Integrand size = 21, antiderivative size = 87 \[ \int \sec ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(7 A+6 C) \tan (c+d x)}{7 d}+\frac {C \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {2 (7 A+6 C) \tan ^3(c+d x)}{21 d}+\frac {(7 A+6 C) \tan ^5(c+d x)}{35 d} \]
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Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4131, 3852} \[ \int \sec ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(7 A+6 C) \tan ^5(c+d x)}{35 d}+\frac {2 (7 A+6 C) \tan ^3(c+d x)}{21 d}+\frac {(7 A+6 C) \tan (c+d x)}{7 d}+\frac {C \tan (c+d x) \sec ^6(c+d x)}{7 d} \]
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Rule 3852
Rule 4131
Rubi steps \begin{align*} \text {integral}& = \frac {C \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {1}{7} (7 A+6 C) \int \sec ^6(c+d x) \, dx \\ & = \frac {C \sec ^6(c+d x) \tan (c+d x)}{7 d}-\frac {(7 A+6 C) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{7 d} \\ & = \frac {(7 A+6 C) \tan (c+d x)}{7 d}+\frac {C \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {2 (7 A+6 C) \tan ^3(c+d x)}{21 d}+\frac {(7 A+6 C) \tan ^5(c+d x)}{35 d} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int \sec ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {A \left (\tan (c+d x)+\frac {2}{3} \tan ^3(c+d x)+\frac {1}{5} \tan ^5(c+d x)\right )}{d}+\frac {C \left (\tan (c+d x)+\tan ^3(c+d x)+\frac {3}{5} \tan ^5(c+d x)+\frac {1}{7} \tan ^7(c+d x)\right )}{d} \]
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Time = 4.59 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {-A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(78\) |
default | \(\frac {-A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(78\) |
parts | \(-\frac {A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}-\frac {C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(80\) |
risch | \(\frac {16 i \left (70 A \,{\mathrm e}^{8 i \left (d x +c \right )}+175 A \,{\mathrm e}^{6 i \left (d x +c \right )}+210 C \,{\mathrm e}^{6 i \left (d x +c \right )}+147 A \,{\mathrm e}^{4 i \left (d x +c \right )}+126 C \,{\mathrm e}^{4 i \left (d x +c \right )}+49 A \,{\mathrm e}^{2 i \left (d x +c \right )}+42 C \,{\mathrm e}^{2 i \left (d x +c \right )}+7 A +6 C \right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}\) | \(111\) |
parallelrisch | \(\frac {\left (1176 A +1008 C \right ) \sin \left (3 d x +3 c \right )+\left (392 A +336 C \right ) \sin \left (5 d x +5 c \right )+\left (56 A +48 C \right ) \sin \left (7 d x +7 c \right )+840 \sin \left (d x +c \right ) \left (A +2 C \right )}{105 d \left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right )}\) | \(113\) |
norman | \(\frac {-\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}+\frac {4 \left (5 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {4 \left (5 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}+\frac {8 \left (91 A +53 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}-\frac {2 \left (113 A +129 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}-\frac {2 \left (113 A +129 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{15 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{7}}\) | \(169\) |
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Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \sec ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (8 \, {\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{2} + 15 \, C\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{7}} \]
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\[ \int \sec ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{6}{\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.69 \[ \int \sec ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, C \tan \left (d x + c\right )^{7} + 21 \, {\left (A + 3 \, C\right )} \tan \left (d x + c\right )^{5} + 35 \, {\left (2 \, A + 3 \, C\right )} \tan \left (d x + c\right )^{3} + 105 \, {\left (A + C\right )} \tan \left (d x + c\right )}{105 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91 \[ \int \sec ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, C \tan \left (d x + c\right )^{7} + 21 \, A \tan \left (d x + c\right )^{5} + 63 \, C \tan \left (d x + c\right )^{5} + 70 \, A \tan \left (d x + c\right )^{3} + 105 \, C \tan \left (d x + c\right )^{3} + 105 \, A \tan \left (d x + c\right ) + 105 \, C \tan \left (d x + c\right )}{105 \, d} \]
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Time = 15.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \sec ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\frac {C\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}+\left (\frac {A}{5}+\frac {3\,C}{5}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^5+\left (\frac {2\,A}{3}+C\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (A+C\right )\,\mathrm {tan}\left (c+d\,x\right )}{d} \]
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