Integrand size = 15, antiderivative size = 69 \[ \int \frac {x^2}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {x^3}{4 \sqrt {\text {csch}(2 \log (c x))}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5671, 5669, 272, 43, 65, 212} \[ \int \frac {x^2}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {x^3}{4 \sqrt {\text {csch}(2 \log (c x))}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}} \]
[In]
[Out]
Rule 43
Rule 65
Rule 212
Rule 272
Rule 5669
Rule 5671
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2}{\sqrt {\text {csch}(2 \log (x))}} \, dx,x,c x\right )}{c^3} \\ & = \frac {\text {Subst}\left (\int \sqrt {1-\frac {1}{x^4}} x^3 \, dx,x,c x\right )}{c^4 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ & = -\frac {\text {Subst}\left (\int \frac {\sqrt {1-x}}{x^2} \, dx,x,\frac {1}{c^4 x^4}\right )}{4 c^4 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ & = \frac {x^3}{4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{c^4 x^4}\right )}{8 c^4 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ & = \frac {x^3}{4 \sqrt {\text {csch}(2 \log (c x))}}-\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ & = \frac {x^3}{4 \sqrt {\text {csch}(2 \log (c x))}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {x \left (c^2 x^2 \sqrt {1-c^4 x^4}+\arcsin \left (c^2 x^2\right )\right )}{4 c^2 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.41
method | result | size |
risch | \(\frac {\sqrt {2}\, x^{3}}{8 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}-\frac {\ln \left (\frac {c^{4} x^{2}}{\sqrt {c^{4}}}+\sqrt {c^{4} x^{4}-1}\right ) \sqrt {2}\, x}{8 \sqrt {c^{4}}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}\, \sqrt {c^{4} x^{4}-1}}\) | \(97\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.33 \[ \int \frac {x^2}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {2 \, \sqrt {2} {\left (c^{5} x^{5} - c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} + \sqrt {2} \log \left (2 \, c^{4} x^{4} - 2 \, {\left (c^{5} x^{5} - c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 1\right )}{16 \, c^{3}} \]
[In]
[Out]
\[ \int \frac {x^2}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int \frac {x^{2}}{\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {x^2}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int { \frac {x^{2}}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int { \frac {x^{2}}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^2}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int \frac {x^2}{\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \]
[In]
[Out]