Integrand size = 12, antiderivative size = 144 \[ \int \frac {1}{(a+a \sec (c+d x))^5} \, dx=\frac {x}{a^5}-\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {34 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac {173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {488 \tan (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )} \]
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Time = 0.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3862, 4007, 4004, 3879} \[ \int \frac {1}{(a+a \sec (c+d x))^5} \, dx=-\frac {488 \tan (c+d x)}{315 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac {x}{a^5}-\frac {173 \tan (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac {34 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}-\frac {13 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac {\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
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Rule 3862
Rule 3879
Rule 4004
Rule 4007
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {\int \frac {-9 a+4 a \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx}{9 a^2} \\ & = -\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac {\int \frac {63 a^2-39 a^2 \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{63 a^4} \\ & = -\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {34 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac {\int \frac {-315 a^3+204 a^3 \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{315 a^6} \\ & = -\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {34 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac {173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {945 a^4-519 a^4 \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{945 a^8} \\ & = \frac {x}{a^5}-\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {34 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac {173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {488 \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{315 a^4} \\ & = \frac {x}{a^5}-\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {34 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac {173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {488 \tan (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )} \\ \end{align*}
Time = 4.87 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(a+a \sec (c+d x))^5} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \left (39690 d x \cos \left (\frac {d x}{2}\right )+39690 d x \cos \left (c+\frac {d x}{2}\right )+26460 d x \cos \left (c+\frac {3 d x}{2}\right )+26460 d x \cos \left (2 c+\frac {3 d x}{2}\right )+11340 d x \cos \left (2 c+\frac {5 d x}{2}\right )+11340 d x \cos \left (3 c+\frac {5 d x}{2}\right )+2835 d x \cos \left (3 c+\frac {7 d x}{2}\right )+2835 d x \cos \left (4 c+\frac {7 d x}{2}\right )+315 d x \cos \left (4 c+\frac {9 d x}{2}\right )+315 d x \cos \left (5 c+\frac {9 d x}{2}\right )-116676 \sin \left (\frac {d x}{2}\right )+100800 \sin \left (c+\frac {d x}{2}\right )-88284 \sin \left (c+\frac {3 d x}{2}\right )+56700 \sin \left (2 c+\frac {3 d x}{2}\right )-43236 \sin \left (2 c+\frac {5 d x}{2}\right )+18900 \sin \left (3 c+\frac {5 d x}{2}\right )-12384 \sin \left (3 c+\frac {7 d x}{2}\right )+3150 \sin \left (4 c+\frac {7 d x}{2}\right )-1726 \sin \left (4 c+\frac {9 d x}{2}\right )\right )}{161280 a^5 d} \]
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Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.53
method | result | size |
parallelrisch | \(\frac {-35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+270 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-1008 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2730 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+5040 d x -9765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5040 d \,a^{5}}\) | \(77\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(85\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(85\) |
norman | \(\frac {\frac {x}{a}-\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}+\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{56 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{144 a d}}{a^{4}}\) | \(104\) |
risch | \(\frac {x}{a^{5}}-\frac {2 i \left (1575 \,{\mathrm e}^{8 i \left (d x +c \right )}+9450 \,{\mathrm e}^{7 i \left (d x +c \right )}+28350 \,{\mathrm e}^{6 i \left (d x +c \right )}+50400 \,{\mathrm e}^{5 i \left (d x +c \right )}+58338 \,{\mathrm e}^{4 i \left (d x +c \right )}+44142 \,{\mathrm e}^{3 i \left (d x +c \right )}+21618 \,{\mathrm e}^{2 i \left (d x +c \right )}+6192 \,{\mathrm e}^{i \left (d x +c \right )}+863\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(119\) |
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Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(a+a \sec (c+d x))^5} \, dx=\frac {315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x - {\left (863 \, \cos \left (d x + c\right )^{4} + 2740 \, \cos \left (d x + c\right )^{3} + 3549 \, \cos \left (d x + c\right )^{2} + 2125 \, \cos \left (d x + c\right ) + 488\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
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\[ \int \frac {1}{(a+a \sec (c+d x))^5} \, dx=\frac {\int \frac {1}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec {\left (c + d x \right )} + 1}\, dx}{a^{5}} \]
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Time = 0.39 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(a+a \sec (c+d x))^5} \, dx=-\frac {\frac {\frac {9765 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2730 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {10080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(a+a \sec (c+d x))^5} \, dx=\frac {\frac {5040 \, {\left (d x + c\right )}}{a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 270 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1008 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2730 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9765 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]
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Time = 13.44 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a+a \sec (c+d x))^5} \, dx=\frac {x}{a^5}-\frac {\frac {863\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{315}-\frac {356\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{315}+\frac {169\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{420}-\frac {41\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{504}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{144}}{a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
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