Integrand size = 33, antiderivative size = 156 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a^3 (5 A+6 C) x+\frac {3 a^3 C \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}-\frac {(5 A-6 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d} \]
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Time = 0.48 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4172, 4102, 4103, 4081, 3855} \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {(5 A-6 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {1}{2} a^3 x (5 A+6 C)+\frac {3 a^3 C \text {arctanh}(\sin (c+d x))}{d}+\frac {A \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 a d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d} \]
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Rule 3855
Rule 4081
Rule 4102
Rule 4103
Rule 4172
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 (3 a A-a (A-3 C) \sec (c+d x)) \, dx}{3 a} \\ & = \frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x))^2 \left (2 a^2 (5 A+3 C)-a^2 (5 A-6 C) \sec (c+d x)\right ) \, dx}{6 a} \\ & = \frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}-\frac {(5 A-6 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^3 A+18 a^3 C \sec (c+d x)\right ) \, dx}{6 a} \\ & = \frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}-\frac {(5 A-6 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}-\frac {\int \left (-3 a^4 (5 A+6 C)-18 a^4 C \sec (c+d x)\right ) \, dx}{6 a} \\ & = \frac {1}{2} a^3 (5 A+6 C) x+\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}-\frac {(5 A-6 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\left (3 a^3 C\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} a^3 (5 A+6 C) x+\frac {3 a^3 C \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}-\frac {(5 A-6 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1014\) vs. \(2(156)=312\).
Time = 8.29 (sec) , antiderivative size = 1014, normalized size of antiderivative = 6.50 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^3 \left (\frac {(5 A+6 C) x \cos ^2(c+d x) (1+\cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right )}{8 (A+2 C+A \cos (2 c+2 d x))}-\frac {3 C \cos ^2(c+d x) (1+\cos (c+d x))^3 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right )}{4 d (A+2 C+A \cos (2 c+2 d x))}+\frac {3 C \cos ^2(c+d x) (1+\cos (c+d x))^3 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right )}{4 d (A+2 C+A \cos (2 c+2 d x))}+\frac {(15 A+4 C) \cos (d x) \cos ^2(c+d x) (1+\cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right ) \sin (c)}{16 d (A+2 C+A \cos (2 c+2 d x))}+\frac {3 A \cos (2 d x) \cos ^2(c+d x) (1+\cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right ) \sin (2 c)}{16 d (A+2 C+A \cos (2 c+2 d x))}+\frac {A \cos (3 d x) \cos ^2(c+d x) (1+\cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right ) \sin (3 c)}{48 d (A+2 C+A \cos (2 c+2 d x))}+\frac {(15 A+4 C) \cos (c) \cos ^2(c+d x) (1+\cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right ) \sin (d x)}{16 d (A+2 C+A \cos (2 c+2 d x))}+\frac {3 A \cos (2 c) \cos ^2(c+d x) (1+\cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right ) \sin (2 d x)}{16 d (A+2 C+A \cos (2 c+2 d x))}+\frac {A \cos (3 c) \cos ^2(c+d x) (1+\cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right ) \sin (3 d x)}{48 d (A+2 C+A \cos (2 c+2 d x))}+\frac {C \cos ^2(c+d x) (1+\cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right ) \sin \left (\frac {d x}{2}\right )}{4 d (A+2 C+A \cos (2 c+2 d x)) \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {C \cos ^2(c+d x) (1+\cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+C \sec ^2(c+d x)\right ) \sin \left (\frac {d x}{2}\right )}{4 d (A+2 C+A \cos (2 c+2 d x)) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}\right ) \]
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Time = 0.41 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(\frac {3 a^{3} \left (-8 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+8 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+\left (\frac {46 A}{9}+\frac {4 C}{3}\right ) \sin \left (2 d x +2 c \right )+A \sin \left (3 d x +3 c \right )+\frac {A \sin \left (4 d x +4 c \right )}{9}+\frac {20 \left (A +\frac {6 C}{5}\right ) x d \cos \left (d x +c \right )}{3}+\sin \left (d x +c \right ) \left (A +\frac {8 C}{3}\right )\right )}{8 d \cos \left (d x +c \right )}\) | \(127\) |
derivativedivides | \(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{3} C \sin \left (d x +c \right )+3 a^{3} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} C \left (d x +c \right )+3 a^{3} A \sin \left (d x +c \right )+3 a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} A \left (d x +c \right )+a^{3} C \tan \left (d x +c \right )}{d}\) | \(131\) |
default | \(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{3} C \sin \left (d x +c \right )+3 a^{3} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} C \left (d x +c \right )+3 a^{3} A \sin \left (d x +c \right )+3 a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} A \left (d x +c \right )+a^{3} C \tan \left (d x +c \right )}{d}\) | \(131\) |
risch | \(\frac {5 a^{3} A x}{2}+3 a^{3} x C -\frac {3 i a^{3} A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {15 i a^{3} A \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{3} C}{2 d}+\frac {15 i a^{3} A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{3} C}{2 d}+\frac {3 i a^{3} A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i a^{3} C}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {a^{3} A \sin \left (3 d x +3 c \right )}{12 d}\) | \(215\) |
norman | \(\frac {\left (\frac {5}{2} a^{3} A +3 a^{3} C \right ) x +\left (-\frac {15}{2} a^{3} A -9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {15}{2} a^{3} A -9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-\frac {5}{2} a^{3} A -3 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {5}{2} a^{3} A -3 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {5}{2} a^{3} A +3 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {15}{2} a^{3} A +9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {15}{2} a^{3} A +9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {a^{3} \left (11 A +4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {5 a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}+\frac {8 a^{3} \left (2 A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {4 a^{3} \left (5 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {4 a^{3} \left (23 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {a^{3} \left (37 A -12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}+\frac {a^{3} \left (53 A -24 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {3 a^{3} C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {3 a^{3} C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(444\) |
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Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.88 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (5 \, A + 6 \, C\right )} a^{3} d x \cos \left (d x + c\right ) + 9 \, C a^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, C a^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, A a^{3} \cos \left (d x + c\right )^{3} + 9 \, A a^{3} \cos \left (d x + c\right )^{2} + 2 \, {\left (11 \, A + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 6 \, C a^{3}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.88 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 12 \, {\left (d x + c\right )} A a^{3} - 36 \, {\left (d x + c\right )} C a^{3} - 18 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, A a^{3} \sin \left (d x + c\right ) - 12 \, C a^{3} \sin \left (d x + c\right ) - 12 \, C a^{3} \tan \left (d x + c\right )}{12 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.35 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {18 \, C a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 18 \, C a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + 3 \, {\left (5 \, A a^{3} + 6 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
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Time = 15.67 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.21 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {11\,A\,a^3\,\sin \left (c+d\,x\right )}{3\,d}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,C\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {3\,A\,a^3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \]
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