Integrand size = 29, antiderivative size = 101 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\frac {17 a^4 x}{2}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2951, 2729, 2727, 2718, 2715, 8} \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=-\frac {4 a^4 \cos (c+d x)}{d}-\frac {a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {17 a^4 x}{2} \]
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Rule 8
Rule 2715
Rule 2718
Rule 2727
Rule 2729
Rule 2951
Rubi steps \begin{align*} \text {integral}& = a^4 \int \left (8+\frac {4}{(-1+\sin (c+d x))^2}+\frac {12}{-1+\sin (c+d x)}+4 \sin (c+d x)+\sin ^2(c+d x)\right ) \, dx \\ & = 8 a^4 x+a^4 \int \sin ^2(c+d x) \, dx+\left (4 a^4\right ) \int \frac {1}{(-1+\sin (c+d x))^2} \, dx+\left (4 a^4\right ) \int \sin (c+d x) \, dx+\left (12 a^4\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx \\ & = 8 a^4 x-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {12 a^4 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a^4 \int 1 \, dx-\frac {1}{3} \left (4 a^4\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx \\ & = \frac {17 a^4 x}{2}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 7.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.56 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=-\frac {a^4 \left (-3 (161+204 c+204 d x) \cos \left (\frac {1}{2} (c+d x)\right )+(647+204 c+204 d x) \cos \left (\frac {3}{2} (c+d x)\right )-39 \cos \left (\frac {5}{2} (c+d x)\right )+3 \cos \left (\frac {7}{2} (c+d x)\right )+6 (146+136 c+136 d x+(-59+68 c+68 d x) \cos (c+d x)-14 \cos (2 (c+d x))-\cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.31
method | result | size |
risch | \(\frac {17 a^{4} x}{2}+\frac {i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {2 a^{4} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {2 a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{d}-\frac {i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {8 \left (-15 i a^{4} {\mathrm e}^{i \left (d x +c \right )}+9 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-8 a^{4}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}\) | \(132\) |
parallelrisch | \(-\frac {a^{4} \left (-204 d x \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-204 d x \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-612 d x \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+612 d x \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+3 \sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+39 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+575 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+39 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-3 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )-207 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+267 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-837 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \left (\sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )\right )}\) | \(193\) |
derivativedivides | \(\frac {a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+6 a^{4} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+4 a^{4} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(268\) |
default | \(\frac {a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+6 a^{4} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+4 a^{4} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(268\) |
norman | \(\frac {-\frac {17 a^{4} x}{2}+\frac {80 a^{4}}{3 d}+\frac {17 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {26 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {307 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {188 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {307 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {26 a^{4} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {17 a^{4} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {17 a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {51 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {51 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {51 a^{4} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {51 a^{4} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {17 a^{4} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {17 a^{4} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {16 a^{4} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {96 a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {80 a^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {304 a^{4} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {544 a^{4} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(392\) |
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Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (89) = 178\).
Time = 0.26 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.95 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\frac {3 \, a^{4} \cos \left (d x + c\right )^{4} - 18 \, a^{4} \cos \left (d x + c\right )^{3} - 102 \, a^{4} d x - 8 \, a^{4} + 17 \, {\left (3 \, a^{4} d x + 5 \, a^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (51 \, a^{4} d x - 98 \, a^{4}\right )} \cos \left (d x + c\right ) - {\left (3 \, a^{4} \cos \left (d x + c\right )^{3} - 102 \, a^{4} d x + 21 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - {\left (51 \, a^{4} d x - 106 \, a^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
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Timed out. \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.56 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\frac {2 \, a^{4} \tan \left (d x + c\right )^{3} + {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{4} + 12 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4} - 8 \, a^{4} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {8 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{4}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.34 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\frac {51 \, {\left (d x + c\right )} a^{4} + \frac {6 \, {\left (a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {16 \, {\left (6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \]
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Time = 17.13 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.84 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\frac {17\,a^4\,x}{2}+\frac {\frac {17\,a^4\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {51\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (153\,c+153\,d\,x-378\right )}{6}\right )-\frac {a^4\,\left (51\,c+51\,d\,x-160\right )}{6}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {51\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (153\,c+153\,d\,x-102\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {85\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (255\,c+255\,d\,x-306\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {85\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (255\,c+255\,d\,x-494\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {119\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (357\,c+357\,d\,x-460\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {119\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (357\,c+357\,d\,x-660\right )}{6}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
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