Integrand size = 6, antiderivative size = 16 \[ \int x \coth ^2(x) \, dx=\frac {x^2}{2}-x \coth (x)+\log (\sinh (x)) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3801, 3556, 30} \[ \int x \coth ^2(x) \, dx=\frac {x^2}{2}-x \coth (x)+\log (\sinh (x)) \]
[In]
[Out]
Rule 30
Rule 3556
Rule 3801
Rubi steps \begin{align*} \text {integral}& = -x \coth (x)+\int x \, dx+\int \coth (x) \, dx \\ & = \frac {x^2}{2}-x \coth (x)+\log (\sinh (x)) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int x \coth ^2(x) \, dx=\frac {x^2}{2}-x \coth (x)+\log (\sinh (x)) \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.75
method | result | size |
risch | \(\frac {x^{2}}{2}-2 x -\frac {2 x}{{\mathrm e}^{2 x}-1}+\ln \left ({\mathrm e}^{2 x}-1\right )\) | \(28\) |
parallelrisch | \(\frac {-2 \ln \left (1-\tanh \left (x \right )\right ) \tanh \left (x \right )+2 \ln \left (\tanh \left (x \right )\right ) \tanh \left (x \right )+x \left (-2+\left (-2+x \right ) \tanh \left (x \right )\right )}{2 \tanh \left (x \right )}\) | \(36\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (14) = 28\).
Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 5.94 \[ \int x \coth ^2(x) \, dx=\frac {{\left (x^{2} - 4 \, x\right )} \cosh \left (x\right )^{2} + 2 \, {\left (x^{2} - 4 \, x\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (x^{2} - 4 \, x\right )} \sinh \left (x\right )^{2} - x^{2} + 2 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{2 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}} \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int x \coth ^2(x) \, dx=\frac {x^{2}}{2} + x - \frac {x}{\tanh {\left (x \right )}} - \log {\left (\tanh {\left (x \right )} + 1 \right )} + \log {\left (\tanh {\left (x \right )} \right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (14) = 28\).
Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.31 \[ \int x \coth ^2(x) \, dx=-\frac {x e^{\left (2 \, x\right )}}{e^{\left (2 \, x\right )} - 1} - \frac {x^{2} - {\left (x^{2} - 2 \, x\right )} e^{\left (2 \, x\right )}}{2 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} + \log \left (e^{x} + 1\right ) + \log \left (e^{x} - 1\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (14) = 28\).
Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.31 \[ \int x \coth ^2(x) \, dx=\frac {x^{2} e^{\left (2 \, x\right )} - x^{2} - 4 \, x e^{\left (2 \, x\right )} + 2 \, e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 2 \, \log \left (e^{\left (2 \, x\right )} - 1\right )}{2 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} \]
[In]
[Out]
Time = 0.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int x \coth ^2(x) \, dx=\ln \left ({\mathrm {e}}^{2\,x}-1\right )-2\,x-\frac {2\,x}{{\mathrm {e}}^{2\,x}-1}+\frac {x^2}{2} \]
[In]
[Out]