Integrand size = 41, antiderivative size = 204 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {(B-4 C) \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {(6 A-55 B+244 C) \tan (c+d x)}{105 a^4 d}+\frac {(3 A+25 B-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(B-4 C) \tan (c+d x)}{a^4 d (1+\sec (c+d x))}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(2 A+5 B-12 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3} \]
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Time = 0.73 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4169, 4104, 4093, 3872, 3855, 3852, 8} \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {(6 A-55 B+244 C) \tan (c+d x)}{105 a^4 d}+\frac {(3 A+25 B-88 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {(B-4 C) \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {(B-4 C) \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac {(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {(2 A+5 B-12 C) \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4093
Rule 4104
Rule 4169
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\sec ^4(c+d x) (a (3 A+4 B-4 C)+a (A-B+8 C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(2 A+5 B-12 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^3(c+d x) \left (3 a^2 (2 A+5 B-12 C)+a^2 (3 A-10 B+52 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = \frac {(3 A+25 B-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(2 A+5 B-12 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^2(c+d x) \left (2 a^3 (3 A+25 B-88 C)+a^3 (6 A-55 B+244 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = \frac {(3 A+25 B-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(2 A+5 B-12 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(B-4 C) \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {\int \sec (c+d x) \left (-105 a^4 (B-4 C)-a^4 (6 A-55 B+244 C) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = \frac {(3 A+25 B-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(2 A+5 B-12 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(B-4 C) \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(B-4 C) \int \sec (c+d x) \, dx}{a^4}+\frac {(6 A-55 B+244 C) \int \sec ^2(c+d x) \, dx}{105 a^4} \\ & = \frac {(B-4 C) \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {(3 A+25 B-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(2 A+5 B-12 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(B-4 C) \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(6 A-55 B+244 C) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d} \\ & = \frac {(B-4 C) \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {(6 A-55 B+244 C) \tan (c+d x)}{105 a^4 d}+\frac {(3 A+25 B-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(2 A+5 B-12 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(B-4 C) \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1208\) vs. \(2(204)=408\).
Time = 9.41 (sec) , antiderivative size = 1208, normalized size of antiderivative = 5.92 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {32 (-B+4 C) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {32 (-B+4 C) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {4 \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (A \sin \left (\frac {c}{2}\right )-B \sin \left (\frac {c}{2}\right )+C \sin \left (\frac {c}{2}\right )\right )}{7 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {8 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (3 A \sin \left (\frac {c}{2}\right )-10 B \sin \left (\frac {c}{2}\right )+17 C \sin \left (\frac {c}{2}\right )\right )}{35 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {16 \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (6 A \sin \left (\frac {c}{2}\right )-55 B \sin \left (\frac {c}{2}\right )+139 C \sin \left (\frac {c}{2}\right )\right )}{105 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {4 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (A \sin \left (\frac {d x}{2}\right )-B \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{7 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {8 \cos ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (3 A \sin \left (\frac {d x}{2}\right )-10 B \sin \left (\frac {d x}{2}\right )+17 C \sin \left (\frac {d x}{2}\right )\right )}{35 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {16 \cos ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (6 A \sin \left (\frac {d x}{2}\right )-55 B \sin \left (\frac {d x}{2}\right )+139 C \sin \left (\frac {d x}{2}\right )\right )}{105 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {32 \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (6 A \sin \left (\frac {d x}{2}\right )-160 B \sin \left (\frac {d x}{2}\right )+559 C \sin \left (\frac {d x}{2}\right )\right )}{105 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {32 C \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (c) \sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (d x)}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4} \]
