Integrand size = 27, antiderivative size = 123 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {4 \tan (c+d x)}{21 a^3 d}+\frac {4 \tan ^3(c+d x)}{63 a^3 d} \]
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Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2938, 2751, 3852} \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {4 \tan ^3(c+d x)}{63 a^3 d}+\frac {4 \tan (c+d x)}{21 a^3 d}-\frac {\sec ^3(c+d x)}{21 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {\sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}+\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3} \]
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Rule 2751
Rule 2938
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}+\frac {\int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{3 a} \\ & = \frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}+\frac {5 \int \frac {\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{21 a^2} \\ & = \frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {4 \int \sec ^4(c+d x) \, dx}{21 a^3} \\ & = \frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{21 a^3 d} \\ & = \frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {4 \tan (c+d x)}{21 a^3 d}+\frac {4 \tan ^3(c+d x)}{63 a^3 d} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.50 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {10752+900 \cos (c+d x)-6912 \cos (2 (c+d x))+50 \cos (3 (c+d x))-3072 \cos (4 (c+d x))-150 \cos (5 (c+d x))+256 \cos (6 (c+d x))+9216 \sin (c+d x)+675 \sin (2 (c+d x))+512 \sin (3 (c+d x))+300 \sin (4 (c+d x))-1536 \sin (5 (c+d x))-25 \sin (6 (c+d x))}{64512 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a+a \sin (c+d x))^3} \]
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Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {16 i \left (2 i {\mathrm e}^{3 i \left (d x +c \right )}-12 \,{\mathrm e}^{2 i \left (d x +c \right )}-6 i {\mathrm e}^{i \left (d x +c \right )}-27 \,{\mathrm e}^{4 i \left (d x +c \right )}+1+36 i {\mathrm e}^{5 i \left (d x +c \right )}+42 \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{63 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9} d \,a^{3}}\) | \(109\) |
parallelrisch | \(\frac {-\frac {2}{63}+\frac {4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {256 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63}-\frac {36 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {8 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {50 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{21}-6 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}\) | \(152\) |
derivativedivides | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {5}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {64}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {40}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {27}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {39}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {59}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {13}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {5}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{3}}\) | \(190\) |
default | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {5}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {64}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {40}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {27}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {39}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {59}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {13}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {5}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{3}}\) | \(190\) |
norman | \(\frac {\frac {4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {2}{63 a d}-\frac {4 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {6 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {36 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{21 d a}+\frac {8 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}-\frac {256 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 d a}-\frac {50 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}\) | \(209\) |
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Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {8 \, \cos \left (d x + c\right )^{6} - 36 \, \cos \left (d x + c\right )^{4} + 15 \, \cos \left (d x + c\right )^{2} - {\left (24 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} - 7\right )} \sin \left (d x + c\right ) + 14}{63 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\sin {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (113) = 226\).
Time = 0.23 (sec) , antiderivative size = 442, normalized size of antiderivative = 3.59 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2 \, {\left (\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {75 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {128 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {162 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {42 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {189 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {126 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {63 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 1\right )}}{63 \, {\left (a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {36 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} d} \]
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Time = 0.55 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.40 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {21 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 13\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 756 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 4200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 11340 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 14994 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 13356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6768 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2196 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 209}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{2016 \, d} \]
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Time = 16.00 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.27 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{63}+\frac {4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}+\frac {50\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{21}+\frac {256\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{63}+\frac {36\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}-\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{7}-\frac {4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{a^3\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^9} \]
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