Optimal. Leaf size=22 \[ \frac{\tan ^{-1}\left (\frac{\tanh (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0401457, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3675, 203} \[ \frac{\tan ^{-1}\left (\frac{\tanh (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3675
Rule 203
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(2+3 x)}{1+2 \coth ^2(2+3 x)} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\coth (2+3 x)\right )\right )\\ &=\frac{\tan ^{-1}\left (\frac{\tanh (2+3 x)}{\sqrt{2}}\right )}{3 \sqrt{2}}\\ \end{align*}
Mathematica [B] time = 0.105043, 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 = 0.093, size = 132, normalized size = 6. \begin{align*}{\frac{\sqrt{3}}{3\,\sqrt{6}-3\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( 1+3/2\,x \right ) }{\sqrt{6}-\sqrt{2}}} \right ) }-{\frac{1}{3\,\sqrt{6}-3\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( 1+3/2\,x \right ) }{\sqrt{6}-\sqrt{2}}} \right ) }-{\frac{\sqrt{3}}{3\,\sqrt{6}+3\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( 1+3/2\,x \right ) }{\sqrt{6}+\sqrt{2}}} \right ) }-{\frac{1}{3\,\sqrt{6}+3\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( 1+3/2\,x \right ) }{\sqrt{6}+\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.64696, 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 = 2.10221, size = 147, normalized size = 6.68 \begin{align*} -\frac{1}{6} \, \sqrt{2} \arctan \left (-\frac{2 \, \sqrt{2} \cosh \left (3 \, x + 2\right ) + \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{2}{\left (3 x + 2 \right )}}{2 \coth ^{2}{\left (3 x + 2 \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]