Optimal. Leaf size=21 \[ \log (2 \cos (x)-\sin (x))-\log (\cos (x)-\sin (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 21, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 31
Rule 616
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 = 0.03, 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.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 29, normalized size = 1.38 \[ \frac {1}{2} \, \log \left (\frac {3}{4} \, \cos \relax (x)^{2} - \cos \relax (x) \sin \relax (x) + \frac {1}{4}\right ) - \frac {1}{2} \, \log \left (-2 \, \cos \relax (x) \sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.96, size = 15, normalized size = 0.71 \[ -\log \left ({\left | \tan \relax (x) - 1 \right |}\right ) + \log \left ({\left | \tan \relax (x) - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.13, size = 14, normalized size = 0.67 \[ \ln \left (\tan \relax (x )-2\right )-\ln \left (\tan \relax (x )-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 13, normalized size = 0.62 \[ -\log \left (\tan \relax (x) - 1\right ) + \log \left (\tan \relax (x) - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.81, size = 9, normalized size = 0.43 \[ -2\,\mathrm {atanh}\left (2\,\mathrm {tan}\relax (x)-3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\relax (x )}}{- 3 \tan {\relax (x )} + \sec ^{2}{\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________