Integrand size = 19, antiderivative size = 24 \[ \int \cos (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {C \text {arctanh}(\sin (c+d x))}{d}+\frac {A \sin (c+d x)}{d} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {4130, 3855} \[ \int \cos (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {A \sin (c+d x)}{d}+\frac {C \text {arctanh}(\sin (c+d x))}{d} \]
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Rule 3855
Rule 4130
Rubi steps \begin{align*} \text {integral}& = \frac {A \sin (c+d x)}{d}+C \int \sec (c+d x) \, dx \\ & = \frac {C \text {arctanh}(\sin (c+d x))}{d}+\frac {A \sin (c+d x)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \cos (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {C \text {arctanh}(\sin (c+d x))}{d}+\frac {A \cos (d x) \sin (c)}{d}+\frac {A \cos (c) \sin (d x)}{d} \]
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Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\frac {A \sin \left (d x +c \right )+C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(30\) |
default | \(\frac {A \sin \left (d x +c \right )+C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(30\) |
parallelrisch | \(\frac {A \sin \left (d x +c \right )-C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(43\) |
risch | \(-\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(71\) |
norman | \(\frac {-\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(101\) |
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Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \cos (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {C \log \left (\sin \left (d x + c\right ) + 1\right ) - C \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A \sin \left (d x + c\right )}{2 \, d} \]
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\[ \int \cos (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \cos (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {C {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A \sin \left (d x + c\right )}{2 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \cos (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {C \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - C \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 2 \, A \sin \left (d x + c\right )}{2 \, d} \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \cos (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {A\,\sin \left (c+d\,x\right )+C\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{d} \]
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