Optimal. Leaf size=31 \[ \frac {\sec (x) \tanh ^{-1}(\sin (x))}{\sqrt {a \sec ^2(x)}}-\frac {\tan (x)}{\sqrt {a \sec ^2(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3657, 4125, 2592, 321, 206} \[ \frac {\sec (x) \tanh ^{-1}(\sin (x))}{\sqrt {a \sec ^2(x)}}-\frac {\tan (x)}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 321
Rule 2592
Rule 3657
Rule 4125
Rubi steps
\begin {align*} \int \frac {\tan ^2(x)}{\sqrt {a+a \tan ^2(x)}} \, dx &=\int \frac {\tan ^2(x)}{\sqrt {a \sec ^2(x)}} \, dx\\ &=\frac {\sec (x) \int \sin (x) \tan (x) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=\frac {\sec (x) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (x)\right )}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {\tan (x)}{\sqrt {a \sec ^2(x)}}+\frac {\sec (x) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right )}{\sqrt {a \sec ^2(x)}}\\ &=\frac {\tanh ^{-1}(\sin (x)) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {\tan (x)}{\sqrt {a \sec ^2(x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 49, normalized size = 1.58 \[ -\frac {\sec (x) \left (\sin (x)+\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.42, size = 64, normalized size = 2.06 \[ \frac {{\left (\tan \relax (x)^{2} + 1\right )} \sqrt {a} \log \left (2 \, a \tan \relax (x)^{2} + 2 \, \sqrt {a \tan \relax (x)^{2} + a} \sqrt {a} \tan \relax (x) + a\right ) - 2 \, \sqrt {a \tan \relax (x)^{2} + a} \tan \relax (x)}{2 \, {\left (a \tan \relax (x)^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.39, size = 40, normalized size = 1.29 \[ -\frac {\log \left ({\left | -\sqrt {a} \tan \relax (x) + \sqrt {a \tan \relax (x)^{2} + a} \right |}\right )}{\sqrt {a}} - \frac {\tan \relax (x)}{\sqrt {a \tan \relax (x)^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.23, size = 38, normalized size = 1.23 \[ \frac {\ln \left (\sqrt {a}\, \tan \relax (x )+\sqrt {a +a \left (\tan ^{2}\relax (x )\right )}\right )}{\sqrt {a}}-\frac {\tan \relax (x )}{\sqrt {a +a \left (\tan ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.88, size = 42, normalized size = 1.35 \[ \frac {\log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) - \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) - 2 \, \sin \relax (x)}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {tan}\relax (x)}^2}{\sqrt {a\,{\mathrm {tan}\relax (x)}^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{2}{\relax (x )}}{\sqrt {a \left (\tan ^{2}{\relax (x )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________