Integrand size = 17, antiderivative size = 21 \[ \int \frac {\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx=-\log (\cos (x)-\sin (x))+\log (2 \cos (x)-\sin (x)) \]
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Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {630, 31} \[ \int \frac {\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx=\log (2 \cos (x)-\sin (x))-\log (\cos (x)-\sin (x)) \]
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Rule 31
Rule 630
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{2-3 x+x^2} \, dx,x,\tan (x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{-2+x} \, dx,x,\tan (x)\right )-\text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\tan (x)\right ) \\ & = -\log (1-\tan (x))+\log (2-\tan (x)) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx=2 \left (-\frac {1}{2} \log (\cos (x)-\sin (x))+\frac {1}{2} \log (2 \cos (x)-\sin (x))\right ) \]
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Time = 0.62 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
default | \(-\ln \left (\tan \left (x \right )-1\right )+\ln \left (-2+\tan \left (x \right )\right )\) | \(14\) |
risch | \(\ln \left ({\mathrm e}^{2 i x}+\frac {3}{5}-\frac {4 i}{5}\right )-\ln \left ({\mathrm e}^{2 i x}-i\right )\) | \(23\) |
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Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx=\frac {1}{2} \, \log \left (\frac {3}{4} \, \cos \left (x\right )^{2} - \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{4}\right ) - \frac {1}{2} \, \log \left (-2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]
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\[ \int \frac {\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx=\int \frac {\sec ^{2}{\left (x \right )}}{- 3 \tan {\left (x \right )} + \sec ^{2}{\left (x \right )} + 1}\, dx \]
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none
Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx=-\log \left (\tan \left (x\right ) - 1\right ) + \log \left (\tan \left (x\right ) - 2\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx=-\log \left ({\left | \tan \left (x\right ) - 1 \right |}\right ) + \log \left ({\left | \tan \left (x\right ) - 2 \right |}\right ) \]
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Time = 27.38 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.43 \[ \int \frac {\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx=-2\,\mathrm {atanh}\left (2\,\mathrm {tan}\left (x\right )-3\right ) \]
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