Integrand size = 9, antiderivative size = 17 \[ \int \cos ^3(x) \sin ^4(x) \, dx=\frac {\sin ^5(x)}{5}-\frac {\sin ^7(x)}{7} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2644, 14} \[ \int \cos ^3(x) \sin ^4(x) \, dx=\frac {\sin ^5(x)}{5}-\frac {\sin ^7(x)}{7} \]
[In]
[Out]
Rule 14
Rule 2644
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (x)\right ) \\ & = \text {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (x)\right ) \\ & = \frac {\sin ^5(x)}{5}-\frac {\sin ^7(x)}{7} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \cos ^3(x) \sin ^4(x) \, dx=\frac {3 \sin (x)}{64}-\frac {1}{64} \sin (3 x)-\frac {1}{320} \sin (5 x)+\frac {1}{448} \sin (7 x) \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\left (\sin ^{5}\left (x \right )\right )}{5}-\frac {\left (\sin ^{7}\left (x \right )\right )}{7}\) | \(14\) |
default | \(\frac {\left (\sin ^{5}\left (x \right )\right )}{5}-\frac {\left (\sin ^{7}\left (x \right )\right )}{7}\) | \(14\) |
risch | \(\frac {3 \sin \left (x \right )}{64}+\frac {\sin \left (7 x \right )}{448}-\frac {\sin \left (5 x \right )}{320}-\frac {\sin \left (3 x \right )}{64}\) | \(24\) |
parallelrisch | \(\frac {3 \sin \left (x \right )}{64}+\frac {\sin \left (7 x \right )}{448}-\frac {\sin \left (5 x \right )}{320}-\frac {\sin \left (3 x \right )}{64}\) | \(24\) |
norman | \(\frac {\frac {32 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {192 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{35}+\frac {32 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{5}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{7}}\) | \(37\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \cos ^3(x) \sin ^4(x) \, dx=\frac {1}{35} \, {\left (5 \, \cos \left (x\right )^{6} - 8 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \cos ^3(x) \sin ^4(x) \, dx=- \frac {\sin ^{7}{\left (x \right )}}{7} + \frac {\sin ^{5}{\left (x \right )}}{5} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos ^3(x) \sin ^4(x) \, dx=-\frac {1}{7} \, \sin \left (x\right )^{7} + \frac {1}{5} \, \sin \left (x\right )^{5} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos ^3(x) \sin ^4(x) \, dx=-\frac {1}{7} \, \sin \left (x\right )^{7} + \frac {1}{5} \, \sin \left (x\right )^{5} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \cos ^3(x) \sin ^4(x) \, dx=-\frac {{\sin \left (x\right )}^5\,\left (5\,{\sin \left (x\right )}^2-7\right )}{35} \]
[In]
[Out]