Integrand size = 23, antiderivative size = 111 \[ \int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=a^3 A x+\frac {a^3 (7 A+5 B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {5 a^3 (A+B) \tan (c+d x)}{2 d}+\frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d} \]
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Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4002, 3999, 3852, 8, 3855} \[ \int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 (7 A+5 B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {5 a^3 (A+B) \tan (c+d x)}{2 d}+\frac {(3 A+5 B) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+a^3 A x+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3999
Rule 4002
Rubi steps \begin{align*} \text {integral}& = \frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{3} \int (a+a \sec (c+d x))^2 (3 a A+a (3 A+5 B) \sec (c+d x)) \, dx \\ & = \frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}+\frac {1}{6} \int (a+a \sec (c+d x)) \left (6 a^2 A+15 a^2 (A+B) \sec (c+d x)\right ) \, dx \\ & = a^3 A x+\frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}+\frac {1}{2} \left (5 a^3 (A+B)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{2} \left (a^3 (7 A+5 B)\right ) \int \sec (c+d x) \, dx \\ & = a^3 A x+\frac {a^3 (7 A+5 B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}-\frac {\left (5 a^3 (A+B)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 d} \\ & = a^3 A x+\frac {a^3 (7 A+5 B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {5 a^3 (A+B) \tan (c+d x)}{2 d}+\frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d} \\ \end{align*}
Time = 1.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.63 \[ \int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 \left (6 A d x+3 (7 A+5 B) \text {arctanh}(\sin (c+d x))+3 (6 A+8 B+(A+3 B) \sec (c+d x)) \tan (c+d x)+2 B \tan ^3(c+d x)\right )}{6 d} \]
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Time = 3.64 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.22
method | result | size |
parts | \(a^{3} A x +\frac {\left (a^{3} A +3 B \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (3 a^{3} A +B \,a^{3}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (3 a^{3} A +3 B \,a^{3}\right ) \tan \left (d x +c \right )}{d}-\frac {B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(135\) |
parallelrisch | \(\frac {\left (-\frac {21 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (A +\frac {5 B}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}+\frac {21 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (A +\frac {5 B}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}+A x d \cos \left (3 d x +3 c \right )+\left (A +3 B \right ) \sin \left (2 d x +2 c \right )+\left (3 A +\frac {11 B}{3}\right ) \sin \left (3 d x +3 c \right )+3 A x d \cos \left (d x +c \right )+3 \sin \left (d x +c \right ) \left (A +\frac {5 B}{3}\right )\right ) a^{3}}{d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(172\) |
derivativedivides | \(\frac {a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \tan \left (d x +c \right )+3 B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \tan \left (d x +c \right )+a^{3} A \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(176\) |
default | \(\frac {a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \tan \left (d x +c \right )+3 B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \tan \left (d x +c \right )+a^{3} A \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(176\) |
norman | \(\frac {a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-a^{3} A x +3 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {5 a^{3} \left (A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {a^{3} \left (7 A +11 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 a^{3} \left (9 A +10 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-\frac {a^{3} \left (7 A +5 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{3} \left (7 A +5 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(205\) |
risch | \(a^{3} A x -\frac {i a^{3} \left (3 A \,{\mathrm e}^{5 i \left (d x +c \right )}+9 B \,{\mathrm e}^{5 i \left (d x +c \right )}-18 A \,{\mathrm e}^{4 i \left (d x +c \right )}-18 B \,{\mathrm e}^{4 i \left (d x +c \right )}-36 A \,{\mathrm e}^{2 i \left (d x +c \right )}-48 B \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )} A -9 B \,{\mathrm e}^{i \left (d x +c \right )}-18 A -22 B \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}+\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}\) | \(221\) |
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Time = 0.30 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.27 \[ \int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {12 \, A a^{3} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (7 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (7 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (9 \, A + 11 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) + 2 \, B a^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
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\[ \int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=a^{3} \left (\int A\, dx + \int 3 A \sec {\left (c + d x \right )}\, dx + \int 3 A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec {\left (c + d x \right )}\, dx + \int 3 B \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.23 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.78 \[ \int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {12 \, {\left (d x + c\right )} A a^{3} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} - 3 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 9 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 12 \, B a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 36 \, A a^{3} \tan \left (d x + c\right ) + 36 \, B a^{3} \tan \left (d x + c\right )}{12 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.70 \[ \int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {6 \, {\left (d x + c\right )} A a^{3} + 3 \, {\left (7 \, A a^{3} + 5 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (7 \, A a^{3} + 5 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
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Time = 13.77 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.88 \[ \int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {2\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {5\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,A\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {11\,B\,a^3\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {B\,a^3\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3} \]
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