Integrand size = 24, antiderivative size = 69 \[ \int \frac {\sin ^6(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {5 x}{2 a^2}-\frac {5 \tan (c+d x)}{2 a^2 d}+\frac {5 \tan ^3(c+d x)}{6 a^2 d}-\frac {\sin ^2(c+d x) \tan ^3(c+d x)}{2 a^2 d} \]
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Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3254, 2671, 294, 308, 209} \[ \int \frac {\sin ^6(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {5 \tan ^3(c+d x)}{6 a^2 d}-\frac {5 \tan (c+d x)}{2 a^2 d}-\frac {\sin ^2(c+d x) \tan ^3(c+d x)}{2 a^2 d}+\frac {5 x}{2 a^2} \]
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Rule 209
Rule 294
Rule 308
Rule 2671
Rule 3254
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin ^2(c+d x) \tan ^4(c+d x) \, dx}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{a^2 d} \\ & = -\frac {\sin ^2(c+d x) \tan ^3(c+d x)}{2 a^2 d}+\frac {5 \text {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 a^2 d} \\ & = -\frac {\sin ^2(c+d x) \tan ^3(c+d x)}{2 a^2 d}+\frac {5 \text {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{2 a^2 d} \\ & = -\frac {5 \tan (c+d x)}{2 a^2 d}+\frac {5 \tan ^3(c+d x)}{6 a^2 d}-\frac {\sin ^2(c+d x) \tan ^3(c+d x)}{2 a^2 d}+\frac {5 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 a^2 d} \\ & = \frac {5 x}{2 a^2}-\frac {5 \tan (c+d x)}{2 a^2 d}+\frac {5 \tan ^3(c+d x)}{6 a^2 d}-\frac {\sin ^2(c+d x) \tan ^3(c+d x)}{2 a^2 d} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.67 \[ \int \frac {\sin ^6(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {30 (c+d x)-3 \sin (2 (c+d x))+4 \left (-7+\sec ^2(c+d x)\right ) \tan (c+d x)}{12 a^2 d} \]
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Time = 0.74 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 \tan \left (d x +c \right )-\frac {\tan \left (d x +c \right )}{2 \left (1+\tan ^{2}\left (d x +c \right )\right )}+\frac {5 \arctan \left (\tan \left (d x +c \right )\right )}{2}}{d \,a^{2}}\) | \(56\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 \tan \left (d x +c \right )-\frac {\tan \left (d x +c \right )}{2 \left (1+\tan ^{2}\left (d x +c \right )\right )}+\frac {5 \arctan \left (\tan \left (d x +c \right )\right )}{2}}{d \,a^{2}}\) | \(56\) |
parallelrisch | \(\frac {180 d x \cos \left (d x +c \right )+60 d x \cos \left (3 d x +3 c \right )-65 \sin \left (3 d x +3 c \right )-30 \sin \left (d x +c \right )-3 \sin \left (5 d x +5 c \right )}{24 a^{2} d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(83\) |
risch | \(\frac {5 x}{2 a^{2}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{2}}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{2}}-\frac {2 i \left (9 \,{\mathrm e}^{4 i \left (d x +c \right )}+12 \,{\mathrm e}^{2 i \left (d x +c \right )}+7\right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(90\) |
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Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^6(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {15 \, d x \cos \left (d x + c\right )^{3} - {\left (3 \, \cos \left (d x + c\right )^{4} + 14 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right )}{6 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1275 vs. \(2 (63) = 126\).
Time = 16.73 (sec) , antiderivative size = 1275, normalized size of antiderivative = 18.48 \[ \int \frac {\sin ^6(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
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Time = 0.36 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {\sin ^6(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {\frac {3 \, \tan \left (d x + c\right )}{a^{2} \tan \left (d x + c\right )^{2} + a^{2}} - \frac {2 \, {\left (\tan \left (d x + c\right )^{3} - 6 \, \tan \left (d x + c\right )\right )}}{a^{2}} - \frac {15 \, {\left (d x + c\right )}}{a^{2}}}{6 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int \frac {\sin ^6(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {15 \, {\left (d x + c\right )}}{a^{2}} - \frac {3 \, \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )} a^{2}} + \frac {2 \, {\left (a^{4} \tan \left (d x + c\right )^{3} - 6 \, a^{4} \tan \left (d x + c\right )\right )}}{a^{6}}}{6 \, d} \]
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Time = 13.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {\sin ^6(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {5\,x}{2\,a^2}-\frac {\mathrm {tan}\left (c+d\,x\right )}{2\,d\,\left (a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+a^2\right )}-\frac {2\,\mathrm {tan}\left (c+d\,x\right )}{a^2\,d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,a^2\,d} \]
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