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Time = 0.34 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.86
method | result | size |
parallelrisch | \(\frac {-3360 \cos \left (d x +c \right ) \left (B -4 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+3360 \cos \left (d x +c \right ) \left (B -4 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+24 \left (\left (\frac {15 A}{4}-65 B +275 C \right ) \cos \left (2 d x +2 c \right )+\left (A -\frac {535 B}{24}+\frac {559 C}{6}\right ) \cos \left (3 d x +3 c \right )+\left (\frac {A}{8}-\frac {10 B}{3}+\frac {83 C}{6}\right ) \cos \left (4 d x +4 c \right )+\left (9 A -\frac {2645 B}{24}+\frac {2861 C}{6}\right ) \cos \left (d x +c \right )+\frac {29 A}{8}-\frac {185 B}{3}+\frac {836 C}{3}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3360 d \,a^{4} \cos \left (d x +c \right )}\) | \(175\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B +\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (32 C -8 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-32 C +8 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{4}}\) | \(242\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B +\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (32 C -8 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-32 C +8 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{4}}\) | \(242\) |
risch | \(-\frac {2 i \left (105 B \,{\mathrm e}^{8 i \left (d x +c \right )}-420 C \,{\mathrm e}^{8 i \left (d x +c \right )}+735 B \,{\mathrm e}^{7 i \left (d x +c \right )}-2940 C \,{\mathrm e}^{7 i \left (d x +c \right )}+2275 B \,{\mathrm e}^{6 i \left (d x +c \right )}-9100 C \,{\mathrm e}^{6 i \left (d x +c \right )}-210 A \,{\mathrm e}^{5 i \left (d x +c \right )}+4165 B \,{\mathrm e}^{5 i \left (d x +c \right )}-16660 C \,{\mathrm e}^{5 i \left (d x +c \right )}-126 A \,{\mathrm e}^{4 i \left (d x +c \right )}+4795 B \,{\mathrm e}^{4 i \left (d x +c \right )}-20524 C \,{\mathrm e}^{4 i \left (d x +c \right )}-252 A \,{\mathrm e}^{3 i \left (d x +c \right )}+4445 B \,{\mathrm e}^{3 i \left (d x +c \right )}-18788 C \,{\mathrm e}^{3 i \left (d x +c \right )}-132 A \,{\mathrm e}^{2 i \left (d x +c \right )}+2785 B \,{\mathrm e}^{2 i \left (d x +c \right )}-11668 C \,{\mathrm e}^{2 i \left (d x +c \right )}-42 A \,{\mathrm e}^{i \left (d x +c \right )}+1015 B \,{\mathrm e}^{i \left (d x +c \right )}-4228 C \,{\mathrm e}^{i \left (d x +c \right )}-6 A +160 B -664 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{4} d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a^{4} d}\) | \(386\) |
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Time = 0.29 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.61 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, {\left ({\left (B - 4 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (B - 4 \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (B - 4 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (B - 4 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (B - 4 \, C\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (B - 4 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (B - 4 \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (B - 4 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (B - 4 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (B - 4 \, C\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (3 \, A - 80 \, B + 332 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (24 \, A - 535 \, B + 2236 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (39 \, A - 620 \, B + 2636 \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (9 \, A - 65 \, B + 296 \, C\right )} \cos \left (d x + c\right ) + 105 \, C\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \sec ^{4}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{5}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{6}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (196) = 392\).
Time = 0.22 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.01 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {C {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - 5 \, B {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.40 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {840 \, {\left (B - 4 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {840 \, {\left (B - 4 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {1680 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 16.04 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.23 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A-3\,B+5\,C}{40\,a^4}+\frac {A-B+C}{20\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-3\,B+5\,C}{12\,a^4}-\frac {2\,A+2\,B-10\,C}{24\,a^4}+\frac {A-B+C}{8\,a^4}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (A-3\,B+5\,C\right )}{8\,a^4}+\frac {2\,B-2\,A+10\,C}{8\,a^4}-\frac {2\,A+2\,B-10\,C}{4\,a^4}+\frac {A-B+C}{2\,a^4}\right )}{d}+\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (B-4\,C\right )}{a^4\,d}-\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d} \]
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