Integrand size = 31, antiderivative size = 163 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 (15 A+13 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 (15 A+13 B) \tan (c+d x)}{5 d}+\frac {3 a^3 (15 A+13 B) \sec (c+d x) \tan (c+d x)}{40 d}+\frac {(5 A-B) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac {a^3 (15 A+13 B) \tan ^3(c+d x)}{60 d} \]
[Out]
Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4095, 4086, 3876, 3855, 3852, 8, 3853} \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 (15 A+13 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 (15 A+13 B) \tan ^3(c+d x)}{60 d}+\frac {a^3 (15 A+13 B) \tan (c+d x)}{5 d}+\frac {3 a^3 (15 A+13 B) \tan (c+d x) \sec (c+d x)}{40 d}+\frac {(5 A-B) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac {B \tan (c+d x) (a \sec (c+d x)+a)^4}{5 a d} \]
[In]
[Out]
Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3876
Rule 4086
Rule 4095
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^3 (4 a B+a (5 A-B) \sec (c+d x)) \, dx}{5 a} \\ & = \frac {(5 A-B) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac {1}{20} (15 A+13 B) \int \sec (c+d x) (a+a \sec (c+d x))^3 \, dx \\ & = \frac {(5 A-B) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac {1}{20} (15 A+13 B) \int \left (a^3 \sec (c+d x)+3 a^3 \sec ^2(c+d x)+3 a^3 \sec ^3(c+d x)+a^3 \sec ^4(c+d x)\right ) \, dx \\ & = \frac {(5 A-B) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac {1}{20} \left (a^3 (15 A+13 B)\right ) \int \sec (c+d x) \, dx+\frac {1}{20} \left (a^3 (15 A+13 B)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{20} \left (3 a^3 (15 A+13 B)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{20} \left (3 a^3 (15 A+13 B)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {a^3 (15 A+13 B) \text {arctanh}(\sin (c+d x))}{20 d}+\frac {3 a^3 (15 A+13 B) \sec (c+d x) \tan (c+d x)}{40 d}+\frac {(5 A-B) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac {1}{40} \left (3 a^3 (15 A+13 B)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^3 (15 A+13 B)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{20 d}-\frac {\left (3 a^3 (15 A+13 B)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{20 d} \\ & = \frac {a^3 (15 A+13 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 (15 A+13 B) \tan (c+d x)}{5 d}+\frac {3 a^3 (15 A+13 B) \sec (c+d x) \tan (c+d x)}{40 d}+\frac {(5 A-B) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac {a^3 (15 A+13 B) \tan ^3(c+d x)}{60 d} \\ \end{align*}
Time = 2.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.63 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 \left (15 (15 A+13 B) \text {arctanh}(\sin (c+d x))+\left (8 (45 A+38 B)+15 (15 A+13 B) \sec (c+d x)+8 (15 A+19 B) \sec ^2(c+d x)+30 (A+3 B) \sec ^3(c+d x)+24 B \sec ^4(c+d x)\right ) \tan (c+d x)\right )}{120 d} \]
[In]
[Out]
Time = 5.02 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.20
method | result | size |
parts | \(\frac {\left (a^{3} A +3 B \,a^{3}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {\left (3 a^{3} A +B \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (3 a^{3} A +3 B \,a^{3}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}-\frac {B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{3} A \tan \left (d x +c \right )}{d}\) | \(196\) |
norman | \(\frac {-\frac {32 a^{3} \left (15 A +13 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}+\frac {7 a^{3} \left (15 A +13 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}-\frac {a^{3} \left (15 A +13 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {a^{3} \left (49 A +51 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (183 A +133 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}-\frac {a^{3} \left (15 A +13 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{3} \left (15 A +13 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(201\) |
parallelrisch | \(\frac {10 \left (-\frac {15 \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \left (A +\frac {13 B}{15}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}+\frac {15 \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \left (A +\frac {13 B}{15}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8}+\left (\frac {19 A}{20}+\frac {5 B}{4}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {13 A}{10}+\frac {19 B}{15}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {3 A}{8}+\frac {13 B}{40}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {3 A}{10}+\frac {19 B}{75}\right ) \sin \left (5 d x +5 c \right )+\sin \left (d x +c \right ) \left (A +\frac {4 B}{3}\right )\right ) a^{3}}{d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(217\) |
derivativedivides | \(\frac {a^{3} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-3 a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{3} A \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(271\) |
default | \(\frac {a^{3} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-3 a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{3} A \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(271\) |
risch | \(-\frac {i a^{3} \left (225 A \,{\mathrm e}^{9 i \left (d x +c \right )}+195 B \,{\mathrm e}^{9 i \left (d x +c \right )}-120 A \,{\mathrm e}^{8 i \left (d x +c \right )}+570 A \,{\mathrm e}^{7 i \left (d x +c \right )}+750 B \,{\mathrm e}^{7 i \left (d x +c \right )}-1200 A \,{\mathrm e}^{6 i \left (d x +c \right )}-720 B \,{\mathrm e}^{6 i \left (d x +c \right )}-2400 A \,{\mathrm e}^{4 i \left (d x +c \right )}-2320 B \,{\mathrm e}^{4 i \left (d x +c \right )}-570 A \,{\mathrm e}^{3 i \left (d x +c \right )}-750 B \,{\mathrm e}^{3 i \left (d x +c \right )}-1680 A \,{\mathrm e}^{2 i \left (d x +c \right )}-1520 B \,{\mathrm e}^{2 i \left (d x +c \right )}-225 \,{\mathrm e}^{i \left (d x +c \right )} A -195 B \,{\mathrm e}^{i \left (d x +c \right )}-360 A -304 B \right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{8 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{8 d}\) | \(299\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.01 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (15 \, A + 13 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (15 \, A + 13 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (45 \, A + 38 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (15 \, A + 13 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (15 \, A + 19 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 30 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) + 24 \, B a^{3}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
[In]
[Out]
\[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=a^{3} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{4}{\left (c + d x \right )}\, dx + \int A \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{6}{\left (c + d x \right )}\, dx\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (151) = 302\).
Time = 0.22 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.07 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{3} + 240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} - 15 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 45 \, B a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, A a^{3} \tan \left (d x + c\right )}{240 \, d} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.51 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (15 \, A a^{3} + 13 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (15 \, A a^{3} + 13 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (225 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 195 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1050 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 910 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1920 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1664 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1830 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1330 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 735 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 765 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]
[In]
[Out]
Time = 16.11 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.37 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (15\,A+13\,B\right )}{4\,d}-\frac {\left (\frac {15\,A\,a^3}{4}+\frac {13\,B\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {35\,A\,a^3}{2}-\frac {91\,B\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (32\,A\,a^3+\frac {416\,B\,a^3}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {61\,A\,a^3}{2}-\frac {133\,B\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {49\,A\,a^3}{4}+\frac {51\,B\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
[In]
[Out]