Integrand size = 27, antiderivative size = 35 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx=(A b+a B) x+\frac {a A \text {arctanh}(\sin (c+d x))}{d}+\frac {b B \sin (c+d x)}{d} \]
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Time = 0.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3047, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {a A \text {arctanh}(\sin (c+d x))}{d}+x (a B+A b)+\frac {b B \sin (c+d x)}{d} \]
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Rule 2814
Rule 3047
Rule 3102
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {b B \sin (c+d x)}{d}+\int (a A+(A b+a B) \cos (c+d x)) \sec (c+d x) \, dx \\ & = (A b+a B) x+\frac {b B \sin (c+d x)}{d}+(a A) \int \sec (c+d x) \, dx \\ & = (A b+a B) x+\frac {a A \text {arctanh}(\sin (c+d x))}{d}+\frac {b B \sin (c+d x)}{d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx=A b x+a B x+\frac {a A \text {arctanh}(\sin (c+d x))}{d}+\frac {b B \cos (d x) \sin (c)}{d}+\frac {b B \cos (c) \sin (d x)}{d} \]
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Time = 1.62 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37
method | result | size |
derivativedivides | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B a \left (d x +c \right )+A b \left (d x +c \right )+B \sin \left (d x +c \right ) b}{d}\) | \(48\) |
default | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B a \left (d x +c \right )+A b \left (d x +c \right )+B \sin \left (d x +c \right ) b}{d}\) | \(48\) |
parts | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (A b +B a \right ) \left (d x +c \right )}{d}+\frac {b B \sin \left (d x +c \right )}{d}\) | \(50\) |
parallelrisch | \(\frac {-a A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+a A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+B \sin \left (d x +c \right ) b +\left (A b +B a \right ) x d}{d}\) | \(56\) |
risch | \(x A b +a B x -\frac {i B b \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i B b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(83\) |
norman | \(\frac {\left (A b +B a \right ) x +\left (A b +B a \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 A b +2 B a \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 B b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 B b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {a A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(141\) |
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Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {2 \, {\left (B a + A b\right )} d x + A a \log \left (\sin \left (d x + c\right ) + 1\right ) - A a \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, B b \sin \left (d x + c\right )}{2 \, d} \]
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\[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx=\int \left (A + B \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.34 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {{\left (d x + c\right )} B a + {\left (d x + c\right )} A b + A a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + B b \sin \left (d x + c\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (35) = 70\).
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.26 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (B a + A b\right )} {\left (d x + c\right )} + \frac {2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]
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Time = 0.52 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.86 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {B\,b\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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