Optimal. Leaf size=3 \[ \tan ^{-1}(\tanh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0172277, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {203} \[ \tan ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\cosh ^2(x)+\sinh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tanh (x)\right )\\ &=\tan ^{-1}(\tanh (x))\\ \end{align*}
Mathematica [A] time = 0.0047023, size = 3, normalized size = 1. \[ \tan ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.039, size = 116, normalized size = 38.7 \begin{align*} -2\,{\frac{\sqrt{2}}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{2+2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{2+2\,\sqrt{2}}} \right ) }+2\,{\frac{\sqrt{2}}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{-2+2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{-2+2\,\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.61923, size = 47, normalized size = 15.67 \begin{align*} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{\left (-x\right )}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.29802, size = 69, normalized size = 23. \begin{align*} -\arctan \left (-\frac{\cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 11.3414, size = 117, normalized size = 39. \begin{align*} \frac{\operatorname{atan}{\left (\frac{\tanh{\left (\frac{x}{2} \right )}}{\sqrt{3 - 2 \sqrt{2}}} \right )}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left (\frac{\tanh{\left (\frac{x}{2} \right )}}{\sqrt{2 \sqrt{2} + 3}} \right )}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10793, size = 7, normalized size = 2.33 \begin{align*} \arctan \left (e^{\left (2 \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]