Integrand size = 29, antiderivative size = 151 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {1}{8} a^4 (48 A+35 B) x+\frac {a^4 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d} \]
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Time = 0.46 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3055, 3047, 3102, 2814, 3855} \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {a^4 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (48 A+35 B)+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
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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))^3 \sin (c+d x)}{4 d}+\frac {1}{4} \int (a+a \cos (c+d x))^3 (4 a A+a (4 A+7 B) \cos (c+d x)) \sec (c+d x) \, dx \\ & = \frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {1}{12} \int (a+a \cos (c+d x))^2 \left (12 a^2 A+a^2 (32 A+35 B) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {1}{24} \int (a+a \cos (c+d x)) \left (24 a^3 A+15 a^3 (8 A+7 B) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {1}{24} \int \left (24 a^4 A+\left (24 a^4 A+15 a^4 (8 A+7 B)\right ) \cos (c+d x)+15 a^4 (8 A+7 B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {1}{24} \int \left (24 a^4 A+3 a^4 (48 A+35 B) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {1}{8} a^4 (48 A+35 B) x+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^4 A\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{8} a^4 (48 A+35 B) x+\frac {a^4 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d} \\ \end{align*}
Time = 1.74 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.91 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {a^4 \left (576 A d x+420 B d x-96 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 (27 A+28 B) \sin (c+d x)+24 (4 A+7 B) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+32 B \sin (3 (c+d x))+3 B \sin (4 (c+d x))\right )}{96 d} \]
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Time = 3.46 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(-\frac {\left (A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-A -\frac {7 B}{4}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {A}{12}-\frac {B}{3}\right ) \sin \left (3 d x +3 c \right )-\frac {\sin \left (4 d x +4 c \right ) B}{32}+\left (-\frac {27 A}{4}-7 B \right ) \sin \left (d x +c \right )-6 x \left (A +\frac {35 B}{48}\right ) d \right ) a^{4}}{d}\) | \(108\) |
parts | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(181\) |
derivativedivides | \(\frac {\frac {a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \sin \left (d x +c \right )+6 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} A \left (d x +c \right )+4 B \,a^{4} \sin \left (d x +c \right )+a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \left (d x +c \right )}{d}\) | \(208\) |
default | \(\frac {\frac {a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \sin \left (d x +c \right )+6 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} A \left (d x +c \right )+4 B \,a^{4} \sin \left (d x +c \right )+a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \left (d x +c \right )}{d}\) | \(208\) |
risch | \(6 a^{4} x A +\frac {35 a^{4} B x}{8}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} a^{4} A}{8 d}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} A}{8 d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} A}{12 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{4}}{3 d}+\frac {\sin \left (2 d x +2 c \right ) a^{4} A}{d}+\frac {7 \sin \left (2 d x +2 c \right ) B \,a^{4}}{4 d}\) | \(224\) |
norman | \(\frac {\left (6 a^{4} A +\frac {35}{8} B \,a^{4}\right ) x +\left (6 a^{4} A +\frac {35}{8} B \,a^{4}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{4} A +\frac {175}{8} B \,a^{4}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{4} A +\frac {175}{8} B \,a^{4}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 a^{4} A +\frac {175}{4} B \,a^{4}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 a^{4} A +\frac {175}{4} B \,a^{4}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5 a^{4} \left (8 A +7 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {4 a^{4} \left (59 A +56 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (272 A +245 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{4} \left (368 A +395 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {3 a^{4} \left (31 B +24 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {a^{4} A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{4} A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(333\) |
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Time = 0.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.78 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {3 \, {\left (48 \, A + 35 \, B\right )} a^{4} d x + 12 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, B a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 3 \, {\left (16 \, A + 27 \, B\right )} a^{4} \cos \left (d x + c\right ) + 160 \, {\left (A + B\right )} a^{4}\right )} \sin \left (d x + c\right )}{24 \, d} \]
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\[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=a^{4} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 4 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
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Time = 0.22 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.31 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=-\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 384 \, {\left (d x + c\right )} A a^{4} + 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 96 \, {\left (d x + c\right )} B a^{4} - 96 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 576 \, A a^{4} \sin \left (d x + c\right ) - 384 \, B a^{4} \sin \left (d x + c\right )}{96 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.42 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {24 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (48 \, A a^{4} + 35 \, B a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (120 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 424 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 520 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 511 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 216 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 279 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
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Time = 0.74 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.25 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {144\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+24\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+105\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+12\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )+A\,a^4\,\sin \left (3\,c+3\,d\,x\right )+21\,B\,a^4\,\sin \left (2\,c+2\,d\,x\right )+4\,B\,a^4\,\sin \left (3\,c+3\,d\,x\right )+\frac {3\,B\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{8}+81\,A\,a^4\,\sin \left (c+d\,x\right )+84\,B\,a^4\,\sin \left (c+d\,x\right )}{12\,d} \]
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