Integrand size = 29, antiderivative size = 82 \[ \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {1}{2} a^2 (4 A+3 B) x+\frac {a^2 A \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 (2 A+3 B) \sin (c+d x)}{2 d}+\frac {B \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d} \]
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Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3055, 3047, 3102, 2814, 3855} \[ \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {a^2 A \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 (2 A+3 B) \sin (c+d x)}{2 d}+\frac {1}{2} a^2 x (4 A+3 B)+\frac {B \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{2 d} \]
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Rule 2814
Rule 3047
Rule 3055
Rule 3102
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {B \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {1}{2} \int (a+a \cos (c+d x)) (2 a A+a (2 A+3 B) \cos (c+d x)) \sec (c+d x) \, dx \\ & = \frac {B \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a^2 A+\left (2 a^2 A+a^2 (2 A+3 B)\right ) \cos (c+d x)+a^2 (2 A+3 B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {a^2 (2 A+3 B) \sin (c+d x)}{2 d}+\frac {B \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a^2 A+a^2 (4 A+3 B) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {1}{2} a^2 (4 A+3 B) x+\frac {a^2 (2 A+3 B) \sin (c+d x)}{2 d}+\frac {B \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\left (a^2 A\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} a^2 (4 A+3 B) x+\frac {a^2 A \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 (2 A+3 B) \sin (c+d x)}{2 d}+\frac {B \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.17 \[ \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {a^2 \left (8 A d x+6 B d x-4 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 (A+2 B) \sin (c+d x)+B \sin (2 (c+d x))\right )}{4 d} \]
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Time = 1.53 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(-\frac {\left (A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {\sin \left (2 d x +2 c \right ) B}{4}+\left (-A -2 B \right ) \sin \left (d x +c \right )-2 d x \left (A +\frac {3 B}{4}\right )\right ) a^{2}}{d}\) | \(74\) |
derivativedivides | \(\frac {A \,a^{2} \sin \left (d x +c \right )+B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A \,a^{2} \left (d x +c \right )+2 B \,a^{2} \sin \left (d x +c \right )+A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{2} \left (d x +c \right )}{d}\) | \(96\) |
default | \(\frac {A \,a^{2} \sin \left (d x +c \right )+B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A \,a^{2} \left (d x +c \right )+2 B \,a^{2} \sin \left (d x +c \right )+A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{2} \left (d x +c \right )}{d}\) | \(96\) |
parts | \(\frac {A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (A \,a^{2}+2 B \,a^{2}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (2 A \,a^{2}+B \,a^{2}\right ) \left (d x +c \right )}{d}+\frac {B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(97\) |
risch | \(2 a^{2} x A +\frac {3 a^{2} B x}{2}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,a^{2}}{2 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B \,a^{2}}{d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{2}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{2}}{d}+\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{2}}{4 d}\) | \(153\) |
norman | \(\frac {\left (2 A \,a^{2}+\frac {3}{2} B \,a^{2}\right ) x +\left (2 A \,a^{2}+\frac {3}{2} B \,a^{2}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 A \,a^{2}+\frac {9}{2} B \,a^{2}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 A \,a^{2}+\frac {9}{2} B \,a^{2}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a^{2} \left (2 A +3 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a^{2} \left (2 A +5 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 a^{2} \left (A +2 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(225\) |
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Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96 \[ \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {{\left (4 \, A + 3 \, B\right )} a^{2} d x + A a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - A a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (B a^{2} \cos \left (d x + c\right ) + 2 \, {\left (A + 2 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{2 \, d} \]
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\[ \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=a^{2} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 2 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 2 B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
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Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15 \[ \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {8 \, {\left (d x + c\right )} A a^{2} + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 4 \, {\left (d x + c\right )} B a^{2} + 4 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, A a^{2} \sin \left (d x + c\right ) + 8 \, B a^{2} \sin \left (d x + c\right )}{4 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.77 \[ \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {2 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (4 \, A a^{2} + 3 \, B a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
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Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.72 \[ \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {A\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {2\,B\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {4\,A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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