Integrand size = 12, antiderivative size = 136 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \, dx=-\frac {4 \left (a-\frac {1}{x}\right )}{a^5 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {6 \sqrt {1-\frac {1}{a^2 x^2}} x}{a^3}+\frac {19 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{8 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {51 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a^4} \] Output:
(-4*a+4/x)/a^5/(1-1/a^2/x^2)^(1/2)-6*(1-1/a^2/x^2)^(1/2)*x/a^3+19/8*(1-1/a ^2/x^2)^(1/2)*x^2/a^2-(1-1/a^2/x^2)^(1/2)*x^3/a+1/4*(1-1/a^2/x^2)^(1/2)*x^ 4+51/8*arctanh((1-1/a^2/x^2)^(1/2))/a^4
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.61 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-80-29 a x+11 a^2 x^2-6 a^3 x^3+2 a^4 x^4\right )}{1+a x}+51 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{8 a^4} \] Input:
Integrate[x^3/E^(3*ArcCoth[a*x]),x]
Output:
((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-80 - 29*a*x + 11*a^2*x^2 - 6*a^3*x^3 + 2*a^4 *x^4))/(1 + a*x) + 51*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(8*a^4)
Time = 1.10 (sec) , antiderivative size = 136, 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.
| \(\displaystyle \int x^3 e^{-3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6719 |
| \(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right )^2 x^5}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1+\frac {1}{a x}\right )}d\frac {1}{x}\) |
| \(\Big \downarrow \) 2353 |
| \(\displaystyle -\int \left (\frac {x^5}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 x^4}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x^3}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 x^2}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {19 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{8 a^2}+\frac {1}{4} x^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {51 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a^4}-\frac {6 x \sqrt {1-\frac {1}{a^2 x^2}}}{a^3}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^3 \left (a+\frac {1}{x}\right )}\) |
Input:
Int[x^3/E^(3*ArcCoth[a*x]),x]
Output:
(-4*Sqrt[1 - 1/(a^2*x^2)])/(a^3*(a + x^(-1))) - (6*Sqrt[1 - 1/(a^2*x^2)]*x )/a^3 + (19*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(8*a^2) - (Sqrt[1 - 1/(a^2*x^2)]*x^ 3)/a + (Sqrt[1 - 1/(a^2*x^2)]*x^4)/4 + (51*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]) /(8*a^4)
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 = 0.10 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(\frac {\left (2 a^{3} x^{3}-8 a^{2} x^{2}+19 a x -48\right ) \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{8 a^{4}}+\frac {\left (\frac {51 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{8 a^{3} \sqrt {a^{2}}}-\frac {4 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{5} \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(153\) |
| default | \(-\frac {\left (-2 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{3} x^{3}+8 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-4 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}-21 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}+16 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +72 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-2 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -42 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+21 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-72 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-8 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+144 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -21 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +42 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -144 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +72 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+21 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -72 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{8 a^{4} \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(539\) |
Input:
int(x^3*((a*x-1)/(a*x+1))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/8*(2*a^3*x^3-8*a^2*x^2+19*a*x-48)*(a*x+1)/a^4*((a*x-1)/(a*x+1))^(1/2)+(5 1/8/a^3*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)-4/a^5/(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 = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.68 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \, dx=\frac {{\left (2 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 11 \, a^{2} x^{2} - 29 \, a x - 80\right )} \sqrt {\frac {a x - 1}{a x + 1}} + 51 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 51 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{8 \, a^{4}} \] Input:
integrate(x^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")
Output:
1/8*((2*a^4*x^4 - 6*a^3*x^3 + 11*a^2*x^2 - 29*a*x - 80)*sqrt((a*x - 1)/(a* x + 1)) + 51*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 51*log(sqrt((a*x - 1)/(a *x + 1)) - 1))/a^4
\[ \int e^{-3 \coth ^{-1}(a x)} x^3 \, dx=\int x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**3*((a*x-1)/(a*x+1))**(3/2),x)
Output:
Integral(x**3*((a*x - 1)/(a*x + 1))**(3/2), x)
Time = 0.04 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.64 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \, dx=-\frac {1}{8} \, a {\left (\frac {2 \, {\left (77 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 149 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 123 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 35 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {4 \, {\left (a x - 1\right )} a^{5}}{a x + 1} - \frac {6 \, {\left (a x - 1\right )}^{2} a^{5}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3} a^{5}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{5}}{{\left (a x + 1\right )}^{4}} - a^{5}} - \frac {51 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{5}} + \frac {51 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{5}} + \frac {32 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{5}}\right )} \] Input:
integrate(x^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")
Output:
-1/8*a*(2*(77*((a*x - 1)/(a*x + 1))^(7/2) - 149*((a*x - 1)/(a*x + 1))^(5/2 ) + 123*((a*x - 1)/(a*x + 1))^(3/2) - 35*sqrt((a*x - 1)/(a*x + 1)))/(4*(a* x - 1)*a^5/(a*x + 1) - 6*(a*x - 1)^2*a^5/(a*x + 1)^2 + 4*(a*x - 1)^3*a^5/( a*x + 1)^3 - (a*x - 1)^4*a^5/(a*x + 1)^4 - a^5) - 51*log(sqrt((a*x - 1)/(a *x + 1)) + 1)/a^5 + 51*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^5 + 32*sqrt((a *x - 1)/(a*x + 1))/a^5)
\[ \int e^{-3 \coth ^{-1}(a x)} x^3 \, dx=\int { x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")
Output:
undef
Time = 23.42 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.41 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \, dx=\frac {51\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a^4}-\frac {\frac {35\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {123\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}+\frac {149\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{4}-\frac {77\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{a^4+\frac {6\,a^4\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {4\,a^4\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {a^4\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {4\,a^4\,\left (a\,x-1\right )}{a\,x+1}}-\frac {4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a^4} \] Input:
int(x^3*((a*x - 1)/(a*x + 1))^(3/2),x)
Output:
(51*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(4*a^4) - ((35*((a*x - 1)/(a*x + 1 ))^(1/2))/4 - (123*((a*x - 1)/(a*x + 1))^(3/2))/4 + (149*((a*x - 1)/(a*x + 1))^(5/2))/4 - (77*((a*x - 1)/(a*x + 1))^(7/2))/4)/(a^4 + (6*a^4*(a*x - 1 )^2)/(a*x + 1)^2 - (4*a^4*(a*x - 1)^3)/(a*x + 1)^3 + (a^4*(a*x - 1)^4)/(a* x + 1)^4 - (4*a^4*(a*x - 1))/(a*x + 1)) - (4*((a*x - 1)/(a*x + 1))^(1/2))/ a^4
Time = 0.16 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.12 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \, dx=\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{4} x^{4}-6 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+11 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-29 \sqrt {a x +1}\, \sqrt {a x -1}\, a x -80 \sqrt {a x +1}\, \sqrt {a x -1}+102 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x +102 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )-57 a x -57}{8 a^{4} \left (a x +1\right )} \] Input:
int(x^3*((a*x-1)/(a*x+1))^(3/2),x)
Output:
(2*sqrt(a*x + 1)*sqrt(a*x - 1)*a**4*x**4 - 6*sqrt(a*x + 1)*sqrt(a*x - 1)*a **3*x**3 + 11*sqrt(a*x + 1)*sqrt(a*x - 1)*a**2*x**2 - 29*sqrt(a*x + 1)*sqr t(a*x - 1)*a*x - 80*sqrt(a*x + 1)*sqrt(a*x - 1) + 102*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x + 102*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqr t(2)) - 57*a*x - 57)/(8*a**4*(a*x + 1))