Optimal. Leaf size=37 \[ -\frac{3 x}{2}+\frac{\coth ^3(x)}{2 (\coth (x)+1)}-\coth ^2(x)+\frac{3 \coth (x)}{2}+2 \log (\sinh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0677348, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3550, 3528, 3525, 3475} \[ -\frac{3 x}{2}+\frac{\coth ^3(x)}{2 (\coth (x)+1)}-\coth ^2(x)+\frac{3 \coth (x)}{2}+2 \log (\sinh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3550
Rule 3528
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \frac{\coth ^4(x)}{1+\coth (x)} \, dx &=\frac{\coth ^3(x)}{2 (1+\coth (x))}-\frac{1}{2} \int (3-4 \coth (x)) \coth ^2(x) \, dx\\ &=-\coth ^2(x)+\frac{\coth ^3(x)}{2 (1+\coth (x))}+\frac{1}{2} i \int (-4 i+3 i \coth (x)) \coth (x) \, dx\\ &=-\frac{3 x}{2}+\frac{3 \coth (x)}{2}-\coth ^2(x)+\frac{\coth ^3(x)}{2 (1+\coth (x))}+2 \int \coth (x) \, dx\\ &=-\frac{3 x}{2}+\frac{3 \coth (x)}{2}-\coth ^2(x)+\frac{\coth ^3(x)}{2 (1+\coth (x))}+2 \log (\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.0575645, size = 33, normalized size = 0.89 \[ \frac{1}{4} \left (-6 x-\sinh (2 x)+\cosh (2 x)+4 \coth (x)-2 \text{csch}^2(x)+8 \log (\sinh (x))\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.02, size = 32, normalized size = 0.9 \begin{align*} -{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{2}}{2}}+{\rm coth} \left (x\right )-{\frac{1}{2+2\,{\rm coth} \left (x\right )}}-{\frac{7\,\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{4}}-{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.11246, size = 73, normalized size = 1.97 \begin{align*} \frac{1}{2} \, x + \frac{2 \,{\left (2 \, e^{\left (-2 \, x\right )} - 1\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac{1}{4} \, e^{\left (-2 \, x\right )} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) + 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 3.04278, size = 1168, normalized size = 31.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.85889, size = 197, normalized size = 5.32 \begin{align*} \frac{x \tanh ^{3}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} + \frac{x \tanh ^{2}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} - \frac{4 \log{\left (\tanh{\left (x \right )} + 1 \right )} \tanh ^{3}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} - \frac{4 \log{\left (\tanh{\left (x \right )} + 1 \right )} \tanh ^{2}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} + \frac{4 \log{\left (\tanh{\left (x \right )} \right )} \tanh ^{3}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} + \frac{4 \log{\left (\tanh{\left (x \right )} \right )} \tanh ^{2}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} + \frac{3 \tanh ^{2}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} + \frac{\tanh{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} - \frac{1}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11409, size = 54, normalized size = 1.46 \begin{align*} -\frac{7}{2} \, x + \frac{{\left (e^{\left (4 \, x\right )} - 10 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}}{4 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + 2 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]