Integrand size = 25, antiderivative size = 147 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^3 A x+\frac {a^3 (28 A+15 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^3 (4 A+3 C) \tan (c+d x)}{8 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {C \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{8 d} \]
[Out]
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4140, 4002, 3999, 3852, 8, 3855} \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (28 A+15 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^3 (4 A+3 C) \tan (c+d x)}{8 d}+\frac {(4 A+5 C) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{8 d}+a^3 A x+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{4 a d}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d} \]
[In]
[Out]
Rule 8
Rule 3852
Rule 3855
Rule 3999
Rule 4002
Rule 4140
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\int (a+a \sec (c+d x))^3 (4 a A+3 a C \sec (c+d x)) \, dx}{4 a} \\ & = \frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {C \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{4 a d}+\frac {\int (a+a \sec (c+d x))^2 \left (12 a^2 A+3 a^2 (4 A+5 C) \sec (c+d x)\right ) \, dx}{12 a} \\ & = \frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {C \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{8 d}+\frac {\int (a+a \sec (c+d x)) \left (24 a^3 A+15 a^3 (4 A+3 C) \sec (c+d x)\right ) \, dx}{24 a} \\ & = a^3 A x+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {C \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{8 d}+\frac {1}{8} \left (5 a^3 (4 A+3 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (a^3 (28 A+15 C)\right ) \int \sec (c+d x) \, dx \\ & = a^3 A x+\frac {a^3 (28 A+15 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {C \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{8 d}-\frac {\left (5 a^3 (4 A+3 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{8 d} \\ & = a^3 A x+\frac {a^3 (28 A+15 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^3 (4 A+3 C) \tan (c+d x)}{8 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {C \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{8 d} \\ \end{align*}
Time = 2.93 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.63 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 \left (8 A d x+(28 A+15 C) \text {arctanh}(\sin (c+d x))+\left (24 A+26 C+(4 A+15 C) \sec (c+d x)+6 C \sec ^2(c+d x)+2 C \sec ^3(c+d x)\right ) \tan (c+d x)+2 C \tan ^3(c+d x)\right )}{8 d} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.22
method | result | size |
parts | \(a^{3} A x +\frac {\left (a^{3} A +3 a^{3} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (3 a^{3} A +a^{3} C \right ) \tan \left (d x +c \right )}{d}+\frac {a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {3 A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{3}}{d}-\frac {3 a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(179\) |
derivativedivides | \(\frac {a^{3} A \left (d x +c \right )+a^{3} C \tan \left (d x +c \right )+3 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} A \tan \left (d x +c \right )-3 a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(205\) |
default | \(\frac {a^{3} A \left (d x +c \right )+a^{3} C \tan \left (d x +c \right )+3 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} A \tan \left (d x +c \right )-3 a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(205\) |
parallelrisch | \(\frac {6 a^{3} \left (-\frac {7 \left (A +\frac {15 C}{28}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{3}+\frac {7 \left (A +\frac {15 C}{28}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{3}+\frac {2 d x A \cos \left (2 d x +2 c \right )}{3}+\frac {d x A \cos \left (4 d x +4 c \right )}{6}+\left (A +\frac {5 C}{3}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (\frac {A}{3}+\frac {5 C}{4}\right ) \sin \left (3 d x +3 c \right )}{2}+\frac {\left (A +C \right ) \sin \left (4 d x +4 c \right )}{2}+\frac {\left (A +\frac {23 C}{4}\right ) \sin \left (d x +c \right )}{6}+\frac {d x A}{2}\right )}{d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(209\) |
norman | \(\frac {a^{3} A x +a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-4 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {5 a^{3} \left (4 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {7 a^{3} \left (4 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (68 A +55 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {a^{3} \left (76 A +73 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {a^{3} \left (28 A +15 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{3} \left (28 A +15 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(252\) |
risch | \(a^{3} A x -\frac {i a^{3} \left (4 A \,{\mathrm e}^{7 i \left (d x +c \right )}+15 C \,{\mathrm e}^{7 i \left (d x +c \right )}-24 A \,{\mathrm e}^{6 i \left (d x +c \right )}-8 C \,{\mathrm e}^{6 i \left (d x +c \right )}+4 A \,{\mathrm e}^{5 i \left (d x +c \right )}+23 C \,{\mathrm e}^{5 i \left (d x +c \right )}-72 A \,{\mathrm e}^{4 i \left (d x +c \right )}-72 C \,{\mathrm e}^{4 i \left (d x +c \right )}-4 A \,{\mathrm e}^{3 i \left (d x +c \right )}-23 C \,{\mathrm e}^{3 i \left (d x +c \right )}-72 A \,{\mathrm e}^{2 i \left (d x +c \right )}-88 C \,{\mathrm e}^{2 i \left (d x +c \right )}-4 A \,{\mathrm e}^{i \left (d x +c \right )}-15 C \,{\mathrm e}^{i \left (d x +c \right )}-24 A -24 C \right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\) | \(293\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {16 \, A a^{3} d x \cos \left (d x + c\right )^{4} + {\left (28 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (28 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, {\left (A + C\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (4 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 8 \, C a^{3} \cos \left (d x + c\right ) + 2 \, C a^{3}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \]
[In]
[Out]
\[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\int A\, dx + \int 3 A \sec {\left (c + d x \right )}\, dx + \int 3 A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sec ^{3}{\left (c + d x \right )}\, dx + \int C \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.70 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {16 \, {\left (d x + c\right )} A a^{3} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 48 \, A a^{3} \tan \left (d x + c\right ) + 16 \, C a^{3} \tan \left (d x + c\right )}{16 \, d} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.51 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {8 \, {\left (d x + c\right )} A a^{3} + {\left (28 \, A a^{3} + 15 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (28 \, A a^{3} + 15 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (20 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 68 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 55 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 76 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 73 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 28 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 49 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{8 \, d} \]
[In]
[Out]
Time = 15.89 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.57 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {15\,C\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {3\,A\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {3\,C\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {15\,C\,a^3\,\sin \left (c+d\,x\right )}{8\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^3}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{4\,d\,{\cos \left (c+d\,x\right )}^4} \]
[In]
[Out]