Integrand size = 38, antiderivative size = 35 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=(b B+a C) x+\frac {b C \text {arctanh}(\sin (c+d x))}{d}+\frac {a B \sin (c+d x)}{d} \]
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Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {4157, 4081, 3855} \[ \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=x (a C+b B)+\frac {a B \sin (c+d x)}{d}+\frac {b C \text {arctanh}(\sin (c+d x))}{d} \]
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Rule 3855
Rule 4081
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \cos (c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx \\ & = \frac {a B \sin (c+d x)}{d}-\int (-b B-a C-b C \sec (c+d x)) \, dx \\ & = (b B+a C) x+\frac {a B \sin (c+d x)}{d}+(b C) \int \sec (c+d x) \, dx \\ & = (b B+a C) x+\frac {b C \text {arctanh}(\sin (c+d x))}{d}+\frac {a B \sin (c+d x)}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=b B x+a C x+\frac {b C \text {arctanh}(\sin (c+d x))}{d}+\frac {a B \cos (d x) \sin (c)}{d}+\frac {a B \cos (c) \sin (d x)}{d} \]
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Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37
method | result | size |
derivativedivides | \(\frac {a B \sin \left (d x +c \right )+C a \left (d x +c \right )+B b \left (d x +c \right )+C b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(48\) |
default | \(\frac {a B \sin \left (d x +c \right )+C a \left (d x +c \right )+B b \left (d x +c \right )+C b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(48\) |
parallelrisch | \(\frac {-C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b +C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b +a B \sin \left (d x +c \right )+\left (B b +C a \right ) x d}{d}\) | \(56\) |
risch | \(B b x +a x C -\frac {i a B \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a B \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(83\) |
norman | \(\frac {\left (B b +C a \right ) x +\left (-2 B b -2 C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (B b +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {2 a B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {2 a B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+\frac {2 a B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}+\frac {C b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {C b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(192\) |
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Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (C a + B b\right )} d x + C b \log \left (\sin \left (d x + c\right ) + 1\right ) - C b \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, B a \sin \left (d x + c\right )}{2 \, d} \]
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\[ \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (d x + c\right )} C a + 2 \, {\left (d x + c\right )} B b + C b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a \sin \left (d x + c\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (35) = 70\).
Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.26 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {C b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - C b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (C a + B b\right )} {\left (d x + c\right )} + \frac {2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]
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Time = 16.81 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.86 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B\,a\,\sin \left (c+d\,x\right )}{d}+\frac {2\,B\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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