Integrand size = 16, antiderivative size = 59 \[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5527, 4267, 2317, 2438} \[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac {x^2 \text {csch}(a+b x)}{b} \]
[In]
[Out]
Rule 2317
Rule 2438
Rule 4267
Rule 5527
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \text {csch}(a+b x)}{b}+\frac {2 \int x \text {csch}(a+b x) \, dx}{b} \\ & = -\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \int \log \left (1-e^{a+b x}\right ) \, dx}{b^2}+\frac {2 \int \log \left (1+e^{a+b x}\right ) \, dx}{b^2} \\ & = -\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {2 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^3} \\ & = -\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.17 \[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=\frac {-b x \left (b x \text {csch}(a+b x)-2 \log \left (1-e^{a+b x}\right )+2 \log \left (1+e^{a+b x}\right )\right )-2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(133\) vs. \(2(56)=112\).
Time = 0.41 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.27
method | result | size |
risch | \(-\frac {2 x^{2} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{2}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{3}}-\frac {2 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(134\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (54) = 108\).
Time = 0.25 (sec) , antiderivative size = 367, normalized size of antiderivative = 6.22 \[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + {\left (a \cosh \left (b x + a\right )^{2} + 2 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a \sinh \left (b x + a\right )^{2} - a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )^{2} - b x - a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )\right )}}{b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} - b^{3}} \]
[In]
[Out]
\[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=\int x^{2} \cosh {\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.41 \[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, x^{2} e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {2 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{3}} + \frac {2 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{3}} \]
[In]
[Out]
\[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=\int { x^{2} \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx=\int \frac {x^2\,\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]
[In]
[Out]