Integrand size = 12, antiderivative size = 118 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a-\frac {1}{x}\right )}+\frac {14 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {11 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
[Out]
Time = 0.85 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6304, 6874, 665, 277, 270, 272, 44, 65, 214} \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\frac {3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}+\frac {14 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a-\frac {1}{x}\right )}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {11 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
[In]
[Out]
Rule 44
Rule 65
Rule 214
Rule 270
Rule 272
Rule 277
Rule 665
Rule 6304
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^2}{x^4 \left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {4}{a^3 (a-x) \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{x^4 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{a x^3 \sqrt {1-\frac {x^2}{a^2}}}+\frac {4}{a^2 x^2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {4}{a^3 x \sqrt {1-\frac {x^2}{a^2}}}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {4 \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^3}-\frac {4 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^3}-\frac {4 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}-\frac {3 \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\text {Subst}\left (\int \frac {1}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a-\frac {1}{x}\right )}+\frac {4 \sqrt {1-\frac {1}{a^2 x^2}} x}{a^2}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a^3}-\frac {2 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 a^2}-\frac {3 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a} \\ & = -\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a-\frac {1}{x}\right )}+\frac {14 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}+\frac {3 \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 {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a^3}+\frac {4 \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \\ & = -\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a-\frac {1}{x}\right )}+\frac {14 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}+\frac {3 \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 {4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^3}+\frac {3 \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 {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a-\frac {1}{x}\right )}+\frac {14 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}+\frac {3 \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 {11 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.64 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-52+19 a x+7 a^2 x^2+2 a^3 x^3\right )}{-1+a x}+33 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{6 a^3} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.28
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}+9 a x +28\right ) \left (a x -1\right )}{6 a^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {11 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{2 a^{2} \sqrt {a^{2}}}-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{a^{4} \left (x -\frac {1}{a}\right )}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(151\) |
default | \(\frac {9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-18 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-4 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +42 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+42 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +18 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -10 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-84 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -84 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a +42 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+42 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )}{6 a^{3} \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(471\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.95 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\frac {33 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 33 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 33 \, a x - 52\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, {\left (a^{4} x - a^{3}\right )}} \]
[In]
[Out]
\[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\int \frac {x^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.54 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=-\frac {1}{6} \, a {\left (\frac {2 \, {\left (\frac {75 \, {\left (a x - 1\right )}}{a x + 1} - \frac {88 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {33 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 12\right )}}{a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 3 \, a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 3 \, a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{4} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{4}} + \frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{4}}\right )} \]
[In]
[Out]
\[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\int { \frac {x^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Time = 4.13 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.31 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\frac {11\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^3}-\frac {\frac {88\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {11\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {25\,\left (a\,x-1\right )}{a\,x+1}+4}{a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}-3\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}+3\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]
[In]
[Out]