Integrand size = 31, antiderivative size = 150 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {1}{2} a^4 (13 A+12 B) x+\frac {a^4 (4 A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d} \]
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Time = 0.50 (sec) , antiderivative size = 150, 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 = {3054, 3055, 3047, 3102, 2814, 3855} \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {a^4 (4 A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{6 d}+\frac {1}{2} a^4 x (13 A+12 B)-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d} \]
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Rule 2814
Rule 3047
Rule 3054
Rule 3055
Rule 3102
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\int (a+a \cos (c+d x))^3 (a (4 A+B)-a (3 A-B) \cos (c+d x)) \sec (c+d x) \, dx \\ & = -\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{3} \int (a+a \cos (c+d x))^2 \left (3 a^2 (4 A+B)-a^2 (3 A-8 B) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{6} \int (a+a \cos (c+d x)) \left (6 a^3 (4 A+B)+15 a^3 (A+2 B) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{6} \int \left (6 a^4 (4 A+B)+\left (6 a^4 (4 A+B)+15 a^4 (A+2 B)\right ) \cos (c+d x)+15 a^4 (A+2 B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{6} \int \left (6 a^4 (4 A+B)+3 a^4 (13 A+12 B) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {1}{2} a^4 (13 A+12 B) x+\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\left (a^4 (4 A+B)\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} a^4 (13 A+12 B) x+\frac {a^4 (4 A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(312\) vs. \(2(150)=300\).
Time = 4.16 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.08 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {1}{192} a^4 (1+\cos (c+d x))^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (78 A x+72 B x-\frac {12 (4 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {12 (4 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {3 (16 A+27 B) \cos (d x) \sin (c)}{d}+\frac {3 (A+4 B) \cos (2 d x) \sin (2 c)}{d}+\frac {B \cos (3 d x) \sin (3 c)}{d}+\frac {3 (16 A+27 B) \cos (c) \sin (d x)}{d}+\frac {3 (A+4 B) \cos (2 c) \sin (2 d x)}{d}+\frac {B \cos (3 c) \sin (3 d x)}{d}+\frac {12 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {12 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right ) \]
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Time = 3.51 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93
method | result | size |
parallelrisch | \(\frac {\left (-32 \cos \left (d x +c \right ) \left (A +\frac {B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+32 \cos \left (d x +c \right ) \left (A +\frac {B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (16 A +\frac {82 B}{3}\right ) \sin \left (2 d x +2 c \right )+\left (A +4 B \right ) \sin \left (3 d x +3 c \right )+\frac {\sin \left (4 d x +4 c \right ) B}{3}+52 \left (A +\frac {12 B}{13}\right ) x d \cos \left (d x +c \right )+9 \left (A +\frac {4 B}{9}\right ) \sin \left (d x +c \right )\right ) a^{4}}{8 d \cos \left (d x +c \right )}\) | \(140\) |
parts | \(\frac {a^{4} A \tan \left (d x +c \right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(154\) |
derivativedivides | \(\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} A \left (d x +c \right )+6 B \,a^{4} \sin \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \left (d x +c \right )+a^{4} A \tan \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(179\) |
default | \(\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} A \left (d x +c \right )+6 B \,a^{4} \sin \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \left (d x +c \right )+a^{4} A \tan \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(179\) |
risch | \(\frac {13 a^{4} x A}{2}+6 a^{4} B x +\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B \,a^{4}}{2 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} A}{8 d}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{8 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} A}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{4} A}{d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} A}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {2 i a^{4} A}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {4 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}+\frac {4 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{4}}{12 d}\) | \(296\) |
norman | \(\frac {\left (-\frac {13}{2} a^{4} A -6 B \,a^{4}\right ) x +\left (-\frac {65}{2} a^{4} A -30 B \,a^{4}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {13}{2} a^{4} A +6 B \,a^{4}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {65}{2} a^{4} A +30 B \,a^{4}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-26 a^{4} A -24 B \,a^{4}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (26 a^{4} A +24 B \,a^{4}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5 a^{4} \left (A +2 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{4} \left (3 A -50 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{4} \left (11 A +18 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {5 a^{4} \left (21 A +26 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (39 A +106 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a^{4} \left (51 A +26 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a^{4} \left (4 A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{4} \left (4 A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(380\) |
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Time = 0.34 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {3 \, {\left (13 \, A + 12 \, B\right )} a^{4} d x \cos \left (d x + c\right ) + 3 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, B a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, A + 5 \, B\right )} a^{4} \cos \left (d x + c\right ) + 6 \, A a^{4}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \]
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\[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=a^{4} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 A \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int A \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 B \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.23 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.25 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 72 \, {\left (d x + c\right )} A a^{4} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 48 \, {\left (d x + c\right )} B a^{4} + 24 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{4} \sin \left (d x + c\right ) + 72 \, B a^{4} \sin \left (d x + c\right ) + 12 \, A a^{4} \tan \left (d x + c\right )}{12 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.51 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=-\frac {\frac {12 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (13 \, A a^{4} + 12 \, B a^{4}\right )} {\left (d x + c\right )} - 6 \, {\left (4 \, A a^{4} + B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6 \, {\left (4 \, A a^{4} + B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (21 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 76 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 54 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
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Time = 0.49 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.61 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {4\,A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {20\,B\,a^4\,\sin \left (c+d\,x\right )}{3\,d}+\frac {13\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {12\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {A\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {2\,B\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d} \]
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