Optimal. Leaf size=22 \[ \frac{\tan ^{-1}\left (\sqrt{2} \tanh (3 x+2)\right )}{3 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0212245, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {203} \[ \frac{\tan ^{-1}\left (\sqrt{2} \tanh (3 x+2)\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\cosh ^2(2+3 x)+2 \sinh ^2(2+3 x)} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\tanh (2+3 x)\right )\\ &=\frac{\tan ^{-1}\left (\sqrt{2} \tanh (2+3 x)\right )}{3 \sqrt{2}}\\ \end{align*}
Mathematica [B] time = 0.0665493, size = 47, normalized size = 2.14 \[ \frac{\tan ^{-1}\left (\frac{\left (3+2 e^4+3 e^8\right ) \tanh (3 x)+3 \left (e^8-1\right )}{4 \sqrt{2} e^4}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.078, size = 156, normalized size = 7.1 \begin{align*} -{\frac{\sqrt{6}}{6\,\sqrt{3}+6\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( 1+3/2\,x \right ) }{2\,\sqrt{3}+2\,\sqrt{2}}} \right ) }-{\frac{2}{6\,\sqrt{3}+6\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( 1+3/2\,x \right ) }{2\,\sqrt{3}+2\,\sqrt{2}}} \right ) }+{\frac{\sqrt{6}}{6\,\sqrt{3}-6\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( 1+3/2\,x \right ) }{2\,\sqrt{3}-2\,\sqrt{2}}} \right ) }-{\frac{2}{6\,\sqrt{3}-6\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( 1+3/2\,x \right ) }{2\,\sqrt{3}-2\,\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.52181, size = 28, normalized size = 1.27 \begin{align*} -\frac{1}{6} \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (3 \, e^{\left (-6 \, x - 4\right )} - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.08918, size = 147, normalized size = 6.68 \begin{align*} -\frac{1}{6} \, \sqrt{2} \arctan \left (-\frac{\sqrt{2} \cosh \left (3 \, x + 2\right ) + 2 \, \sqrt{2} \sinh \left (3 \, x + 2\right )}{2 \,{\left (\cosh \left (3 \, x + 2\right ) - \sinh \left (3 \, x + 2\right )\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 9.60681, size = 280, normalized size = 12.73 \begin{align*} \frac{2 \sqrt{6} \operatorname{atan}{\left (\frac{\tanh{\left (\frac{3 x}{2} + 1 \right )}}{\sqrt{5 - 2 \sqrt{6}}} \right )}}{66 \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{6} \sqrt{5 - 2 \sqrt{6}}} + \frac{5 \operatorname{atan}{\left (\frac{\tanh{\left (\frac{3 x}{2} + 1 \right )}}{\sqrt{5 - 2 \sqrt{6}}} \right )}}{66 \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{6} \sqrt{5 - 2 \sqrt{6}}} - \frac{5 \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5} \operatorname{atan}{\left (\frac{\tanh{\left (\frac{3 x}{2} + 1 \right )}}{\sqrt{2 \sqrt{6} + 5}} \right )}}{66 \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{6} \sqrt{5 - 2 \sqrt{6}}} - \frac{2 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5} \operatorname{atan}{\left (\frac{\tanh{\left (\frac{3 x}{2} + 1 \right )}}{\sqrt{2 \sqrt{6} + 5}} \right )}}{66 \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{6} \sqrt{5 - 2 \sqrt{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27923, size = 28, normalized size = 1.27 \begin{align*} \frac{1}{6} \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (3 \, e^{\left (6 \, x + 4\right )} - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]