Integrand size = 20, antiderivative size = 46 \[ \int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \sin ^3(a+b x)}{3 b}-\frac {64 \sin ^5(a+b x)}{5 b}+\frac {32 \sin ^7(a+b x)}{7 b} \]
[Out]
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} \[ \int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \sin ^7(a+b x)}{7 b}-\frac {64 \sin ^5(a+b x)}{5 b}+\frac {32 \sin ^3(a+b x)}{3 b} \]
[In]
[Out]
Rule 276
Rule 2644
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 32 \int \cos ^5(a+b x) \sin ^2(a+b x) \, dx \\ & = \frac {32 \text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {32 \text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {32 \sin ^3(a+b x)}{3 b}-\frac {64 \sin ^5(a+b x)}{5 b}+\frac {32 \sin ^7(a+b x)}{7 b} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {4 (157+108 \cos (2 (a+b x))+15 \cos (4 (a+b x))) \sin ^3(a+b x)}{105 b} \]
[In]
[Out]
Time = 12.74 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {\frac {32 \sin \left (x b +a \right )^{7}}{7}-\frac {64 \sin \left (x b +a \right )^{5}}{5}+\frac {32 \sin \left (x b +a \right )^{3}}{3}}{b}\) | \(37\) |
risch | \(\frac {5 \sin \left (x b +a \right )}{2 b}-\frac {\sin \left (7 x b +7 a \right )}{14 b}-\frac {3 \sin \left (5 x b +5 a \right )}{10 b}-\frac {\sin \left (3 x b +3 a \right )}{6 b}\) | \(55\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32 \, {\left (15 \, \cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} - 4 \, \cos \left (b x + a\right )^{2} - 8\right )} \sin \left (b x + a\right )}{105 \, b} \]
[In]
[Out]
Timed out. \[ \int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.02 \[ \int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {15 \, \sin \left (7 \, b x + 7 \, a\right ) + 63 \, \sin \left (5 \, b x + 5 \, a\right ) + 35 \, \sin \left (3 \, b x + 3 \, a\right ) - 525 \, \sin \left (b x + a\right )}{210 \, b} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \, {\left (15 \, \sin \left (b x + a\right )^{7} - 42 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3}\right )}}{105 \, b} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32\,\left (15\,{\sin \left (a+b\,x\right )}^7-42\,{\sin \left (a+b\,x\right )}^5+35\,{\sin \left (a+b\,x\right )}^3\right )}{105\,b} \]
[In]
[Out]