Integrand size = 29, antiderivative size = 32 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=a B x+\frac {a (A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d} \]
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Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3047, 3100, 2814, 3855} \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {a (A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d}+a B x \]
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Rule 2814
Rule 3047
Rule 3100
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a A+(a A+a B) \cos (c+d x)+a B \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {a A \tan (c+d x)}{d}+\int (a (A+B)+a B \cos (c+d x)) \sec (c+d x) \, dx \\ & = a B x+\frac {a A \tan (c+d x)}{d}+(a (A+B)) \int \sec (c+d x) \, dx \\ & = a B x+\frac {a (A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=a B x+\frac {a A \text {arctanh}(\sin (c+d x))}{d}+\frac {a B \text {arctanh}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d} \]
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Time = 2.33 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56
method | result | size |
parts | \(\frac {a A \tan \left (d x +c \right )}{d}+\frac {\left (a A +B a \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {B a \left (d x +c \right )}{d}\) | \(50\) |
derivativedivides | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B a \left (d x +c \right )+a A \tan \left (d x +c \right )+B a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(57\) |
default | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B a \left (d x +c \right )+a A \tan \left (d x +c \right )+B a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(57\) |
parallelrisch | \(-\frac {\left (\left (A +B \right ) \cos \left (d x +c \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (A +B \right ) \cos \left (d x +c \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-d x B \cos \left (d x +c \right )-A \sin \left (d x +c \right )\right ) a}{d \cos \left (d x +c \right )}\) | \(81\) |
risch | \(a B x +\frac {2 i a A}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}-\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}\) | \(105\) |
norman | \(\frac {a B x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a B x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a B x -\frac {2 a A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 a A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-a B x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a \left (A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(178\) |
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).
Time = 0.39 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.47 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {2 \, B a d x \cos \left (d x + c\right ) + {\left (A + B\right )} a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A + B\right )} a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A a \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
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\[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=a \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (32) = 64\).
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.28 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {2 \, {\left (d x + c\right )} B a + A a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + B a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a \tan \left (d x + c\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (32) = 64\).
Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.62 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {{\left (d x + c\right )} B a + {\left (A a + B a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (A a + B a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
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Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.12 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {A\,a\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {2\,A\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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