Integrand size = 12, antiderivative size = 88 \[ \int \frac {1}{(a+a \sec (c+d x))^3} \, dx=\frac {x}{a^3}-\frac {\tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {7 \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {22 \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )} \]
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Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3862, 4007, 4004, 3879} \[ \int \frac {1}{(a+a \sec (c+d x))^3} \, dx=-\frac {22 \tan (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac {x}{a^3}-\frac {7 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac {\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rule 3862
Rule 3879
Rule 4004
Rule 4007
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {\int \frac {-5 a+2 a \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {\tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {7 \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {\int \frac {15 a^2-7 a^2 \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = \frac {x}{a^3}-\frac {\tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {7 \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {22 \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{15 a^2} \\ & = \frac {x}{a^3}-\frac {\tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {7 \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {22 \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(a+a \sec (c+d x))^3} \, dx=\frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (60 d x \cos ^5\left (\frac {1}{2} (c+d x)\right )-3 \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+26 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )-128 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )-3 \cos \left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )+26 \cos ^3\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )\right )}{15 a^3 d (1+\sec (c+d x))^3} \]
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Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.58
method | result | size |
parallelrisch | \(\frac {-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+60 d x -105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{60 a^{3} d}\) | \(51\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(59\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(59\) |
norman | \(\frac {\frac {x}{a}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}}{a^{2}}\) | \(66\) |
risch | \(\frac {x}{a^{3}}-\frac {2 i \left (45 \,{\mathrm e}^{4 i \left (d x +c \right )}+135 \,{\mathrm e}^{3 i \left (d x +c \right )}+185 \,{\mathrm e}^{2 i \left (d x +c \right )}+115 \,{\mathrm e}^{i \left (d x +c \right )}+32\right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(75\) |
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Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(a+a \sec (c+d x))^3} \, dx=\frac {15 \, d x \cos \left (d x + c\right )^{3} + 45 \, d x \cos \left (d x + c\right )^{2} + 45 \, d x \cos \left (d x + c\right ) + 15 \, d x - {\left (32 \, \cos \left (d x + c\right )^{2} + 51 \, \cos \left (d x + c\right ) + 22\right )} \sin \left (d x + c\right )}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
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\[ \int \frac {1}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {1}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {60 \, {\left (d x + c\right )}}{a^{3}} - \frac {3 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 20 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
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Time = 13.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(a+a \sec (c+d x))^3} \, dx=\frac {x}{a^3}-\frac {\frac {32\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}-\frac {13\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{30}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{20}}{a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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