Integrand size = 10, antiderivative size = 13 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=7 \sin ^5(x)-5 \sin ^7(x) \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {12, 2644, 14} \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=7 \sin ^5(x)-5 \sin ^7(x) \]
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Rule 12
Rule 14
Rule 2644
Rubi steps \begin{align*} \text {integral}& = 35 \int \cos ^3(x) \sin ^4(x) \, dx \\ & = 35 \text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (x)\right ) \\ & = 35 \text {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (x)\right ) \\ & = 7 \sin ^5(x)-5 \sin ^7(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(33\) vs. \(2(13)=26\).
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.54 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=35 \left (\frac {3 \sin (x)}{64}-\frac {1}{64} \sin (3 x)-\frac {1}{320} \sin (5 x)+\frac {1}{448} \sin (7 x)\right ) \]
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Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(7 \sin \left (x \right )^{5}-5 \sin \left (x \right )^{7}\) | \(14\) |
default | \(7 \sin \left (x \right )^{5}-5 \sin \left (x \right )^{7}\) | \(14\) |
risch | \(\frac {105 \sin \left (x \right )}{64}+\frac {5 \sin \left (7 x \right )}{64}-\frac {7 \sin \left (5 x \right )}{64}-\frac {35 \sin \left (3 x \right )}{64}\) | \(24\) |
parallelrisch | \(\frac {105 \sin \left (x \right )}{64}+\frac {5 \sin \left (7 x \right )}{64}-\frac {7 \sin \left (5 x \right )}{64}-\frac {35 \sin \left (3 x \right )}{64}\) | \(24\) |
norman | \(\frac {224 \tan \left (\frac {x}{2}\right )^{5}-192 \tan \left (\frac {x}{2}\right )^{7}+224 \tan \left (\frac {x}{2}\right )^{9}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{7}}\) | \(37\) |
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Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.62 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx={\left (5 \, \cos \left (x\right )^{6} - 8 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=- 5 \sin ^{7}{\left (x \right )} + 7 \sin ^{5}{\left (x \right )} \]
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Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=-5 \, \sin \left (x\right )^{7} + 7 \, \sin \left (x\right )^{5} \]
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Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=-5 \, \sin \left (x\right )^{7} + 7 \, \sin \left (x\right )^{5} \]
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Time = 27.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=7\,{\sin \left (x\right )}^5-5\,{\sin \left (x\right )}^7 \]
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