Integrand size = 31, antiderivative size = 138 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 (8 A+7 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 (8 A+7 B) \tan (c+d x)}{6 d}+\frac {a^2 (8 A+7 B) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 A-B) (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d} \]
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Time = 0.38 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4095, 4086, 3873, 3852, 8, 4131, 3855} \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 (8 A+7 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 (8 A+7 B) \tan (c+d x)}{6 d}+\frac {a^2 (8 A+7 B) \tan (c+d x) \sec (c+d x)}{24 d}+\frac {(4 A-B) \tan (c+d x) (a \sec (c+d x)+a)^2}{12 d}+\frac {B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 a d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3873
Rule 4086
Rule 4095
Rule 4131
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^2 (3 a B+a (4 A-B) \sec (c+d x)) \, dx}{4 a} \\ & = \frac {(4 A-B) (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac {1}{12} (8 A+7 B) \int \sec (c+d x) (a+a \sec (c+d x))^2 \, dx \\ & = \frac {(4 A-B) (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac {1}{12} (8 A+7 B) \int \sec (c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx+\frac {1}{6} \left (a^2 (8 A+7 B)\right ) \int \sec ^2(c+d x) \, dx \\ & = \frac {a^2 (8 A+7 B) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 A-B) (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac {1}{8} \left (a^2 (8 A+7 B)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^2 (8 A+7 B)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d} \\ & = \frac {a^2 (8 A+7 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 (8 A+7 B) \tan (c+d x)}{6 d}+\frac {a^2 (8 A+7 B) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 A-B) (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d} \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.62 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 \left (3 (8 A+7 B) \text {arctanh}(\sin (c+d x))+\left (8 (5 A+4 B)+3 (8 A+7 B) \sec (c+d x)+8 (A+2 B) \sec ^2(c+d x)+6 B \sec ^3(c+d x)\right ) \tan (c+d x)\right )}{24 d} \]
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Time = 3.88 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.09
method | result | size |
parts | \(-\frac {\left (A \,a^{2}+2 B \,a^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (2 A \,a^{2}+B \,a^{2}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {A \,a^{2} \tan \left (d x +c \right )}{d}+\frac {B \,a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(150\) |
norman | \(\frac {\frac {11 a^{2} \left (8 A +7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {a^{2} \left (8 A +7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {a^{2} \left (24 A +25 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {a^{2} \left (136 A +83 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {a^{2} \left (8 A +7 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{2} \left (8 A +7 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(175\) |
parallelrisch | \(\frac {14 \left (-\frac {6 \left (A +\frac {7 B}{8}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{7}+\frac {6 \left (A +\frac {7 B}{8}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{7}+\left (A +\frac {8 B}{7}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {3 A}{7}+\frac {3 B}{8}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {5 A}{14}+\frac {2 B}{7}\right ) \sin \left (4 d x +4 c \right )+\frac {3 \left (A +\frac {15 B}{8}\right ) \sin \left (d x +c \right )}{7}\right ) a^{2}}{3 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(178\) |
derivativedivides | \(\frac {-A \,a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+2 A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-2 B \,a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+A \,a^{2} \tan \left (d x +c \right )+B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(187\) |
default | \(\frac {-A \,a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+2 A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-2 B \,a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+A \,a^{2} \tan \left (d x +c \right )+B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(187\) |
risch | \(-\frac {i a^{2} \left (24 A \,{\mathrm e}^{7 i \left (d x +c \right )}+21 B \,{\mathrm e}^{7 i \left (d x +c \right )}-24 A \,{\mathrm e}^{6 i \left (d x +c \right )}+24 A \,{\mathrm e}^{5 i \left (d x +c \right )}+45 B \,{\mathrm e}^{5 i \left (d x +c \right )}-120 A \,{\mathrm e}^{4 i \left (d x +c \right )}-96 B \,{\mathrm e}^{4 i \left (d x +c \right )}-24 A \,{\mathrm e}^{3 i \left (d x +c \right )}-45 B \,{\mathrm e}^{3 i \left (d x +c \right )}-136 A \,{\mathrm e}^{2 i \left (d x +c \right )}-128 B \,{\mathrm e}^{2 i \left (d x +c \right )}-24 \,{\mathrm e}^{i \left (d x +c \right )} A -21 B \,{\mathrm e}^{i \left (d x +c \right )}-40 A -32 B \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{8 d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{8 d}\) | \(274\) |
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Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.05 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {3 \, {\left (8 \, A + 7 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, A + 7 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (5 \, A + 4 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, A + 7 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right ) + 6 \, B a^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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\[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=a^{2} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 A \sec ^{3}{\left (c + d x \right )}\, dx + \int A \sec ^{4}{\left (c + d x \right )}\, dx + \int B \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 B \sec ^{4}{\left (c + d x \right )}\, dx + \int B \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.23 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.67 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} + 32 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} - 3 \, B a^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 24 \, A a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, B a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{2} \tan \left (d x + c\right )}{48 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.54 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {3 \, {\left (8 \, A a^{2} + 7 \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, A a^{2} + 7 \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 21 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 88 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 77 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 136 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 83 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 75 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
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Time = 16.07 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.33 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {\left (-2\,A\,a^2-\frac {7\,B\,a^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {22\,A\,a^2}{3}+\frac {77\,B\,a^2}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {34\,A\,a^2}{3}-\frac {83\,B\,a^2}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (6\,A\,a^2+\frac {25\,B\,a^2}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {2\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A+\frac {7\,B}{8}\right )}{d} \]
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