+\frac {1063 \ln \left ({\mathrm e}^{i x}-1\right )}{262144}+\frac {3603 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{65536}-\frac {3603 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{65536}\) | \(247\) |
Input:
int(1/(sin(3*x)+sin(5*x))^5,x,method=_RETURNVERBOSE)
Output:
0
Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (124) = 248\).
Time = 0.19 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.95 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^5} \, dx=-\frac {279830880 \, \cos \left (x\right )^{20} - 1074470880 \, \cos \left (x\right )^{18} + 1651874448 \, \cos \left (x\right )^{16} - 1293407232 \, \cos \left (x\right )^{14} + 537071702 \, \cos \left (x\right )^{12} - 107755754 \, \cos \left (x\right )^{10} + 6570448 \, \cos \left (x\right )^{8} + 257200 \, \cos \left (x\right )^{6} + 26128 \, \cos \left (x\right )^{4} + 3760 \, \cos \left (x\right )^{2} - 9079560 \, {\left (16 \, \sqrt {2} \cos \left (x\right )^{21} - 64 \, \sqrt {2} \cos \left (x\right )^{19} + 104 \, \sqrt {2} \cos \left (x\right )^{17} - 88 \, \sqrt {2} \cos \left (x\right )^{15} + 41 \, \sqrt {2} \cos \left (x\right )^{13} - 10 \, \sqrt {2} \cos \left (x\right )^{11} + \sqrt {2} \cos \left (x\right )^{9}\right )} \log \left (-\frac {2 \, \cos \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 1}{2 \, \cos \left (x\right )^{2} - 1}\right ) + 334845 \, {\left (16 \, \cos \left (x\right )^{21} - 64 \, \cos \left (x\right )^{19} + 104 \, \cos \left (x\right )^{17} - 88 \, \cos \left (x\right )^{15} + 41 \, \cos \left (x\right )^{13} - 10 \, \cos \left (x\right )^{11} + \cos \left (x\right )^{9}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 334845 \, {\left (16 \, \cos \left (x\right )^{21} - 64 \, \cos \left (x\right )^{19} + 104 \, \cos \left (x\right )^{17} - 88 \, \cos \left (x\right )^{15} + 41 \, \cos \left (x\right )^{13} - 10 \, \cos \left (x\right )^{11} + \cos \left (x\right )^{9}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 560}{165150720 \, {\left (16 \, \cos \left (x\right )^{21} - 64 \, \cos \left (x\right )^{19} + 104 \, \cos \left (x\right )^{17} - 88 \, \cos \left (x\right )^{15} + 41 \, \cos \left (x\right )^{13} - 10 \, \cos \left (x\right )^{11} + \cos \left (x\right )^{9}\right )}} \] Input:
integrate(1/(sin(3*x)+sin(5*x))^5,x, algorithm="fricas")
Output:
-1/165150720*(279830880*cos(x)^20 - 1074470880*cos(x)^18 + 1651874448*cos( x)^16 - 1293407232*cos(x)^14 + 537071702*cos(x)^12 - 107755754*cos(x)^10 + 6570448*cos(x)^8 + 257200*cos(x)^6 + 26128*cos(x)^4 + 3760*cos(x)^2 - 907 9560*(16*sqrt(2)*cos(x)^21 - 64*sqrt(2)*cos(x)^19 + 104*sqrt(2)*cos(x)^17 - 88*sqrt(2)*cos(x)^15 + 41*sqrt(2)*cos(x)^13 - 10*sqrt(2)*cos(x)^11 + sqr t(2)*cos(x)^9)*log(-(2*cos(x)^2 + 2*sqrt(2)*cos(x) + 1)/(2*cos(x)^2 - 1)) + 334845*(16*cos(x)^21 - 64*cos(x)^19 + 104*cos(x)^17 - 88*cos(x)^15 + 41* cos(x)^13 - 10*cos(x)^11 + cos(x)^9)*log(1/2*cos(x) + 1/2) - 334845*(16*co s(x)^21 - 64*cos(x)^19 + 104*cos(x)^17 - 88*cos(x)^15 + 41*cos(x)^13 - 10* cos(x)^11 + cos(x)^9)*log(-1/2*cos(x) + 1/2) + 560)/(16*cos(x)^21 - 64*cos (x)^19 + 104*cos(x)^17 - 88*cos(x)^15 + 41*cos(x)^13 - 10*cos(x)^11 + cos( x)^9)
\[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^5} \, dx=\int \frac {1}{\left (\sin {\left (3 x \right )} + \sin {\left (5 x \right )}\right )^{5}}\, dx \] Input:
integrate(1/(sin(3*x)+sin(5*x))**5,x)
Output:
Integral((sin(3*x) + sin(5*x))**(-5), x)
Leaf count of result is larger than twice the leaf count of optimal. 24329 vs. \(2 (124) = 248\).
