Integrand size = 13, antiderivative size = 11 \[ \int \frac {\sec (x) \tan (x)}{a+b \sec (x)} \, dx=\frac {\log (a+b \sec (x))}{b} \]
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Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4424, 36, 29, 31} \[ \int \frac {\sec (x) \tan (x)}{a+b \sec (x)} \, dx=\frac {\log (a \cos (x)+b)}{b}-\frac {\log (\cos (x))}{b} \]
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Rule 29
Rule 31
Rule 36
Rule 4424
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x (b+a x)} \, dx,x,\cos (x)\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,\cos (x)\right )}{b}+\frac {a \text {Subst}\left (\int \frac {1}{b+a x} \, dx,x,\cos (x)\right )}{b} \\ & = -\frac {\log (\cos (x))}{b}+\frac {\log (b+a \cos (x))}{b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.82 \[ \int \frac {\sec (x) \tan (x)}{a+b \sec (x)} \, dx=-\frac {\log (\cos (x))}{b}+\frac {\log (b+a \cos (x))}{b} \]
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Time = 0.35 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\ln \left (a +b \sec \left (x \right )\right )}{b}\) | \(12\) |
default | \(\frac {\ln \left (a +b \sec \left (x \right )\right )}{b}\) | \(12\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 b \,{\mathrm e}^{i x}}{a}+1\right )}{b}-\frac {\ln \left ({\mathrm e}^{2 i x}+1\right )}{b}\) | \(38\) |
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none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \frac {\sec (x) \tan (x)}{a+b \sec (x)} \, dx=\frac {\log \left (a \cos \left (x\right ) + b\right ) - \log \left (-\cos \left (x\right )\right )}{b} \]
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Time = 0.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27 \[ \int \frac {\sec (x) \tan (x)}{a+b \sec (x)} \, dx=\begin {cases} \frac {\log {\left (\frac {a}{b} + \sec {\left (x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {\sec {\left (x \right )}}{a} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (x) \tan (x)}{a+b \sec (x)} \, dx=\frac {\log \left (b \sec \left (x\right ) + a\right )}{b} \]
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none
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 2.00 \[ \int \frac {\sec (x) \tan (x)}{a+b \sec (x)} \, dx=\frac {\log \left ({\left | a \cos \left (x\right ) + b \right |}\right )}{b} - \frac {\log \left ({\left | \cos \left (x\right ) \right |}\right )}{b} \]
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Time = 26.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 4.36 \[ \int \frac {\sec (x) \tan (x)}{a+b \sec (x)} \, dx=\frac {\mathrm {atan}\left (\frac {b\,{\sin \left (\frac {x}{2}\right )}^2}{a\,{\cos \left (\frac {x}{2}\right )}^2\,1{}\mathrm {i}+b\,{\cos \left (\frac {x}{2}\right )}^2\,1{}\mathrm {i}-a\,{\sin \left (\frac {x}{2}\right )}^2\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{b} \]
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