Optimal. Leaf size=54 \[ x-\frac{\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac{\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac{7 \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0884717, antiderivative size = 54, 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, 578, 522, 206} \[ x-\frac{\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac{\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac{7 \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 470
Rule 578
Rule 522
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (\coth ^2(x)+\text{csch}^2(x)\right )^3} \, dx &=\operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right ) \left (2-x^2\right )^3} \, dx,x,\tanh (x)\right )\\ &=-\frac{\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2 \left (6-2 x^2\right )}{\left (1-x^2\right ) \left (2-x^2\right )^2} \, dx,x,\tanh (x)\right )\\ &=-\frac{\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac{\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac{1}{8} \operatorname{Subst}\left (\int \frac{-2-6 x^2}{\left (1-x^2\right ) \left (2-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=-\frac{\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac{\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac{7}{4} \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\tanh (x)\right )+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=x-\frac{7 \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{4 \sqrt{2}}-\frac{\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac{\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}\\ \end{align*}
Mathematica [A] time = 0.187356, size = 66, normalized size = 1.22 \[ \frac{76 x-2 \sinh (2 x)-3 \sinh (4 x)+48 x \cosh (2 x)+4 x \cosh (4 x)-7 \sqrt{2} (\cosh (2 x)+3)^2 \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{8 (\cosh (2 x)+3)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.049, size = 145, normalized size = 2.7 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +2\,{\frac{-1/8\, \left ( \tanh \left ( x/2 \right ) \right ) ^{7}-5/8\, \left ( \tanh \left ( x/2 \right ) \right ) ^{5}-5/8\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-1/8\,\tanh \left ( x/2 \right ) }{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+1 \right ) ^{2}}}-{\frac{7\,\sqrt{2}}{32}\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{7\,\sqrt{2}}{32}\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 = 1.77191, size = 113, normalized size = 2.09 \begin{align*} \frac{7}{16} \, \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{19 \, e^{\left (-2 \, x\right )} + 57 \, e^{\left (-4 \, x\right )} + 17 \, e^{\left (-6 \, x\right )} + 3}{2 \,{\left (12 \, e^{\left (-2 \, x\right )} + 38 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.41457, size = 2363, normalized size = 43.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
Giac [A] time = 1.13614, size = 97, normalized size = 1.8 \begin{align*} -\frac{7}{16} \, \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{17 \, e^{\left (6 \, x\right )} + 57 \, e^{\left (4 \, x\right )} + 19 \, e^{\left (2 \, x\right )} + 3}{2 \,{\left (e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]