Integrand size = 12, antiderivative size = 118 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a-\frac {1}{x}\right )}+\frac {14 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {11 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
11/2*arctanh((1-1/a^2/x^2)^(1/2))/a^3-4*(1-1/a^2/x^2)^(1/2)/a^2/(a-1/x)+14 /3*x*(1-1/a^2/x^2)^(1/2)/a^2+3/2*x^2*(1-1/a^2/x^2)^(1/2)/a+1/3*x^3*(1-1/a^ 2/x^2)^(1/2)
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.64 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-52+19 a x+7 a^2 x^2+2 a^3 x^3\right )}{-1+a x}+33 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{6 a^3} \]
((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-52 + 19*a*x + 7*a^2*x^2 + 2*a^3*x^3))/(-1 + a*x) + 33*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(6*a^3)
Time = 0.62 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6719, 2353, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 e^{3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6719 |
\(\displaystyle -\int \frac {\left (1+\frac {1}{a x}\right )^2 x^4}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 2353 |
\(\displaystyle -\int \left (\frac {x^4}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 x^3}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x^2}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}+\frac {14 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a-\frac {1}{x}\right )}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {11 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3}\) |
(-4*Sqrt[1 - 1/(a^2*x^2)])/(a^2*(a - x^(-1))) + (14*Sqrt[1 - 1/(a^2*x^2)]* x)/(3*a^2) + (3*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(2*a) + (Sqrt[1 - 1/(a^2*x^2)]* x^3)/3 + (11*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a^3)
3.1.18.3.1 Defintions of rubi rules used
Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) ^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x , 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]
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 | \(\frac {9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-18 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-4 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +42 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+42 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +18 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -10 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-84 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -84 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a +42 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+42 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )}{6 a^{3} \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(471\) |
1/6*(2*a^2*x^2+9*a*x+28)*(a*x-1)/a^3/((a*x-1)/(a*x+1))^(1/2)+(11/2/a^2*ln( a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)-4/a^4/(x-1/a)*((x-1/a)^2* a^2+2*(x-1/a)*a)^(1/2))/(a*x+1)/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^ (1/2)
Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.95 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\frac {33 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 33 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 33 \, a x - 52\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, {\left (a^{4} x - a^{3}\right )}} \]
1/6*(33*(a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 33*(a*x - 1)*log(sq rt((a*x - 1)/(a*x + 1)) - 1) + (2*a^4*x^4 + 9*a^3*x^3 + 26*a^2*x^2 - 33*a* x - 52)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x - a^3)
\[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\int \frac {x^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.54 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=-\frac {1}{6} \, a {\left (\frac {2 \, {\left (\frac {75 \, {\left (a x - 1\right )}}{a x + 1} - \frac {88 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {33 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 12\right )}}{a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 3 \, a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 3 \, a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{4} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{4}} + \frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{4}}\right )} \]
-1/6*a*(2*(75*(a*x - 1)/(a*x + 1) - 88*(a*x - 1)^2/(a*x + 1)^2 + 33*(a*x - 1)^3/(a*x + 1)^3 - 12)/(a^4*((a*x - 1)/(a*x + 1))^(7/2) - 3*a^4*((a*x - 1 )/(a*x + 1))^(5/2) + 3*a^4*((a*x - 1)/(a*x + 1))^(3/2) - a^4*sqrt((a*x - 1 )/(a*x + 1))) - 33*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^4 + 33*log(sqrt((a *x - 1)/(a*x + 1)) - 1)/a^4)
\[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\int { \frac {x^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
Time = 4.13 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.31 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \, dx=\frac {11\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^3}-\frac {\frac {88\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {11\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {25\,\left (a\,x-1\right )}{a\,x+1}+4}{a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}-3\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}+3\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]