Integrand size = 24, antiderivative size = 33 \[ \int \frac {\sin ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {\sec (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3254, 2686} \[ \int \frac {\sin ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\sec ^3(c+d x)}{3 a^2 d}-\frac {\sec (c+d x)}{a^2 d} \]
[In]
[Out]
Rule 2686
Rule 3254
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec (c+d x) \tan ^3(c+d x) \, dx}{a^2} \\ & = \frac {\text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = -\frac {\sec (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {\sin ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {-\frac {\sec (c+d x)}{d}+\frac {\sec ^3(c+d x)}{3 d}}{a^2} \]
[In]
[Out]
Time = 0.54 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {1}{3 \cos \left (d x +c \right )^{3}}-\frac {1}{\cos \left (d x +c \right )}}{d \,a^{2}}\) | \(29\) |
default | \(\frac {\frac {1}{3 \cos \left (d x +c \right )^{3}}-\frac {1}{\cos \left (d x +c \right )}}{d \,a^{2}}\) | \(29\) |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}+2 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(56\) |
parallelrisch | \(\frac {-2-6 \cos \left (2 d x +2 c \right )-6 \cos \left (d x +c \right )-2 \cos \left (3 d x +3 c \right )}{3 a^{2} d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(61\) |
norman | \(\frac {-\frac {8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {32 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {4}{3 a d}-\frac {4 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(101\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {\sin ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {3 \, \cos \left (d x + c\right )^{2} - 1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (26) = 52\).
Time = 3.92 (sec) , antiderivative size = 156, normalized size of antiderivative = 4.73 \[ \int \frac {\sin ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\begin {cases} - \frac {12 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} + \frac {4}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\left (c \right )}}{\left (- a \sin ^{2}{\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {\sin ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {3 \, \cos \left (d x + c\right )^{2} - 1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {\sin ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {3 \, \cos \left (d x + c\right )^{2} - 1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {{\cos \left (c+d\,x\right )}^2-\frac {1}{3}}{a^2\,d\,{\cos \left (c+d\,x\right )}^3} \]
[In]
[Out]