Integrand size = 24, antiderivative size = 73 \[ \int \frac {\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {15 x}{8 a}+\frac {15 \tan (c+d x)}{8 a d}-\frac {5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d} \]
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Time = 0.06 (sec) , antiderivative size = 73, 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, 327, 209} \[ \int \frac {\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {15 \tan (c+d x)}{8 a d}-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d}-\frac {5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac {15 x}{8 a} \]
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Rule 209
Rule 294
Rule 327
Rule 2671
Rule 3254
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin ^4(c+d x) \tan ^2(c+d x) \, dx}{a} \\ & = \frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{a d} \\ & = -\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d}+\frac {5 \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 a d} \\ & = -\frac {5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d}+\frac {15 \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 a d} \\ & = \frac {15 \tan (c+d x)}{8 a d}-\frac {5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d}-\frac {15 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 a d} \\ & = -\frac {15 x}{8 a}+\frac {15 \tan (c+d x)}{8 a d}-\frac {5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.60 \[ \int \frac {\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {60 c+60 d x-16 \sin (2 (c+d x))+\sin (4 (c+d x))-32 \tan (c+d x)}{32 a d} \]
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Time = 0.63 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\tan \left (d x +c \right )-\frac {-\frac {9 \left (\tan ^{3}\left (d x +c \right )\right )}{8}-\frac {7 \tan \left (d x +c \right )}{8}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}-\frac {15 \arctan \left (\tan \left (d x +c \right )\right )}{8}}{d a}\) | \(57\) |
default | \(\frac {\tan \left (d x +c \right )-\frac {-\frac {9 \left (\tan ^{3}\left (d x +c \right )\right )}{8}-\frac {7 \tan \left (d x +c \right )}{8}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}-\frac {15 \arctan \left (\tan \left (d x +c \right )\right )}{8}}{d a}\) | \(57\) |
parallelrisch | \(\frac {-120 d x \cos \left (d x +c \right )+80 \sin \left (d x +c \right )-\sin \left (5 d x +5 c \right )+15 \sin \left (3 d x +3 c \right )}{64 a d \cos \left (d x +c \right )}\) | \(58\) |
risch | \(-\frac {15 x}{8 a}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{4 d a}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d a}+\frac {2 i}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {\sin \left (4 d x +4 c \right )}{32 a d}\) | \(83\) |
norman | \(\frac {\frac {15 x}{8 a}-\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}-\frac {35 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {113 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {29 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {113 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {35 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {15 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {75 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {135 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {75 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {75 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {135 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {75 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {15 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(289\) |
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Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \frac {\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {15 \, d x \cos \left (d x + c\right ) + {\left (2 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right )}{8 \, a d \cos \left (d x + c\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1161 vs. \(2 (61) = 122\).
Time = 6.57 (sec) , antiderivative size = 1161, normalized size of antiderivative = 15.90 \[ \int \frac {\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\text {Too large to display} \]
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Time = 0.39 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.99 \[ \int \frac {\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{a \tan \left (d x + c\right )^{4} + 2 \, a \tan \left (d x + c\right )^{2} + a} - \frac {15 \, {\left (d x + c\right )}}{a} + \frac {8 \, \tan \left (d x + c\right )}{a}}{8 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx=-\frac {\frac {15 \, {\left (d x + c\right )}}{a} - \frac {8 \, \tan \left (d x + c\right )}{a} - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2} a}}{8 \, d} \]
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Time = 14.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )}{a\,d}-\frac {15\,x}{8\,a}+\frac {\frac {9\,{\mathrm {tan}\left (c+d\,x\right )}^3}{8}+\frac {7\,\mathrm {tan}\left (c+d\,x\right )}{8}}{d\,\left (a\,{\mathrm {tan}\left (c+d\,x\right )}^4+2\,a\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )} \]
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