Integrand size = 20, antiderivative size = 13 \[ \int \csc ^2(2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {\sec (a+b x)}{4 b} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4373, 2686, 8} \[ \int \csc ^2(2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {\sec (a+b x)}{4 b} \]
[In]
[Out]
Rule 8
Rule 2686
Rule 4373
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \sec (a+b x) \tan (a+b x) \, dx \\ & = \frac {\text {Subst}(\int 1 \, dx,x,\sec (a+b x))}{4 b} \\ & = \frac {\sec (a+b x)}{4 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \csc ^2(2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {\sec (a+b x)}{4 b} \]
[In]
[Out]
Time = 1.72 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {1}{4 \cos \left (x b +a \right ) b}\) | \(14\) |
risch | \(\frac {{\mathrm e}^{i \left (x b +a \right )}}{2 b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}\) | \(28\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \csc ^2(2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {1}{4 \, b \cos \left (b x + a\right )} \]
[In]
[Out]
Timed out. \[ \int \csc ^2(2 a+2 b x) \sin ^3(a+b x) \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (11) = 22\).
Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 6.38 \[ \int \csc ^2(2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {\cos \left (2 \, b x + 2 \, a\right ) \cos \left (b x + a\right ) + \sin \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + \cos \left (b x + a\right )}{2 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (11) = 22\).
Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.15 \[ \int \csc ^2(2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {1}{2 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \csc ^2(2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {1}{4\,b\,\cos \left (a+b\,x\right )} \]
[In]
[Out]