Integrand size = 12, antiderivative size = 116 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \, dx=\frac {4 \left (a-\frac {1}{x}\right )}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {14 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {11 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \] Output:
4*(a-1/x)/a^4/(1-1/a^2/x^2)^(1/2)+14/3*(1-1/a^2/x^2)^(1/2)*x/a^2-3/2*(1-1/ a^2/x^2)^(1/2)*x^2/a+1/3*(1-1/a^2/x^2)^(1/2)*x^3-11/2*arctanh((1-1/a^2/x^2 )^(1/2))/a^3
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (52+19 a x-7 a^2 x^2+2 a^3 x^3\right )}{1+a x}-33 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{6 a^3} \] Input:
Integrate[x^2/E^(3*ArcCoth[a*x]),x]
Output:
((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 = 1.02 (sec) , antiderivative size = 116, 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 |
\(\displaystyle -\frac {3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}+\frac {14 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a+\frac {1}{x}\right )}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {11 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3}\) |
Input:
Int[x^2/E^(3*ArcCoth[a*x]),x]
Output:
(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)
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 = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.25
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}-9 a x +28\right ) \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{6 a^{3}}+\frac {\left (-\frac {11 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{2 a^{2} \sqrt {a^{2}}}+\frac {4 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{4} \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(145\) |
default | \(-\frac {\left (9 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+18 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-4 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -42 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+42 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+9 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -18 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +10 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-84 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +84 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -42 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+42 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{6 a^{3} \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(471\) |
Input:
int(x^2*((a*x-1)/(a*x+1))^(3/2),x,method=_RETURNVERBOSE)
Output:
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)*(a^2*(x+1/ a)^2-2*a*(x+1/a))^(1/2))*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/( a*x-1)
Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.72 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \, dx=\frac {{\left (2 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 19 \, a x + 52\right )} \sqrt {\frac {a x - 1}{a x + 1}} - 33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{6 \, a^{3}} \] Input:
integrate(x^2*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")
Output:
1/6*((2*a^3*x^3 - 7*a^2*x^2 + 19*a*x + 52)*sqrt((a*x - 1)/(a*x + 1)) - 33* log(sqrt((a*x - 1)/(a*x + 1)) + 1) + 33*log(sqrt((a*x - 1)/(a*x + 1)) - 1) )/a^3
\[ \int e^{-3 \coth ^{-1}(a x)} x^2 \, dx=\int x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**2*((a*x-1)/(a*x+1))**(3/2),x)
Output:
Integral(x**2*((a*x - 1)/(a*x + 1))**(3/2), x)
Time = 0.03 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.60 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \, dx=-\frac {1}{6} \, a {\left (\frac {2 \, {\left (39 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 52 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 21 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{4}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{4}} - \frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{4}} - \frac {24 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{4}}\right )} \] Input:
integrate(x^2*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")
Output:
-1/6*a*(2*(39*((a*x - 1)/(a*x + 1))^(5/2) - 52*((a*x - 1)/(a*x + 1))^(3/2) + 21*sqrt((a*x - 1)/(a*x + 1)))/(3*(a*x - 1)*a^4/(a*x + 1) - 3*(a*x - 1)^ 2*a^4/(a*x + 1)^2 + (a*x - 1)^3*a^4/(a*x + 1)^3 - a^4) + 33*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^4 - 33*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^4 - 24* sqrt((a*x - 1)/(a*x + 1))/a^4)
\[ \int e^{-3 \coth ^{-1}(a x)} x^2 \, dx=\int { x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x^2*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")
Output:
undef
Time = 24.33 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.34 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \, dx=\frac {7\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {52\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+13\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\left (a\,x-1\right )}{a\,x+1}}+\frac {4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a^3}-\frac {11\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^3} \] Input:
int(x^2*((a*x - 1)/(a*x + 1))^(3/2),x)
Output:
(7*((a*x - 1)/(a*x + 1))^(1/2) - (52*((a*x - 1)/(a*x + 1))^(3/2))/3 + 13*( (a*x - 1)/(a*x + 1))^(5/2))/(a^3 + (3*a^3*(a*x - 1)^2)/(a*x + 1)^2 - (a^3* (a*x - 1)^3)/(a*x + 1)^3 - (3*a^3*(a*x - 1))/(a*x + 1)) + (4*((a*x - 1)/(a *x + 1))^(1/2))/a^3 - (11*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a^3
Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.14 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \, dx=\frac {8 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-28 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+76 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +208 \sqrt {a x +1}\, \sqrt {a x -1}-264 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -264 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+153 a x +153}{24 a^{3} \left (a x +1\right )} \] Input:
int(x^2*((a*x-1)/(a*x+1))^(3/2),x)
Output:
(8*sqrt(a*x + 1)*sqrt(a*x - 1)*a**3*x**3 - 28*sqrt(a*x + 1)*sqrt(a*x - 1)* a**2*x**2 + 76*sqrt(a*x + 1)*sqrt(a*x - 1)*a*x + 208*sqrt(a*x + 1)*sqrt(a* x - 1) - 264*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x - 264*log((s qrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 153*a*x + 153)/(24*a**3*(a*x + 1) )