Integrand size = 27, antiderivative size = 82 \[ \int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {3 a x}{2}+\frac {2 b \cos (c+d x)}{d}-\frac {b \cos ^3(c+d x)}{3 d}+\frac {b \sec (c+d x)}{d}+\frac {3 a \tan (c+d x)}{2 d}-\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2917, 2671, 294, 327, 209, 2670, 276} \[ \int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=\frac {3 a \tan (c+d x)}{2 d}-\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac {3 a x}{2}-\frac {b \cos ^3(c+d x)}{3 d}+\frac {2 b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d} \]
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Rule 209
Rule 276
Rule 294
Rule 327
Rule 2670
Rule 2671
Rule 2917
Rubi steps \begin{align*} \text {integral}& = a \int \sin ^2(c+d x) \tan ^2(c+d x) \, dx+b \int \sin ^3(c+d x) \tan ^2(c+d x) \, dx \\ & = \frac {a \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}-\frac {b \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d}+\frac {(3 a) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac {b \text {Subst}\left (\int \left (-2+\frac {1}{x^2}+x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {2 b \cos (c+d x)}{d}-\frac {b \cos ^3(c+d x)}{3 d}+\frac {b \sec (c+d x)}{d}+\frac {3 a \tan (c+d x)}{2 d}-\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = -\frac {3 a x}{2}+\frac {2 b \cos (c+d x)}{d}-\frac {b \cos ^3(c+d x)}{3 d}+\frac {b \sec (c+d x)}{d}+\frac {3 a \tan (c+d x)}{2 d}-\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {3 a (c+d x)}{2 d}+\frac {7 b \cos (c+d x)}{4 d}-\frac {b \cos (3 (c+d x))}{12 d}+\frac {b \sec (c+d x)}{d}+\frac {a \sin (2 (c+d x))}{4 d}+\frac {a \tan (c+d x)}{d} \]
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Time = 0.68 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(\frac {20 b \cos \left (2 d x +2 c \right )-\cos \left (4 d x +4 c \right ) b +3 a \sin \left (3 d x +3 c \right )+\left (-36 a x d +64 b \right ) \cos \left (d x +c \right )+27 a \sin \left (d x +c \right )+45 b}{24 d \cos \left (d x +c \right )}\) | \(79\) |
derivativedivides | \(\frac {a \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+b \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(104\) |
default | \(\frac {a \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+b \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(104\) |
risch | \(-\frac {3 a x}{2}-\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {7 b \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {7 b \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i a +2 b \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {b \cos \left (3 d x +3 c \right )}{12 d}\) | \(117\) |
norman | \(\frac {\frac {3 a x}{2}-\frac {16 b}{3 d}-\frac {3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {5 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {5 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+3 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {32 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(171\) |
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Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88 \[ \int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {2 \, b \cos \left (d x + c\right )^{4} + 9 \, a d x \cos \left (d x + c\right ) - 12 \, b \cos \left (d x + c\right )^{2} - 3 \, {\left (a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right ) - 6 \, b}{6 \, d \cos \left (d x + c\right )} \]
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\[ \int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right ) \sin ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {3 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a + 2 \, {\left (\cos \left (d x + c\right )^{3} - \frac {3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} b}{6 \, d} \]
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Time = 0.46 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.45 \[ \int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {9 \, {\left (d x + c\right )} a + \frac {12 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {2 \, {\left (3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 10 \, b\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
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Time = 18.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.37 \[ \int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {3\,a\,x}{2}-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {32\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {16\,b}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \]
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