Integrand size = 27, antiderivative size = 89 \[ \int \sin (c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx=-3 a^2 x+\frac {3 a^2 \cos (c+d x)}{d}-\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {2 a^2 \sec (c+d x)}{d}+\frac {3 a^2 \tan (c+d x)}{d}-\frac {a^2 \sin ^2(c+d x) \tan (c+d x)}{d} \]
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Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2952, 2670, 14, 2671, 294, 327, 209, 276} \[ \int \sin (c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx=-\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {3 a^2 \cos (c+d x)}{d}+\frac {3 a^2 \tan (c+d x)}{d}+\frac {2 a^2 \sec (c+d x)}{d}-\frac {a^2 \sin ^2(c+d x) \tan (c+d x)}{d}-3 a^2 x \]
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Rule 14
Rule 209
Rule 276
Rule 294
Rule 327
Rule 2670
Rule 2671
Rule 2952
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \sin (c+d x) \tan ^2(c+d x)+2 a^2 \sin ^2(c+d x) \tan ^2(c+d x)+a^2 \sin ^3(c+d x) \tan ^2(c+d x)\right ) \, dx \\ & = a^2 \int \sin (c+d x) \tan ^2(c+d x) \, dx+a^2 \int \sin ^3(c+d x) \tan ^2(c+d x) \, dx+\left (2 a^2\right ) \int \sin ^2(c+d x) \tan ^2(c+d x) \, dx \\ & = -\frac {a^2 \text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^2 \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a^2 \sin ^2(c+d x) \tan (c+d x)}{d}-\frac {a^2 \text {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^2 \text {Subst}\left (\int \left (-2+\frac {1}{x^2}+x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {3 a^2 \cos (c+d x)}{d}-\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {2 a^2 \sec (c+d x)}{d}+\frac {3 a^2 \tan (c+d x)}{d}-\frac {a^2 \sin ^2(c+d x) \tan (c+d x)}{d}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -3 a^2 x+\frac {3 a^2 \cos (c+d x)}{d}-\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {2 a^2 \sec (c+d x)}{d}+\frac {3 a^2 \tan (c+d x)}{d}-\frac {a^2 \sin ^2(c+d x) \tan (c+d x)}{d} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.46 \[ \int \sin (c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx=-\frac {a^2 \sec (c+d x) (1+\sin (c+d x))^{5/2} \left (-18 \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (-14+5 \sin (c+d x)+2 \sin ^2(c+d x)+\sin ^3(c+d x)\right )\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4} \]
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Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10
method | result | size |
risch | \(-3 a^{2} x +\frac {11 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {11 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {4 a^{2}}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}-\frac {a^{2} \cos \left (3 d x +3 c \right )}{12 d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{2 d}\) | \(98\) |
parallelrisch | \(-\frac {a^{2} \left (72 d x \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-72 d x \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+113 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+49 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+5 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+27 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+5 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-27 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )\right )}{24 d \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )-\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(145\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )}{d}\) | \(148\) |
default | \(\frac {a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )}{d}\) | \(148\) |
norman | \(\frac {3 a^{2} x -\frac {28 a^{2}}{3 d}-\frac {6 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {10 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {10 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+6 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {56 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(210\) |
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Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.71 \[ \int \sin (c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx=-\frac {a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{3} + 9 \, a^{2} d x - 9 \, a^{2} \cos \left (d x + c\right )^{2} - 6 \, a^{2} + 3 \, {\left (3 \, a^{2} d x - 4 \, a^{2}\right )} \cos \left (d x + c\right ) - {\left (a^{2} \cos \left (d x + c\right )^{3} + 9 \, a^{2} d x + 3 \, a^{2} \cos \left (d x + c\right )^{2} - 6 \, a^{2} \cos \left (d x + c\right ) + 6 \, a^{2}\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx=a^{2} \left (\int \sin ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 \sin ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10 \[ \int \sin (c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx=-\frac {{\left (\cos \left (d x + c\right )^{3} - \frac {3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a^{2} + 3 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{2} - 3 \, a^{2} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{3 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.34 \[ \int \sin (c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx=-\frac {9 \, {\left (d x + c\right )} a^{2} + \frac {12 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} + \frac {2 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
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Time = 15.66 (sec) , antiderivative size = 288, normalized size of antiderivative = 3.24 \[ \int \sin (c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx=-3\,a^2\,x-\frac {3\,a^2\,\left (c+d\,x\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (9\,c+9\,d\,x-10\right )}{3}\right )-\frac {a^2\,\left (9\,c+9\,d\,x-28\right )}{3}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (3\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (9\,c+9\,d\,x-18\right )}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (9\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (27\,c+27\,d\,x-18\right )}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (9\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (27\,c+27\,d\,x-36\right )}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (9\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (27\,c+27\,d\,x-48\right )}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (9\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (27\,c+27\,d\,x-66\right )}{3}\right )}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \]
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