Optimal. Leaf size=21 \[ \log (2 \cos (x)-\sin (x))-\log (\cos (x)-\sin (x)) \]
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Rubi [A] time = 0.115477, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {616, 31} \[ \log (2 \cos (x)-\sin (x))-\log (\cos (x)-\sin (x)) \]
Antiderivative was successfully verified.
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Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{2-3 x+x^2} \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{-2+x} \, dx,x,\tan (x)\right )-\operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,\tan (x)\right )\\ &=-\log (1-\tan (x))+\log (2-\tan (x))\\ \end{align*}
Mathematica [A] time = 0.0308383, size = 29, normalized size = 1.38 \[ 2 \left (\frac{1}{2} \log (2 \cos (x)-\sin (x))-\frac{1}{2} \log (\cos (x)-\sin (x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 14, normalized size = 0.7 \begin{align*} \ln \left ( -2+\tan \left ( x \right ) \right ) -\ln \left ( \tan \left ( x \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961783, size = 18, normalized size = 0.86 \begin{align*} -\log \left (\tan \left (x\right ) - 1\right ) + \log \left (\tan \left (x\right ) - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0796, size = 104, normalized size = 4.95 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (x \right )}}{- 3 \tan{\left (x \right )} + \sec ^{2}{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14781, size = 20, normalized size = 0.95 \begin{align*} -\log \left ({\left | \tan \left (x\right ) - 1 \right |}\right ) + \log \left ({\left | \tan \left (x\right ) - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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