Integrand size = 31, antiderivative size = 181 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=4 a^4 A x+\frac {a^4 (52 A+35 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}+\frac {5 a^4 (4 A+7 C) \tan (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d} \]
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Time = 0.55 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4172, 4002, 3999, 3852, 8, 3855} \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (52 A+35 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^4 (4 A+7 C) \tan (c+d x)}{8 d}-\frac {(12 A-35 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+4 a^4 A x-\frac {(12 A-7 C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{12 d}-\frac {a (4 A-C) \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^4}{d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3999
Rule 4002
Rule 4172
Rubi steps \begin{align*} \text {integral}& = \frac {A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}+\frac {\int (a+a \sec (c+d x))^4 (4 a A-a (4 A-C) \sec (c+d x)) \, dx}{a} \\ & = \frac {A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\int (a+a \sec (c+d x))^3 \left (16 a^2 A-a^2 (12 A-7 C) \sec (c+d x)\right ) \, dx}{4 a} \\ & = \frac {A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}+\frac {\int (a+a \sec (c+d x))^2 \left (48 a^3 A-a^3 (12 A-35 C) \sec (c+d x)\right ) \, dx}{12 a} \\ & = \frac {A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac {\int (a+a \sec (c+d x)) \left (96 a^4 A+15 a^4 (4 A+7 C) \sec (c+d x)\right ) \, dx}{24 a} \\ & = 4 a^4 A x+\frac {A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac {1}{8} \left (5 a^4 (4 A+7 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (a^4 (52 A+35 C)\right ) \int \sec (c+d x) \, dx \\ & = 4 a^4 A x+\frac {a^4 (52 A+35 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}-\frac {\left (5 a^4 (4 A+7 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{8 d} \\ & = 4 a^4 A x+\frac {a^4 (52 A+35 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}+\frac {5 a^4 (4 A+7 C) \tan (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d} \\ \end{align*}
Time = 5.21 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.12 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=4 a^4 A x+\frac {13 a^4 A \text {arctanh}(\sin (c+d x))}{2 d}+\frac {35 a^4 C \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^4 A \sin (c+d x)}{d}+\frac {8 a^4 C \tan (c+d x)}{d}+\frac {a^4 A \sec (c+d x) \tan (c+d x)}{2 d}+\frac {27 a^4 C \sec (c+d x) \tan (c+d x)}{8 d}+\frac {4 a^4 A \sec ^2(c+d x) \tan (c+d x)}{d}+\frac {a^4 C \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {4 a^4 A \tan ^3(c+d x)}{d}+\frac {4 a^4 C \tan ^3(c+d x)}{3 d} \]
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Time = 0.68 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.23
method | result | size |
parallelrisch | \(\frac {8 a^{4} \left (-\frac {13 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {35 C}{52}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4}+\frac {13 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {35 C}{52}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4}+2 d x A \cos \left (2 d x +2 c \right )+\frac {d x A \cos \left (4 d x +4 c \right )}{2}+\left (A +\frac {7 C}{3}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (5 A +\frac {27 C}{2}\right ) \sin \left (3 d x +3 c \right )}{16}+\frac {\left (A +\frac {5 C}{3}\right ) \sin \left (4 d x +4 c \right )}{2}+\frac {A \sin \left (5 d x +5 c \right )}{16}+\frac {\left (A +\frac {35 C}{8}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x A}{2}\right )}{d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(223\) |
derivativedivides | \(\frac {a^{4} A \sin \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \left (d x +c \right )+4 a^{4} C \tan \left (d x +c \right )+6 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} A \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(237\) |
default | \(\frac {a^{4} A \sin \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \left (d x +c \right )+4 a^{4} C \tan \left (d x +c \right )+6 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} A \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(237\) |
risch | \(4 a^{4} A x -\frac {i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a^{4} \left (12 A \,{\mathrm e}^{7 i \left (d x +c \right )}+81 C \,{\mathrm e}^{7 i \left (d x +c \right )}-96 A \,{\mathrm e}^{6 i \left (d x +c \right )}-96 C \,{\mathrm e}^{6 i \left (d x +c \right )}+12 A \,{\mathrm e}^{5 i \left (d x +c \right )}+105 C \,{\mathrm e}^{5 i \left (d x +c \right )}-288 A \,{\mathrm e}^{4 i \left (d x +c \right )}-480 C \,{\mathrm e}^{4 i \left (d x +c \right )}-12 A \,{\mathrm e}^{3 i \left (d x +c \right )}-105 C \,{\mathrm e}^{3 i \left (d x +c \right )}-288 A \,{\mathrm e}^{2 i \left (d x +c \right )}-544 C \,{\mathrm e}^{2 i \left (d x +c \right )}-12 A \,{\mathrm e}^{i \left (d x +c \right )}-81 C \,{\mathrm e}^{i \left (d x +c \right )}-96 A -160 C \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\) | \(332\) |
norman | \(\frac {-4 a^{4} A x +16 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-20 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+20 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-16 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+4 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-\frac {5 a^{4} \left (4 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}-\frac {a^{4} \left (44 A +93 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {7 a^{4} \left (60 A +73 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {a^{4} \left (-203 C +12 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}-\frac {a^{4} \left (53 C +204 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 d}+\frac {a^{4} \left (385 C +156 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {a^{4} \left (52 A +35 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{4} \left (52 A +35 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(339\) |
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Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.94 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {192 \, A a^{4} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (52 \, A + 35 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (52 \, A + 35 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 32 \, {\left (3 \, A + 5 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, A + 27 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 32 \, C a^{4} \cos \left (d x + c\right ) + 6 \, C a^{4}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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\[ \int \cos (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^{4} \left (\int A \cos {\left (c + d x \right )}\, dx + \int 4 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int A \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int C \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 C \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 C \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 C \cos {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}\, dx + \int C \cos {\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.64 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {192 \, {\left (d x + c\right )} A a^{4} + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 3 \, C a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{4} \sin \left (d x + c\right ) + 192 \, A a^{4} \tan \left (d x + c\right ) + 192 \, C a^{4} \tan \left (d x + c\right )}{48 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.40 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {96 \, {\left (d x + c\right )} A a^{4} + \frac {48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 3 \, {\left (52 \, A a^{4} + 35 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (52 \, A a^{4} + 35 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (84 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 276 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 385 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 300 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 511 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 108 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 279 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
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Time = 15.74 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.36 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {8\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {13\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {35\,C\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {4\,A\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {20\,C\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {27\,C\,a^4\,\sin \left (c+d\,x\right )}{8\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {4\,C\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {C\,a^4\,\sin \left (c+d\,x\right )}{4\,d\,{\cos \left (c+d\,x\right )}^4} \]
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