Integrand size = 19, antiderivative size = 24 \[ \int \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B \text {arctanh}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3855, 3852, 8} \[ \int \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B \text {arctanh}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \]
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Rule 8
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = B \int \sec (c+d x) \, dx+C \int \sec ^2(c+d x) \, dx \\ & = \frac {B \text {arctanh}(\sin (c+d x))}{d}-\frac {C \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = \frac {B \text {arctanh}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B \text {arctanh}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \]
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Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \tan \left (d x +c \right )}{d}\) | \(30\) |
default | \(\frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \tan \left (d x +c \right )}{d}\) | \(32\) |
parts | \(\frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \tan \left (d x +c \right )}{d}\) | \(32\) |
risch | \(-\frac {B \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {B \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {2 i C}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(59\) |
parallelrisch | \(\frac {-B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+C \sin \left (d x +c \right )}{d \cos \left (d x +c \right )}\) | \(63\) |
norman | \(-\frac {2 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}+\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(67\) |
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50 \[ \int \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - B \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
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\[ \int \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right )}{d} + \frac {C \tan \left (d x + c\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B {\left (\log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )\right )}}{4 \, d} + \frac {C \tan \left (d x + c\right )}{d} \]
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Time = 15.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,B\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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