Integrand size = 24, antiderivative size = 38 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {x}{a^2}-\frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 3554, 8} \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\tan (c+d x)}{a^2 d}+\frac {x}{a^2} \]
[In]
[Out]
Rule 8
Rule 3254
Rule 3554
Rubi steps \begin{align*} \text {integral}& = \frac {\int \tan ^4(c+d x) \, dx}{a^2} \\ & = \frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\int \tan ^2(c+d x) \, dx}{a^2} \\ & = -\frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {\int 1 \, dx}{a^2} \\ & = \frac {x}{a^2}-\frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {\arctan (\tan (c+d x))}{d}-\frac {\tan (c+d x)}{d}+\frac {\tan ^3(c+d x)}{3 d}}{a^2} \]
[In]
[Out]
Time = 0.67 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )}{d \,a^{2}}\) | \(34\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )}{d \,a^{2}}\) | \(34\) |
risch | \(\frac {x}{a^{2}}-\frac {4 i \left (3 \,{\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}+2\right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(53\) |
parallelrisch | \(\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d -9 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+9 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d -20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 d x +6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(122\) |
norman | \(\frac {\frac {x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x}{a}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {38 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {24 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {38 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {4 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {2 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(290\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {3 \, d x \cos \left (d x + c\right )^{3} - {\left (4 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (31) = 62\).
Time = 7.02 (sec) , antiderivative size = 551, normalized size of antiderivative = 14.50 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\begin {cases} \frac {3 d x \tan ^{6}{\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 {9 d x \tan ^{4}{\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 {9 d x \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 {3 d x}{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 {6 \tan ^{5}{\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 {20 \tan ^{3}{\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 {6 \tan {\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} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\left (c \right )}}{\left (- a \sin ^{2}{\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {\tan \left (d x + c\right )^{3} - 3 \, \tan \left (d x + c\right )}{a^{2}} + \frac {3 \, {\left (d x + c\right )}}{a^{2}}}{3 \, d} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {a^{4} \tan \left (d x + c\right )^{3} - 3 \, a^{4} \tan \left (d x + c\right )}{a^{6}}}{3 \, d} \]
[In]
[Out]
Time = 13.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {x}{a^2}-\frac {\mathrm {tan}\left (c+d\,x\right )-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3}}{a^2\,d} \]
[In]
[Out]