Integrand size = 21, antiderivative size = 159 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^5} \, dx=-\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac {\tan (c+d x)}{21 a^2 d (a+a \sec (c+d x))^3}-\frac {8 \tan (c+d x)}{63 a d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac {\tan (c+d x)}{9 d \left (a^5+a^5 \sec (c+d x)\right )} \]
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Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3896, 3895, 3884, 4085, 3879} \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {\tan (c+d x)}{9 d \left (a^5 \sec (c+d x)+a^5\right )}-\frac {8 \tan (c+d x)}{63 a d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac {\tan (c+d x)}{21 a^2 d (a \sec (c+d x)+a)^3}-\frac {\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}+\frac {5 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4} \]
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Rule 3879
Rule 3884
Rule 3895
Rule 3896
Rule 4085
Rubi steps \begin{align*} \text {integral}& = -\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {5 \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx}{9 a} \\ & = -\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac {5 \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx}{21 a^2} \\ & = -\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac {\tan (c+d x)}{21 a^2 d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec (c+d x) (-3 a+5 a \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{21 a^4} \\ & = -\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac {\tan (c+d x)}{21 a^2 d (a+a \sec (c+d x))^3}-\frac {8 \tan (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{9 a^4} \\ & = -\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac {\tan (c+d x)}{21 a^2 d (a+a \sec (c+d x))^3}-\frac {8 \tan (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}+\frac {\tan (c+d x)}{9 d \left (a^5+a^5 \sec (c+d x)\right )} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.40 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {(65+130 \cos (c+d x)+46 \cos (2 (c+d x))+10 \cos (3 (c+d x))+\cos (4 (c+d x))) \sin (c+d x)}{252 a^5 d (1+\cos (c+d x))^5} \]
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Time = 0.35 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.36
method | result | size |
parallelrisch | \(-\frac {\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{7}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-9\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d \,a^{5}}\) | \(57\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(58\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(58\) |
risch | \(\frac {4 i \left (63 \,{\mathrm e}^{5 i \left (d x +c \right )}+63 \,{\mathrm e}^{4 i \left (d x +c \right )}+84 \,{\mathrm e}^{3 i \left (d x +c \right )}+36 \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{63 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(80\) |
norman | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{16 a d}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{112 a d}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{1008 a d}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{336 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{336 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{144 a d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} a^{4}}\) | \(171\) |
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Time = 0.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.77 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {{\left (2 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )^{2} + 25 \, \cos \left (d x + c\right ) + 5\right )} \sin \left (d x + c\right )}{63 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
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\[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec {\left (c + d x \right )} + 1}\, dx}{a^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.55 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {\frac {63 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {42 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {18 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{1008 \, a^{5} d} \]
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Time = 0.35 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.37 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^5} \, dx=-\frac {7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 42 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{1008 \, a^{5} d} \]
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Time = 13.62 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.36 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+42\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+63\right )}{1008\,a^5\,d} \]
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