Integrand size = 13, antiderivative size = 11 \[ \int \frac {\sec ^2(x)}{a+b \tan (x)} \, dx=\frac {\log (a+b \tan (x))}{b} \]
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Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3587, 31} \[ \int \frac {\sec ^2(x)}{a+b \tan (x)} \, dx=\frac {\log (a+b \tan (x))}{b} \]
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Rule 31
Rule 3587
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (x)\right )}{b} \\ & = \frac {\log (a+b \tan (x))}{b} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^2(x)}{a+b \tan (x)} \, dx=\frac {\log (a+b \tan (x))}{b} \]
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Time = 1.53 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\ln \left (a +b \tan \left (x \right )\right )}{b}\) | \(12\) |
default | \(\frac {\ln \left (a +b \tan \left (x \right )\right )}{b}\) | \(12\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 i x}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{b}\) | \(44\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (11) = 22\).
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 3.64 \[ \int \frac {\sec ^2(x)}{a+b \tan (x)} \, dx=\frac {\log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) - \log \left (\cos \left (x\right )^{2}\right )}{2 \, b} \]
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\[ \int \frac {\sec ^2(x)}{a+b \tan (x)} \, dx=\int \frac {\sec ^{2}{\left (x \right )}}{a + b \tan {\left (x \right )}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^2(x)}{a+b \tan (x)} \, dx=\frac {\log \left (b \tan \left (x\right ) + a\right )}{b} \]
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^2(x)}{a+b \tan (x)} \, dx=\frac {\log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{b} \]
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Time = 26.48 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^2(x)}{a+b \tan (x)} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (x\right )\right )}{b} \]
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