Integrand size = 29, antiderivative size = 111 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {1}{2} a^3 (7 A+5 B) x+\frac {a^3 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac {a B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d} \]
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Time = 0.35 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3055, 3047, 3102, 2814, 3855} \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {a^3 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac {(3 A+5 B) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{6 d}+\frac {1}{2} a^3 x (7 A+5 B)+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
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Rule 2814
Rule 3047
Rule 3055
Rule 3102
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {a B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} \int (a+a \cos (c+d x))^2 (3 a A+a (3 A+5 B) \cos (c+d x)) \sec (c+d x) \, dx \\ & = \frac {a B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {1}{6} \int (a+a \cos (c+d x)) \left (6 a^2 A+15 a^2 (A+B) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {a B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {1}{6} \int \left (6 a^3 A+\left (6 a^3 A+15 a^3 (A+B)\right ) \cos (c+d x)+15 a^3 (A+B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac {a B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {1}{6} \int \left (6 a^3 A+3 a^3 (7 A+5 B) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {1}{2} a^3 (7 A+5 B) x+\frac {5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac {a B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\left (a^3 A\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} a^3 (7 A+5 B) x+\frac {a^3 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac {a B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d} \\ \end{align*}
Time = 1.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {a^3 \left (42 A d x+30 B d x-12 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 (4 A+5 B) \sin (c+d x)+3 (A+3 B) \sin (2 (c+d x))+B \sin (3 (c+d x))\right )}{12 d} \]
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Time = 2.55 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(-\frac {a^{3} \left (A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {\left (-A -3 B \right ) \sin \left (2 d x +2 c \right )}{4}-\frac {B \sin \left (3 d x +3 c \right )}{12}+3 \left (-A -\frac {5 B}{4}\right ) \sin \left (d x +c \right )-\frac {7 \left (A +\frac {5 B}{7}\right ) x d}{2}\right )}{d}\) | \(93\) |
parts | \(\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (A \,a^{3}+3 B \,a^{3}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (3 A \,a^{3}+B \,a^{3}\right ) \left (d x +c \right )}{d}+\frac {\left (3 A \,a^{3}+3 B \,a^{3}\right ) \sin \left (d x +c \right )}{d}+\frac {B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(131\) |
derivativedivides | \(\frac {A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 A \,a^{3} \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \,a^{3} \left (d x +c \right )+3 B \,a^{3} \sin \left (d x +c \right )+A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \left (d x +c \right )}{d}\) | \(147\) |
default | \(\frac {A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 A \,a^{3} \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \,a^{3} \left (d x +c \right )+3 B \,a^{3} \sin \left (d x +c \right )+A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \left (d x +c \right )}{d}\) | \(147\) |
risch | \(\frac {7 a^{3} A x}{2}+\frac {5 a^{3} B x}{2}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A \,a^{3}}{2 d}-\frac {15 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{3}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{3}}{2 d}+\frac {15 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{3}}{8 d}+\frac {A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{3}}{4 d}\) | \(189\) |
norman | \(\frac {\left (\frac {7}{2} A \,a^{3}+\frac {5}{2} B \,a^{3}\right ) x +\left (\frac {7}{2} A \,a^{3}+\frac {5}{2} B \,a^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (14 A \,a^{3}+10 B \,a^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (14 A \,a^{3}+10 B \,a^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 A \,a^{3}+15 B \,a^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a^{3} \left (7 A +11 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {5 a^{3} \left (A +B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a^{3} \left (51 A +55 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{3} \left (57 A +73 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(276\) |
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Time = 0.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.92 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {3 \, {\left (7 \, A + 5 \, B\right )} a^{3} d x + 3 \, A a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, A a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, B a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) + 2 \, {\left (9 \, A + 11 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \]
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\[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx=a^{3} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 3 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
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Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.27 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 36 \, {\left (d x + c\right )} A a^{3} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 12 \, {\left (d x + c\right )} B a^{3} + 12 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 36 \, A a^{3} \sin \left (d x + c\right ) + 36 \, B a^{3} \sin \left (d x + c\right )}{12 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.62 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {6 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (7 \, A a^{3} + 5 \, B a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
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Time = 0.45 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.60 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {3\,A\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {15\,B\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {7\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {5\,B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {3\,B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
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