Integrand size = 24, antiderivative size = 49 \[ \int \frac {\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {3 x}{2 a}+\frac {3 \tan (c+d x)}{2 a d}-\frac {\sin ^2(c+d x) \tan (c+d x)}{2 a d} \]
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Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3254, 2671, 294, 327, 209} \[ \int \frac {\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {3 \tan (c+d x)}{2 a d}-\frac {\sin ^2(c+d x) \tan (c+d x)}{2 a d}-\frac {3 x}{2 a} \]
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Rule 209
Rule 294
Rule 327
Rule 2671
Rule 3254
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin ^2(c+d x) \tan ^2(c+d x) \, dx}{a} \\ & = \frac {\text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{a d} \\ & = -\frac {\sin ^2(c+d x) \tan (c+d x)}{2 a d}+\frac {3 \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 a d} \\ & = \frac {3 \tan (c+d x)}{2 a d}-\frac {\sin ^2(c+d x) \tan (c+d x)}{2 a d}-\frac {3 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 a d} \\ & = -\frac {3 x}{2 a}+\frac {3 \tan (c+d x)}{2 a d}-\frac {\sin ^2(c+d x) \tan (c+d x)}{2 a d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {-6 (c+d x)+\sin (2 (c+d x))+4 \tan (c+d x)}{4 a d} \]
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Time = 0.46 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\tan \left (d x +c \right )+\frac {\tan \left (d x +c \right )}{2+2 \left (\tan ^{2}\left (d x +c \right )\right )}-\frac {3 \arctan \left (\tan \left (d x +c \right )\right )}{2}}{d a}\) | \(44\) |
default | \(\frac {\tan \left (d x +c \right )+\frac {\tan \left (d x +c \right )}{2+2 \left (\tan ^{2}\left (d x +c \right )\right )}-\frac {3 \arctan \left (\tan \left (d x +c \right )\right )}{2}}{d a}\) | \(44\) |
parallelrisch | \(\frac {-12 d x \cos \left (d x +c \right )+9 \sin \left (d x +c \right )+\sin \left (3 d x +3 c \right )}{8 a d \cos \left (d x +c \right )}\) | \(45\) |
risch | \(-\frac {3 x}{2 a}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d a}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d a}+\frac {2 i}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(66\) |
norman | \(\frac {\frac {3 x}{2 a}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {10 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {3 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {9 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {3 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {9 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {3 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(217\) |
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Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int \frac {\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {3 \, d x \cos \left (d x + c\right ) - {\left (\cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right )}{2 \, a d \cos \left (d x + c\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (39) = 78\).
Time = 2.56 (sec) , antiderivative size = 502, normalized size of antiderivative = 10.24 \[ \int \frac {\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\begin {cases} - \frac {3 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} - \frac {3 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} + \frac {3 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} + \frac {3 d x}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} - \frac {6 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} - \frac {4 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} - \frac {6 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\left (c \right )}}{- a \sin ^{2}{\left (c \right )} + a} & \text {otherwise} \end {cases} \]
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Time = 0.38 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )}}{a} - \frac {\tan \left (d x + c\right )}{a \tan \left (d x + c\right )^{2} + a} - \frac {2 \, \tan \left (d x + c\right )}{a}}{2 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )}}{a} - \frac {2 \, \tan \left (d x + c\right )}{a} - \frac {\tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )} a}}{2 \, d} \]
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Time = 13.72 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int \frac {\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )}{2\,d\,\left (a\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}-\frac {3\,x}{2\,a}+\frac {\mathrm {tan}\left (c+d\,x\right )}{a\,d} \]
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