Integrand size = 18, antiderivative size = 46 \[ \int \sin (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \sin ^7(a+b x)}{7 b}-\frac {64 \sin ^9(a+b x)}{9 b}+\frac {32 \sin ^{11}(a+b x)}{11 b} \]
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Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4373, 2644, 276} \[ \int \sin (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \sin ^{11}(a+b x)}{11 b}-\frac {64 \sin ^9(a+b x)}{9 b}+\frac {32 \sin ^7(a+b x)}{7 b} \]
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Rule 276
Rule 2644
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 32 \int \cos ^5(a+b x) \sin ^6(a+b x) \, dx \\ & = \frac {32 \text {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {32 \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {32 \sin ^7(a+b x)}{7 b}-\frac {64 \sin ^9(a+b x)}{9 b}+\frac {32 \sin ^{11}(a+b x)}{11 b} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \sin (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {4 (365+364 \cos (2 (a+b x))+63 \cos (4 (a+b x))) \sin ^7(a+b x)}{693 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(82\) vs. \(2(40)=80\).
Time = 2.86 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.80
method | result | size |
default | \(\frac {5 \sin \left (x b +a \right )}{16 b}-\frac {5 \sin \left (3 x b +3 a \right )}{48 b}-\frac {\sin \left (5 x b +5 a \right )}{32 b}+\frac {5 \sin \left (7 x b +7 a \right )}{224 b}+\frac {\sin \left (9 x b +9 a \right )}{288 b}-\frac {\sin \left (11 x b +11 a \right )}{352 b}\) | \(83\) |
risch | \(\frac {5 \sin \left (x b +a \right )}{16 b}-\frac {5 \sin \left (3 x b +3 a \right )}{48 b}-\frac {\sin \left (5 x b +5 a \right )}{32 b}+\frac {5 \sin \left (7 x b +7 a \right )}{224 b}+\frac {\sin \left (9 x b +9 a \right )}{288 b}-\frac {\sin \left (11 x b +11 a \right )}{352 b}\) | \(83\) |
parallelrisch | \(\frac {\left (-256 \tan \left (x b +a \right )^{9}-1152 \tan \left (x b +a \right )^{7}-2016 \tan \left (x b +a \right )^{5}-1152 \tan \left (x b +a \right )^{3}-256 \tan \left (x b +a \right )\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+\left (512 \tan \left (x b +a \right )^{10}+2304 \tan \left (x b +a \right )^{8}+4032 \tan \left (x b +a \right )^{6}-4032 \tan \left (x b +a \right )^{4}-2304 \tan \left (x b +a \right )^{2}-512\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+256 \tan \left (x b +a \right )^{9}+1152 \tan \left (x b +a \right )^{7}+2016 \tan \left (x b +a \right )^{5}+1152 \tan \left (x b +a \right )^{3}+256 \tan \left (x b +a \right )}{693 b \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right ) \left (1+\tan \left (x b +a \right )^{2}\right )^{5}}\) | \(205\) |
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Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.37 \[ \int \sin (a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32 \, {\left (63 \, \cos \left (b x + a\right )^{10} - 161 \, \cos \left (b x + a\right )^{8} + 113 \, \cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} - 4 \, \cos \left (b x + a\right )^{2} - 8\right )} \sin \left (b x + a\right )}{693 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (39) = 78\).
Time = 4.74 (sec) , antiderivative size = 197, normalized size of antiderivative = 4.28 \[ \int \sin (a+b x) \sin ^5(2 a+2 b x) \, dx=\begin {cases} - \frac {422 \sin {\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{693 b} - \frac {608 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{693 b} - \frac {256 \sin {\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{693 b} + \frac {151 \sin ^{5}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{693 b} + \frac {272 \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{693 b} + \frac {128 \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{693 b} & \text {for}\: b \neq 0 \\x \sin {\left (a \right )} \sin ^{5}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.50 \[ \int \sin (a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {63 \, \sin \left (11 \, b x + 11 \, a\right ) - 77 \, \sin \left (9 \, b x + 9 \, a\right ) - 495 \, \sin \left (7 \, b x + 7 \, a\right ) + 693 \, \sin \left (5 \, b x + 5 \, a\right ) + 2310 \, \sin \left (3 \, b x + 3 \, a\right ) - 6930 \, \sin \left (b x + a\right )}{22176 \, b} \]
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sin (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \, {\left (63 \, \sin \left (b x + a\right )^{11} - 154 \, \sin \left (b x + a\right )^{9} + 99 \, \sin \left (b x + a\right )^{7}\right )}}{693 \, b} \]
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Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sin (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32\,\left (63\,{\sin \left (a+b\,x\right )}^{11}-154\,{\sin \left (a+b\,x\right )}^9+99\,{\sin \left (a+b\,x\right )}^7\right )}{693\,b} \]
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