Time = 4.55 (sec) , antiderivative size = 24329, normalized size of antiderivative = 157.98 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(sin(3*x)+sin(5*x))^5,x, algorithm="maxima")
Output:
-1/165150720*(4*(8744715*cos(41*x) + 40585440*cos(39*x) + 69873594*cos(37* x) + 47900640*cos(35*x) - 34247585*cos(33*x) - 171621760*cos(31*x) - 27484 1576*cos(29*x) - 189119360*cos(27*x) + 43279510*cos(25*x) + 277989440*cos( 23*x) + 404201564*cos(21*x) + 277989440*cos(19*x) + 43279510*cos(17*x) - 1 89119360*cos(15*x) - 274841576*cos(13*x) - 171621760*cos(11*x) - 34247585* cos(9*x) + 47900640*cos(7*x) + 69873594*cos(5*x) + 40585440*cos(3*x) + 874 4715*cos(x))*cos(42*x) + 34978860*(5*cos(40*x) + 10*cos(38*x) + 10*cos(36* x) + cos(34*x) - 19*cos(32*x) - 40*cos(30*x) - 40*cos(28*x) - 14*cos(26*x) + 26*cos(24*x) + 60*cos(22*x) + 60*cos(20*x) + 26*cos(18*x) - 14*cos(16*x ) - 40*cos(14*x) - 40*cos(12*x) - 19*cos(10*x) + cos(8*x) + 10*cos(6*x) + 10*cos(4*x) + 5*cos(2*x) + 1)*cos(41*x) + 20*(40585440*cos(39*x) + 6987359 4*cos(37*x) + 47900640*cos(35*x) - 34247585*cos(33*x) - 171621760*cos(31*x ) - 274841576*cos(29*x) - 189119360*cos(27*x) + 43279510*cos(25*x) + 27798 9440*cos(23*x) + 404201564*cos(21*x) + 277989440*cos(19*x) + 43279510*cos( 17*x) - 189119360*cos(15*x) - 274841576*cos(13*x) - 171621760*cos(11*x) - 34247585*cos(9*x) + 47900640*cos(7*x) + 69873594*cos(5*x) + 40585440*cos(3 *x) + 8744715*cos(x))*cos(40*x) + 162341760*(10*cos(38*x) + 10*cos(36*x) + cos(34*x) - 19*cos(32*x) - 40*cos(30*x) - 40*cos(28*x) - 14*cos(26*x) + 2 6*cos(24*x) + 60*cos(22*x) + 60*cos(20*x) + 26*cos(18*x) - 14*cos(16*x) - 40*cos(14*x) - 40*cos(12*x) - 19*cos(10*x) + cos(8*x) + 10*cos(6*x) + 1...
Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (124) = 248\).
