Integrand size = 17, antiderivative size = 24 \[ \int \sec (c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3567, 3855} \[ \int \sec (c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \]
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Rule 3567
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b \sec (c+d x)}{d}+a \int \sec (c+d x) \, dx \\ & = \frac {a \text {arctanh}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \sec (c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \]
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Time = 0.57 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {\frac {b}{\cos \left (d x +c \right )}+a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(32\) |
default | \(\frac {\frac {b}{\cos \left (d x +c \right )}+a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(32\) |
risch | \(\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(67\) |
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int \sec (c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, b}{2 \, d \cos \left (d x + c\right )} \]
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Time = 2.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \sec (c+d x) (a+b \tan (c+d x)) \, dx=\begin {cases} \frac {a \log {\left (\tan {\left (c + d x \right )} + \sec {\left (c + d x \right )} \right )} + b \sec {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \sec {\left (c \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \sec (c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + \frac {b}{\cos \left (d x + c\right )}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int \sec (c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
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Time = 4.53 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \sec (c+d x) (a+b \tan (c+d x)) \, dx=\frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,b}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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