Optimal. Leaf size=32 \[ x-\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{\tanh (x)}{2-\tanh ^2(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0467893, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {470, 12, 391, 206} \[ x-\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{\tanh (x)}{2-\tanh ^2(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 470
Rule 12
Rule 391
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (\coth ^2(x)+\text{csch}^2(x)\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right ) \left (2-x^2\right )^2} \, dx,x,\tanh (x)\right )\\ &=-\frac{\tanh (x)}{2-\tanh ^2(x)}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{2}{\left (1-x^2\right ) \left (2-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=-\frac{\tanh (x)}{2-\tanh ^2(x)}+\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (2-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=-\frac{\tanh (x)}{2-\tanh ^2(x)}+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (x)\right )-\operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\tanh (x)\right )\\ &=x-\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{\tanh (x)}{2-\tanh ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.130939, size = 64, normalized size = 2. \[ \frac{(\cosh (2 x)+3) \text{csch}^4(x) \left (6 x-2 \sinh (2 x)+2 x \cosh (2 x)-\sqrt{2} (\cosh (2 x)+3) \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )\right )}{8 \left (\coth ^2(x)+\text{csch}^2(x)\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.048, size = 129, normalized size = 4. \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +2\,{\frac{-1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-1/2\,\tanh \left ( x/2 \right ) }{ \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+1}}-{\frac{\sqrt{2}}{8}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{8}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.69692, size = 81, normalized size = 2.53 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (-2 \, x\right )} + 3}\right ) + x - \frac{2 \,{\left (3 \, e^{\left (-2 \, x\right )} + 1\right )}}{6 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.35679, size = 883, normalized size = 27.59 \begin{align*} \frac{4 \, x \cosh \left (x\right )^{4} + 16 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 4 \, x \sinh \left (x\right )^{4} + 24 \,{\left (x + 1\right )} \cosh \left (x\right )^{2} + 24 \,{\left (x \cosh \left (x\right )^{2} + x + 1\right )} \sinh \left (x\right )^{2} +{\left (\sqrt{2} \cosh \left (x\right )^{4} + 4 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt{2} \sinh \left (x\right )^{4} + 6 \,{\left (\sqrt{2} \cosh \left (x\right )^{2} + \sqrt{2}\right )} \sinh \left (x\right )^{2} + 6 \, \sqrt{2} \cosh \left (x\right )^{2} + 4 \,{\left (\sqrt{2} \cosh \left (x\right )^{3} + 3 \, \sqrt{2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt{2}\right )} \log \left (\frac{3 \,{\left (2 \, \sqrt{2} + 3\right )} \cosh \left (x\right )^{2} - 4 \,{\left (3 \, \sqrt{2} + 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \,{\left (2 \, \sqrt{2} + 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt{2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) + 16 \,{\left (x \cosh \left (x\right )^{3} + 3 \,{\left (x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, x + 8}{4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \,{\left (\cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\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{1}{\left (\coth ^{2}{\left (x \right )} + \operatorname{csch}^{2}{\left (x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.15796, size = 81, normalized size = 2.53 \begin{align*} -\frac{1}{4} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (2 \, x\right )} + 3}\right ) + x + \frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]