Time = 0.15 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.54 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^5} \, dx =\text {Too large to display} \] Input:
integrate(1/(sin(3*x)+sin(5*x))^5,x, algorithm="giac")
Output:
3603/65536*sqrt(2)*log(abs(-4*sqrt(2) - 2*(cos(x) - 1)/(cos(x) + 1) - 6)/a bs(4*sqrt(2) - 2*(cos(x) - 1)/(cos(x) + 1) - 6)) + 1/2097152*(128*(cos(x) - 1)/(cos(x) + 1) - 6378*(cos(x) - 1)^2/(cos(x) + 1)^2 - 1)*(cos(x) + 1)^2 /(cos(x) - 1)^2 - 1/16384*(cos(x) - 1)/(cos(x) + 1) + 1/2097152*(cos(x) - 1)^2/(cos(x) + 1)^2 - 1/24576*(53967*(cos(x) - 1)/(cos(x) + 1) + 434951*(c os(x) - 1)^2/(cos(x) + 1)^2 + 1544619*(cos(x) - 1)^3/(cos(x) + 1)^3 + 2131 953*(cos(x) - 1)^4/(cos(x) + 1)^4 + 757005*(cos(x) - 1)^5/(cos(x) + 1)^5 + 106029*(cos(x) - 1)^6/(cos(x) + 1)^6 + 5049*(cos(x) - 1)^7/(cos(x) + 1)^7 + 2459)/(6*(cos(x) - 1)/(cos(x) + 1) + (cos(x) - 1)^2/(cos(x) + 1)^2 + 1) ^4 - 1/5160960*(6495462*(cos(x) - 1)/(cos(x) + 1) + 22589298*(cos(x) - 1)^ 2/(cos(x) + 1)^2 + 45358782*(cos(x) - 1)^3/(cos(x) + 1)^3 + 57467088*(cos( x) - 1)^4/(cos(x) + 1)^4 + 47030130*(cos(x) - 1)^5/(cos(x) + 1)^5 + 243060 30*(cos(x) - 1)^6/(cos(x) + 1)^6 + 7267050*(cos(x) - 1)^7/(cos(x) + 1)^7 + 968625*(cos(x) - 1)^8/(cos(x) + 1)^8 + 829343)/((cos(x) - 1)/(cos(x) + 1) + 1)^9 + 1063/524288*log(-(cos(x) - 1)/(cos(x) + 1))
Time = 21.22 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.36 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^5} \, dx =\text {Too large to display} \] Input:
int(1/(sin(3*x) + sin(5*x))^5,x)
Output:
(1063*log(tan(x/2)))/262144 - ((95*tan(x/2)^2)/2097152 + (21383993*tan(x/2 )^4)/82575360 - (225285883*tan(x/2)^6)/27525120 + (18294266437*tan(x/2)^8) /165150720 - (6680946049*tan(x/2)^10)/7864320 + (22702586729*tan(x/2)^12)/ 5505024 - (371004516437*tan(x/2)^14)/27525120 + (10200248921249*tan(x/2)^1 6)/330301440 - (800654617309*tan(x/2)^18)/15728640 + (1011732457843*tan(x/ 2)^20)/16515072 - (1488663546829*tan(x/2)^22)/27525120 + (639724414253*tan (x/2)^24)/18350080 - (127127888753*tan(x/2)^26)/7864320 + (6879842191*tan( x/2)^28)/1310720 - (299867227*tan(x/2)^30)/262144 + (328817953*tan(x/2)^32 )/2097152 - (25320577*tan(x/2)^34)/2097152 + (805*tan(x/2)^36)/2048 + 1/20 97152)/(tan(x/2)^4 - 33*tan(x/2)^6 + 472*tan(x/2)^8 - 3864*tan(x/2)^10 + 2 0220*tan(x/2)^12 - 71868*tan(x/2)^14 + 180520*tan(x/2)^16 - 329064*tan(x/2 )^18 + 442534*tan(x/2)^20 - 442534*tan(x/2)^22 + 329064*tan(x/2)^24 - 1805 20*tan(x/2)^26 + 71868*tan(x/2)^28 - 20220*tan(x/2)^30 + 3864*tan(x/2)^32 - 472*tan(x/2)^34 + 33*tan(x/2)^36 - tan(x/2)^38) + (3603*2^(1/2)*atanh((1 04566860495*2^(1/2))/(549755813888*((27558878433453*tan(x/2)^2)/1759218604 4416 - 4724773430031/17592186044416)) - (9743982733791*2^(1/2)*tan(x/2)^2) /(8796093022208*((27558878433453*tan(x/2)^2)/17592186044416 - 472477343003 1/17592186044416))))/32768 + tan(x/2)^2/16384 + tan(x/2)^4/2097152
\[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^5} \, dx=\int \frac {1}{\sin \left (5 x \right )^{5}+5 \sin \left (5 x \right )^{4} \sin \left (3 x \right )+10 \sin \left (5 x \right )^{3} \sin \left (3 x \right )^{2}+10 \sin \left (5 x \right )^{2} \sin \left (3 x \right )^{3}+5 \sin \left (5 x \right ) \sin \left (3 x \right )^{4}+\sin \left (3 x \right )^{5}}d x \] Input:
int(1/(sin(3*x)+sin(5*x))^5,x)
Output:
int(1/(sin(5*x)**5 + 5*sin(5*x)**4*sin(3*x) + 10*sin(5*x)**3*sin(3*x)**2 + 10*sin(5*x)**2*sin(3*x)**3 + 5*sin(5*x)*sin(3*x)**4 + sin(3*x)**5),x)