Integrand size = 13, antiderivative size = 18 \[ \int (a+a \cos (x)) (A+B \sec (x)) \, dx=a (A+B) x+a B \text {arctanh}(\sin (x))+a A \sin (x) \]
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Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2907, 3047, 3102, 2814, 3855} \[ \int (a+a \cos (x)) (A+B \sec (x)) \, dx=a x (A+B)+a A \sin (x)+a B \text {arctanh}(\sin (x)) \]
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Rule 2814
Rule 2907
Rule 3047
Rule 3102
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int (a+a \cos (x)) (B+A \cos (x)) \sec (x) \, dx \\ & = \int \left (a B+(a A+a B) \cos (x)+a A \cos ^2(x)\right ) \sec (x) \, dx \\ & = a A \sin (x)+\int (a B+a (A+B) \cos (x)) \sec (x) \, dx \\ & = a (A+B) x+a A \sin (x)+(a B) \int \sec (x) \, dx \\ & = a (A+B) x+a B \text {arctanh}(\sin (x))+a A \sin (x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (a+a \cos (x)) (A+B \sec (x)) \, dx=a A x+a B x+a B \text {arctanh}(\sin (x))+a A \sin (x) \]
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Time = 0.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33
method | result | size |
default | \(a A \sin \left (x \right )+B a x +a A x +B a \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )\) | \(24\) |
parts | \(a A \sin \left (x \right )+B a x +a A x +B a \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )\) | \(24\) |
parallelrisch | \(a \left (-B \ln \left (-\cot \left (x \right )+\csc \left (x \right )-1\right )+B \ln \left (\csc \left (x \right )-\cot \left (x \right )+1\right )+A \sin \left (x \right )+x \left (A +B \right )\right )\) | \(36\) |
risch | \(a A x +B a x -\frac {i A a \,{\mathrm e}^{i x}}{2}+\frac {i A a \,{\mathrm e}^{-i x}}{2}+B a \ln \left (i+{\mathrm e}^{i x}\right )-B a \ln \left ({\mathrm e}^{i x}-i\right )\) | \(55\) |
norman | \(\frac {\left (A a +B a \right ) x +\left (A a +B a \right ) x \tan \left (\frac {x}{2}\right )^{2}+2 A a \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}+B a \ln \left (\tan \left (\frac {x}{2}\right )+1\right )-B a \ln \left (\tan \left (\frac {x}{2}\right )-1\right )\) | \(67\) |
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none
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int (a+a \cos (x)) (A+B \sec (x)) \, dx={\left (A + B\right )} a x + \frac {1}{2} \, B a \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{2} \, B a \log \left (-\sin \left (x\right ) + 1\right ) + A a \sin \left (x\right ) \]
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Time = 0.84 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int (a+a \cos (x)) (A+B \sec (x)) \, dx=A a x + A a \sin {\left (x \right )} + B a x + B a \log {\left (\tan {\left (x \right )} + \sec {\left (x \right )} \right )} \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int (a+a \cos (x)) (A+B \sec (x)) \, dx=A a x + B a x + B a \log \left (\sec \left (x\right ) + \tan \left (x\right )\right ) + A a \sin \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.83 \[ \int (a+a \cos (x)) (A+B \sec (x)) \, dx=B a \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) - B a \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) + {\left (A a + B a\right )} x + \frac {2 \, A a \tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1} \]
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Time = 26.88 (sec) , antiderivative size = 54, normalized size of antiderivative = 3.00 \[ \int (a+a \cos (x)) (A+B \sec (x)) \, dx=2\,A\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+2\,B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+2\,B\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+A\,a\,\sin \left (x\right ) \]
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