Integrand size = 19, antiderivative size = 24 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {A \text {arctanh}(\sin (c+d x))}{d}+\frac {C \sin (c+d x)}{d} \]
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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 = {3093, 3855} \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {A \text {arctanh}(\sin (c+d x))}{d}+\frac {C \sin (c+d x)}{d} \]
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Rule 3093
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {C \sin (c+d x)}{d}+A \int \sec (c+d x) \, dx \\ & = \frac {A \text {arctanh}(\sin (c+d x))}{d}+\frac {C \sin (c+d x)}{d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {A \text {arctanh}(\sin (c+d x))}{d}+\frac {C \cos (d x) \sin (c)}{d}+\frac {C \cos (c) \sin (d x)}{d} \]
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Time = 1.87 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\sin \left (d x +c \right ) C}{d}\) | \(30\) |
default | \(\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\sin \left (d x +c \right ) C}{d}\) | \(30\) |
parts | \(\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \sin \left (d x +c \right )}{d}\) | \(32\) |
parallelrisch | \(\frac {-A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\sin \left (d x +c \right ) C}{d}\) | \(43\) |
risch | \(-\frac {i C \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i C \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(71\) |
norman | \(\frac {\frac {2 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(86\) |
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Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {A \log \left (\sin \left (d x + c\right ) + 1\right ) - A \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d} \]
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\[ \int \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {A \log \left (\sin \left (d x + c\right ) + 1\right ) - A \log \left (\sin \left (d x + c\right ) - 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d} \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {A \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - A \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d} \]
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Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {C\,\sin \left (c+d\,x\right )+A\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{d} \]
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