Integrand size = 20, antiderivative size = 46 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \sin ^9(a+b x)}{9 b}-\frac {64 \sin ^{11}(a+b x)}{11 b}+\frac {32 \sin ^{13}(a+b x)}{13 b} \]
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Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4373, 2644, 276} \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \sin ^{13}(a+b x)}{13 b}-\frac {64 \sin ^{11}(a+b x)}{11 b}+\frac {32 \sin ^9(a+b x)}{9 b} \]
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Rule 276
Rule 2644
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 32 \int \cos ^5(a+b x) \sin ^8(a+b x) \, dx \\ & = \frac {32 \text {Subst}\left (\int x^8 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {32 \text {Subst}\left (\int \left (x^8-2 x^{10}+x^{12}\right ) \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {32 \sin ^9(a+b x)}{9 b}-\frac {64 \sin ^{11}(a+b x)}{11 b}+\frac {32 \sin ^{13}(a+b x)}{13 b} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {4 (505+540 \cos (2 (a+b x))+99 \cos (4 (a+b x))) \sin ^9(a+b x)}{1287 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(40)=80\).
Time = 9.74 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.11
method | result | size |
default | \(\frac {5 \sin \left (x b +a \right )}{32 b}-\frac {25 \sin \left (3 x b +3 a \right )}{384 b}-\frac {\sin \left (5 x b +5 a \right )}{128 b}+\frac {\sin \left (7 x b +7 a \right )}{64 b}-\frac {\sin \left (9 x b +9 a \right )}{576 b}-\frac {3 \sin \left (11 x b +11 a \right )}{1408 b}+\frac {\sin \left (13 x b +13 a \right )}{1664 b}\) | \(97\) |
risch | \(\frac {5 \sin \left (x b +a \right )}{32 b}-\frac {25 \sin \left (3 x b +3 a \right )}{384 b}-\frac {\sin \left (5 x b +5 a \right )}{128 b}+\frac {\sin \left (7 x b +7 a \right )}{64 b}-\frac {\sin \left (9 x b +9 a \right )}{576 b}-\frac {3 \sin \left (11 x b +11 a \right )}{1408 b}+\frac {\sin \left (13 x b +13 a \right )}{1664 b}\) | \(97\) |
parallelrisch | \(\frac {\left (512 \tan \left (x b +a \right )^{7}+1216 \tan \left (x b +a \right )^{5}+512 \tan \left (x b +a \right )^{3}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}+\left (-3072 \tan \left (x b +a \right )^{8}-4992 \tan \left (x b +a \right )^{6}+4992 \tan \left (x b +a \right )^{4}+3072 \tan \left (x b +a \right )^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}+\left (6144 \tan \left (x b +a \right )^{9}-3072 \tan \left (x b +a \right )^{7}-56256 \tan \left (x b +a \right )^{5}-3072 \tan \left (x b +a \right )^{3}+6144 \tan \left (x b +a \right )\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}+\left (-4096 \tan \left (x b +a \right )^{10}+37888 \tan \left (x b +a \right )^{8}+164864 \tan \left (x b +a \right )^{6}-164864 \tan \left (x b +a \right )^{4}-37888 \tan \left (x b +a \right )^{2}+4096\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+\left (-52224 \tan \left (x b +a \right )^{9}-181248 \tan \left (x b +a \right )^{7}-220224 \tan \left (x b +a \right )^{5}-181248 \tan \left (x b +a \right )^{3}-52224 \tan \left (x b +a \right )\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+\left (18432 \tan \left (x b +a \right )^{10}+52224 \tan \left (x b +a \right )^{8}+31872 \tan \left (x b +a \right )^{6}-31872 \tan \left (x b +a \right )^{4}-52224 \tan \left (x b +a \right )^{2}-18432\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+9216 \tan \left (x b +a \right )^{9}+36352 \tan \left (x b +a \right )^{7}+54080 \tan \left (x b +a \right )^{5}+36352 \tan \left (x b +a \right )^{3}+9216 \tan \left (x b +a \right )}{9009 b \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3} \left (1+\tan \left (x b +a \right )^{2}\right )^{5}}\) | \(426\) |
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Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.59 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \, {\left (99 \, \cos \left (b x + a\right )^{12} - 360 \, \cos \left (b x + a\right )^{10} + 458 \, \cos \left (b x + a\right )^{8} - 212 \, \cos \left (b x + a\right )^{6} + 3 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{1287 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (39) = 78\).
Time = 25.24 (sec) , antiderivative size = 447, normalized size of antiderivative = 9.72 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\begin {cases} - \frac {1366 \sin ^{3}{\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {4960 \sin ^{3}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{9009 b} - \frac {256 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1287 b} - \frac {271 \sin ^{2}{\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{3003 b} - \frac {48 \sin ^{2}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{143 b} - \frac {640 \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {1388 \sin {\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {2944 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {512 \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1001 b} + \frac {2234 \sin ^{5}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )}}{9009 b} + \frac {4544 \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{9009 b} + \frac {256 \sin {\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{1001 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} \sin ^{5}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.74 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {99 \, \sin \left (13 \, b x + 13 \, a\right ) - 351 \, \sin \left (11 \, b x + 11 \, a\right ) - 286 \, \sin \left (9 \, b x + 9 \, a\right ) + 2574 \, \sin \left (7 \, b x + 7 \, a\right ) - 1287 \, \sin \left (5 \, b x + 5 \, a\right ) - 10725 \, \sin \left (3 \, b x + 3 \, a\right ) + 25740 \, \sin \left (b x + a\right )}{164736 \, b} \]
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \, {\left (99 \, \sin \left (b x + a\right )^{13} - 234 \, \sin \left (b x + a\right )^{11} + 143 \, \sin \left (b x + a\right )^{9}\right )}}{1287 \, b} \]
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Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32\,\left (99\,{\sin \left (a+b\,x\right )}^{13}-234\,{\sin \left (a+b\,x\right )}^{11}+143\,{\sin \left (a+b\,x\right )}^9\right )}{1287\,b} \]
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