Optimal. Leaf size=26 \[ \frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}}-\frac{1}{4} \tan ^{-1}(\sinh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0268189, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1093, 203} \[ \frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}}-\frac{1}{4} \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1093
Rule 203
Rubi steps
\begin{align*} \int \text{csch}(4 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{4+12 x^2+8 x^4} \, dx,x,\sinh (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{4+8 x^2} \, dx,x,\sinh (x)\right )-2 \operatorname{Subst}\left (\int \frac{1}{8+8 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac{1}{4} \tan ^{-1}(\sinh (x))+\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0214198, size = 26, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}}-\frac{1}{4} \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.04, size = 62, normalized size = 2.4 \begin{align*}{\frac{i}{4}}\ln \left ({{\rm e}^{x}}-i \right ) -{\frac{i}{4}}\ln \left ({{\rm e}^{x}}+i \right ) +{\frac{i}{8}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}+i\sqrt{2}{{\rm e}^{x}}-1 \right ) -{\frac{i}{8}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}-i\sqrt{2}{{\rm e}^{x}}-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.52082, size = 68, normalized size = 2.62 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{\left (-x\right )}\right )}\right ) + \frac{1}{2} \, \arctan \left (e^{\left (-x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.18787, size = 297, normalized size = 11.42 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \cosh \left (x\right ) + \frac{1}{2} \, \sqrt{2} \sinh \left (x\right )\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (-\frac{\sqrt{2} \cosh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{2} \sinh \left (x\right )^{2} + \sqrt{2}}{2 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) - \frac{1}{2} \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\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 \sinh{\left (x \right )} \operatorname{csch}{\left (4 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.2097, size = 59, normalized size = 2.27 \begin{align*} -\frac{1}{8} \, \pi + \frac{1}{8} \, \sqrt{2}{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} - \frac{1}{4} \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]