Integrand size = 31, antiderivative size = 169 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 (7 A+6 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 (10 A+9 B) \tan (c+d x)}{5 d}+\frac {a^2 (7 A+6 B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^2 (5 A+6 B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {B \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{5 d}+\frac {a^2 (10 A+9 B) \tan ^3(c+d x)}{15 d} \]
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Time = 0.39 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4103, 4082, 3872, 3853, 3855, 3852} \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 (7 A+6 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 (10 A+9 B) \tan ^3(c+d x)}{15 d}+\frac {a^2 (10 A+9 B) \tan (c+d x)}{5 d}+\frac {a^2 (5 A+6 B) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac {a^2 (7 A+6 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d} \]
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Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4082
Rule 4103
Rubi steps \begin{align*} \text {integral}& = \frac {B \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{5 d}+\frac {1}{5} \int \sec ^3(c+d x) (a+a \sec (c+d x)) (a (5 A+3 B)+a (5 A+6 B) \sec (c+d x)) \, dx \\ & = \frac {a^2 (5 A+6 B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {B \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{5 d}+\frac {1}{20} \int \sec ^3(c+d x) \left (5 a^2 (7 A+6 B)+4 a^2 (10 A+9 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a^2 (5 A+6 B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {B \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{5 d}+\frac {1}{4} \left (a^2 (7 A+6 B)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{5} \left (a^2 (10 A+9 B)\right ) \int \sec ^4(c+d x) \, dx \\ & = \frac {a^2 (7 A+6 B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^2 (5 A+6 B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {B \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{5 d}+\frac {1}{8} \left (a^2 (7 A+6 B)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^2 (10 A+9 B)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d} \\ & = \frac {a^2 (7 A+6 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 (10 A+9 B) \tan (c+d x)}{5 d}+\frac {a^2 (7 A+6 B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^2 (5 A+6 B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {B \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{5 d}+\frac {a^2 (10 A+9 B) \tan ^3(c+d x)}{15 d} \\ \end{align*}
Time = 1.74 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.60 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 \left (15 (7 A+6 B) \text {arctanh}(\sin (c+d x))+\sec (c+d x) \left (15 (7 A+6 B)+8 (10 A+9 B) (2+\cos (2 (c+d x))) \sec (c+d x)+30 (A+2 B) \sec ^2(c+d x)+24 B \sec ^3(c+d x)\right ) \tan (c+d x)\right )}{120 d} \]
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Time = 4.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.02
method | result | size |
parts | \(\frac {\left (A \,a^{2}+2 B \,a^{2}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (2 A \,a^{2}+B \,a^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {B \,a^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(173\) |
norman | \(\frac {\frac {7 a^{2} \left (7 A +6 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}-\frac {a^{2} \left (7 A +6 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {a^{2} \left (25 A +26 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {8 a^{2} \left (25 A +27 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}+\frac {a^{2} \left (79 A +54 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}-\frac {a^{2} \left (7 A +6 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{2} \left (7 A +6 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(201\) |
parallelrisch | \(\frac {16 \left (-\frac {105 \left (A +\frac {6 B}{7}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{64}+\frac {105 \left (A +\frac {6 B}{7}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{64}+\left (\frac {33 A}{32}+\frac {21 B}{16}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {5 A}{4}+\frac {9 B}{8}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {21 A}{64}+\frac {9 B}{32}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {A}{4}+\frac {9 B}{40}\right ) \sin \left (5 d x +5 c \right )+\sin \left (d x +c \right ) \left (A +\frac {3 B}{2}\right )\right ) a^{2}}{3 d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(217\) |
derivativedivides | \(\frac {A \,a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-2 A \,a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+2 B \,a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(223\) |
default | \(\frac {A \,a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-2 A \,a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+2 B \,a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(223\) |
risch | \(-\frac {i a^{2} \left (105 A \,{\mathrm e}^{9 i \left (d x +c \right )}+90 B \,{\mathrm e}^{9 i \left (d x +c \right )}+330 A \,{\mathrm e}^{7 i \left (d x +c \right )}+420 B \,{\mathrm e}^{7 i \left (d x +c \right )}-480 A \,{\mathrm e}^{6 i \left (d x +c \right )}-240 B \,{\mathrm e}^{6 i \left (d x +c \right )}-1120 A \,{\mathrm e}^{4 i \left (d x +c \right )}-1200 B \,{\mathrm e}^{4 i \left (d x +c \right )}-330 A \,{\mathrm e}^{3 i \left (d x +c \right )}-420 B \,{\mathrm e}^{3 i \left (d x +c \right )}-800 A \,{\mathrm e}^{2 i \left (d x +c \right )}-720 B \,{\mathrm e}^{2 i \left (d x +c \right )}-105 \,{\mathrm e}^{i \left (d x +c \right )} A -90 B \,{\mathrm e}^{i \left (d x +c \right )}-160 A -144 B \right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{4 d}-\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{4 d}\) | \(287\) |
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Time = 0.30 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (7 \, A + 6 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (7 \, A + 6 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (10 \, A + 9 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 15 \, {\left (7 \, A + 6 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (10 \, A + 9 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 30 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right ) + 24 \, B a^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
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\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=a^{2} \left (\int A \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 A \sec ^{4}{\left (c + d x \right )}\, dx + \int A \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int 2 B \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{6}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.22 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.64 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} + 16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{2} + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} - 15 \, A a^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, B a^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, A a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.46 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (7 \, A a^{2} + 6 \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (7 \, A a^{2} + 6 \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (105 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 90 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 490 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 420 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 800 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 864 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 790 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 540 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 375 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 390 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]
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Time = 16.34 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.33 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (7\,A+6\,B\right )}{4\,d}-\frac {\left (\frac {7\,A\,a^2}{4}+\frac {3\,B\,a^2}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {49\,A\,a^2}{6}-7\,B\,a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {40\,A\,a^2}{3}+\frac {72\,B\,a^2}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {79\,A\,a^2}{6}-9\,B\,a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {25\,A\,a^2}{4}+\frac {13\,B\,a^2}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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