Integrand size = 12, antiderivative size = 90 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6304, 849, 821, 272, 65, 214} \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=-\frac {x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}+\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
[In]
[Out]
Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 6304
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1-\frac {x}{a}}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {1}{3} \text {Subst}\left (\int \frac {\frac {3}{a}-\frac {2 x}{a^2}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{6} \text {Subst}\left (\int \frac {\frac {4}{a^2}-\frac {3 x}{a^3}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^3} \\ & = \frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a^3} \\ & = \frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {\text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \\ & = \frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.67 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (4-3 a x+2 a^2 x^2\right )-3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{6 a^3} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.21
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}-3 a x +4\right ) \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{6 a^{3}}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 a^{2} \sqrt {a^{2}}\, \left (a x -1\right )}\) | \(109\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+6 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right )}{6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} \sqrt {a^{2}}}\) | \(173\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {{\left (2 \, a^{3} x^{3} - a^{2} x^{2} + a x + 4\right )} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{6 \, a^{3}} \]
[In]
[Out]
\[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\int x^{2} \sqrt {\frac {a x - 1}{a x + 1}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (74) = 148\).
Time = 0.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.84 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=-\frac {1}{6} \, a {\left (\frac {2 \, {\left (9 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 4 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{4}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{4}} - \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{4}}\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {1}{6} \, \sqrt {a^{2} x^{2} - 1} {\left (x {\left (\frac {2 \, x \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {3 \, \mathrm {sgn}\left (a x + 1\right )}{a^{2}}\right )} + \frac {4 \, \mathrm {sgn}\left (a x + 1\right )}{a^{3}}\right )} + \frac {\log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{2 \, a^{2} {\left | a \right |}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.49 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\left (a\,x-1\right )}{a\,x+1}}-\frac {\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^3} \]
[In]
[Out]