Integrand size = 29, antiderivative size = 44 \[ \int \frac {(A+B \cos (c+d x)) \sec (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {A \text {arctanh}(\sin (c+d x))}{a d}-\frac {(A-B) \sin (c+d x)}{d (a+a \cos (c+d x))} \]
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Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3057, 12, 3855} \[ \int \frac {(A+B \cos (c+d x)) \sec (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {A \text {arctanh}(\sin (c+d x))}{a d}-\frac {(A-B) \sin (c+d x)}{d (a \cos (c+d x)+a)} \]
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Rule 12
Rule 3057
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {\int a A \sec (c+d x) \, dx}{a^2} \\ & = -\frac {(A-B) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {A \int \sec (c+d x) \, dx}{a} \\ & = \frac {A \text {arctanh}(\sin (c+d x))}{a d}-\frac {(A-B) \sin (c+d x)}{d (a+a \cos (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(44)=88\).
Time = 0.47 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.48 \[ \int \frac {(A+B \cos (c+d x)) \sec (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left (A \cos \left (\frac {1}{2} (c+d x)\right ) \left (-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+(-A+B) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )\right )}{a d (1+\cos (c+d x))} \]
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Time = 1.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.23
method | result | size |
parallelrisch | \(\frac {-A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (A -B \right )}{a d}\) | \(54\) |
derivativedivides | \(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}\) | \(61\) |
default | \(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}\) | \(61\) |
risch | \(-\frac {2 i A}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {2 i B}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}-\frac {A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a d}\) | \(91\) |
norman | \(\frac {-\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {\left (A -B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}-\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a d}\) | \(106\) |
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Time = 0.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.68 \[ \int \frac {(A+B \cos (c+d x)) \sec (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {{\left (A \cos \left (d x + c\right ) + A\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A \cos \left (d x + c\right ) + A\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (A - B\right )} \sin \left (d x + c\right )}{2 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
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\[ \int \frac {(A+B \cos (c+d x)) \sec (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\int \frac {A \sec {\left (c + d x \right )}}{\cos {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\cos {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (44) = 88\).
Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.25 \[ \int \frac {(A+B \cos (c+d x)) \sec (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {A {\left (\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + \frac {B \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
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Time = 0.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.61 \[ \int \frac {(A+B \cos (c+d x)) \sec (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\frac {A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a}}{d} \]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B \cos (c+d x)) \sec (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {2\,A\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B\right )}{a\,d} \]
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