Integrand size = 18, antiderivative size = 46 \[ \int \sin (a+b x) \sin ^4(2 a+2 b x) \, dx=-\frac {16 \cos ^5(a+b x)}{5 b}+\frac {32 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^9(a+b x)}{9 b} \]
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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, 2645, 276} \[ \int \sin (a+b x) \sin ^4(2 a+2 b x) \, dx=-\frac {16 \cos ^9(a+b x)}{9 b}+\frac {32 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^5(a+b x)}{5 b} \]
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Rule 276
Rule 2645
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 16 \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx \\ & = -\frac {16 \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {16 \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {16 \cos ^5(a+b x)}{5 b}+\frac {32 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^9(a+b x)}{9 b} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \sin (a+b x) \sin ^4(2 a+2 b x) \, dx=\frac {2 \cos ^5(a+b x) (-249+220 \cos (2 (a+b x))-35 \cos (4 (a+b x)))}{315 b} \]
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Time = 1.55 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.50
method | result | size |
default | \(-\frac {3 \cos \left (x b +a \right )}{8 b}-\frac {\cos \left (3 x b +3 a \right )}{12 b}+\frac {\cos \left (5 x b +5 a \right )}{20 b}+\frac {\cos \left (7 x b +7 a \right )}{112 b}-\frac {\cos \left (9 x b +9 a \right )}{144 b}\) | \(69\) |
risch | \(-\frac {3 \cos \left (x b +a \right )}{8 b}-\frac {\cos \left (3 x b +3 a \right )}{12 b}+\frac {\cos \left (5 x b +5 a \right )}{20 b}+\frac {\cos \left (7 x b +7 a \right )}{112 b}-\frac {\cos \left (9 x b +9 a \right )}{144 b}\) | \(69\) |
parallelrisch | \(\frac {\left (-64 \tan \left (x b +a \right )^{6}-208 \tan \left (x b +a \right )^{4}-64 \tan \left (x b +a \right )^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+\left (256 \tan \left (x b +a \right )^{7}+896 \tan \left (x b +a \right )^{5}-896 \tan \left (x b +a \right )^{3}-256 \tan \left (x b +a \right )\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )-256 \tan \left (x b +a \right )^{8}-960 \tan \left (x b +a \right )^{6}-1328 \tan \left (x b +a \right )^{4}-960 \tan \left (x b +a \right )^{2}-256}{315 b \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right ) \left (1+\tan \left (x b +a \right )^{2}\right )^{4}}\) | \(167\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sin (a+b x) \sin ^4(2 a+2 b x) \, dx=-\frac {16 \, {\left (35 \, \cos \left (b x + a\right )^{9} - 90 \, \cos \left (b x + a\right )^{7} + 63 \, \cos \left (b x + a\right )^{5}\right )}}{315 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (39) = 78\).
Time = 1.99 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.54 \[ \int \sin (a+b x) \sin ^4(2 a+2 b x) \, dx=\begin {cases} - \frac {104 \sin {\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{315 b} - \frac {64 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{315 b} - \frac {107 \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{315 b} - \frac {16 \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{21 b} - \frac {128 \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{315 b} & \text {for}\: b \neq 0 \\x \sin {\left (a \right )} \sin ^{4}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int \sin (a+b x) \sin ^4(2 a+2 b x) \, dx=-\frac {35 \, \cos \left (9 \, b x + 9 \, a\right ) - 45 \, \cos \left (7 \, b x + 7 \, a\right ) - 252 \, \cos \left (5 \, b x + 5 \, a\right ) + 420 \, \cos \left (3 \, b x + 3 \, a\right ) + 1890 \, \cos \left (b x + a\right )}{5040 \, b} \]
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int \sin (a+b x) \sin ^4(2 a+2 b x) \, dx=-\frac {35 \, \cos \left (9 \, b x + 9 \, a\right ) - 45 \, \cos \left (7 \, b x + 7 \, a\right ) - 252 \, \cos \left (5 \, b x + 5 \, a\right ) + 420 \, \cos \left (3 \, b x + 3 \, a\right ) + 1890 \, \cos \left (b x + a\right )}{5040 \, b} \]
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Time = 19.39 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sin (a+b x) \sin ^4(2 a+2 b x) \, dx=-\frac {16\,\left (35\,{\cos \left (a+b\,x\right )}^9-90\,{\cos \left (a+b\,x\right )}^7+63\,{\cos \left (a+b\,x\right )}^5\right )}{315\,b} \]
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