Integrand size = 29, antiderivative size = 117 \[ \int \frac {\sin ^2(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {x}{a}-\frac {\cos (c+d x)}{a d}-\frac {3 \sec (c+d x)}{a d}+\frac {\sec ^3(c+d x)}{a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\tan (c+d x)}{a d}-\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan ^5(c+d x)}{5 a d} \]
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Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2918, 3554, 8, 2670, 276} \[ \int \frac {\sin ^2(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos (c+d x)}{a d}+\frac {\tan ^5(c+d x)}{5 a d}-\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan (c+d x)}{a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\sec ^3(c+d x)}{a d}-\frac {3 \sec (c+d x)}{a d}-\frac {x}{a} \]
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Rule 8
Rule 276
Rule 2670
Rule 2918
Rule 3554
Rubi steps \begin{align*} \text {integral}& = \frac {\int \tan ^6(c+d x) \, dx}{a}-\frac {\int \sin (c+d x) \tan ^6(c+d x) \, dx}{a} \\ & = \frac {\tan ^5(c+d x)}{5 a d}-\frac {\int \tan ^4(c+d x) \, dx}{a}+\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,\cos (c+d x)\right )}{a d} \\ & = -\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan ^5(c+d x)}{5 a d}+\frac {\int \tan ^2(c+d x) \, dx}{a}+\frac {\text {Subst}\left (\int \left (-1+\frac {1}{x^6}-\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = -\frac {\cos (c+d x)}{a d}-\frac {3 \sec (c+d x)}{a d}+\frac {\sec ^3(c+d x)}{a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\tan (c+d x)}{a d}-\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan ^5(c+d x)}{5 a d}-\frac {\int 1 \, dx}{a} \\ & = -\frac {x}{a}-\frac {\cos (c+d x)}{a d}-\frac {3 \sec (c+d x)}{a d}+\frac {\sec ^3(c+d x)}{a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\tan (c+d x)}{a d}-\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan ^5(c+d x)}{5 a d} \\ \end{align*}
Time = 1.18 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.91 \[ \int \frac {\sin ^2(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {1200+18 (-103+40 c+40 d x) \cos (c+d x)+1568 \cos (2 (c+d x))-618 \cos (3 (c+d x))+240 c \cos (3 (c+d x))+240 d x \cos (3 (c+d x))+304 \cos (4 (c+d x))+216 \sin (c+d x)-618 \sin (2 (c+d x))+240 c \sin (2 (c+d x))+240 d x \sin (2 (c+d x))+532 \sin (3 (c+d x))-309 \sin (4 (c+d x))+120 c \sin (4 (c+d x))+120 d x \sin (4 (c+d x))+60 \sin (5 (c+d x))}{960 a d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
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Time = 0.65 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.36
method | result | size |
derivativedivides | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {23}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(159\) |
default | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {23}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(159\) |
risch | \(-\frac {x}{a}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 a d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 a d}-\frac {2 \left (63 \,{\mathrm e}^{3 i \left (d x +c \right )}+105 i {\mathrm e}^{4 i \left (d x +c \right )}+91 i {\mathrm e}^{2 i \left (d x +c \right )}+75 \,{\mathrm e}^{5 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}+23 i+45 i {\mathrm e}^{6 i \left (d x +c \right )}+45 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{15 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} d a}\) | \(159\) |
parallelrisch | \(\frac {\left (-150 d x -480\right ) \cos \left (3 d x +3 c \right )-300 d x \cos \left (d x +c \right )-30 d x \cos \left (5 d x +5 c \right )-15 \cos \left (6 d x +6 c \right )+100 \sin \left (d x +c \right )+50 \sin \left (3 d x +3 c \right )+46 \sin \left (5 d x +5 c \right )-960 \cos \left (d x +c \right )-705 \cos \left (2 d x +2 c \right )-270 \cos \left (4 d x +4 c \right )-96 \cos \left (5 d x +5 c \right )-546}{30 a d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(163\) |
norman | \(\frac {\frac {x}{a}-\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {88 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {4 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-\frac {2 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {4 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {4 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {2 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {32}{5 a d}-\frac {20 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {2 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {54 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5 d a}+\frac {28 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {16 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {182 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {208 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(415\) |
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Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \frac {\sin ^2(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {15 \, d x \cos \left (d x + c\right )^{3} + 38 \, \cos \left (d x + c\right )^{4} + 11 \, \cos \left (d x + c\right )^{2} + {\left (15 \, d x \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )^{4} + 22 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 1}{15 \, {\left (a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{3}\right )}} \]
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Timed out. \[ \int \frac {\sin ^2(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (111) = 222\).
Time = 0.31 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.42 \[ \int \frac {\sin ^2(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 \, {\left (\frac {\frac {81 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {78 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {172 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {26 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {22 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {70 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {20 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {30 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {15 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 48}{a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {4 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}\right )}}{15 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.27 \[ \int \frac {\sin ^2(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {120 \, {\left (d x + c\right )}}{a} + \frac {240}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a} - \frac {5 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 23\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2570 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 413}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \]
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Time = 20.42 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.47 \[ \int \frac {\sin ^2(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}-\frac {52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}-\frac {344\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}-\frac {52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {54\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+\frac {32}{5}}{a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {x}{a} \]
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