Integrand size = 12, antiderivative size = 90 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \] Output:
2/3*(1-1/a^2/x^2)^(1/2)*x/a^2-1/2*(1-1/a^2/x^2)^(1/2)*x^2/a+1/3*(1-1/a^2/x ^2)^(1/2)*x^3-1/2*arctanh((1-1/a^2/x^2)^(1/2))/a^3
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.67 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (4-3 a x+2 a^2 x^2\right )-3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{6 a^3} \] Input:
Integrate[x^2/E^ArcCoth[a*x],x]
Output:
(a*Sqrt[1 - 1/(a^2*x^2)]*x*(4 - 3*a*x + 2*a^2*x^2) - 3*Log[(1 + Sqrt[1 - 1 /(a^2*x^2)])*x])/(6*a^3)
Time = 0.50 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6719, 539, 27, 539, 27, 534, 243, 73, 221}
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.
\(\displaystyle \int x^2 e^{-\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6719 |
\(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right ) x^4}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\) |
\(\Big \downarrow \) 539 |
\(\displaystyle \frac {1}{3} \int \frac {\left (3 a-\frac {2}{x}\right ) x^3}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\left (3 a-\frac {2}{x}\right ) x^3}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{3 a^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 539 |
\(\displaystyle \frac {-\frac {1}{2} \int \frac {\left (4 a-\frac {3}{x}\right ) x^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {3}{2} a x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {\int \frac {\left (4 a-\frac {3}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{2 a}-\frac {3}{2} a x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle \frac {-\frac {-3 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-4 a x \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}-\frac {3}{2} a x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {-\frac {-\frac {3}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-4 a x \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}-\frac {3}{2} a x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {-\frac {3 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-4 a x \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}-\frac {3}{2} a x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-4 a x \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}-\frac {3}{2} a x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}\) |
Input:
Int[x^2/E^ArcCoth[a*x],x]
Output:
(Sqrt[1 - 1/(a^2*x^2)]*x^3)/3 + ((-3*a*Sqrt[1 - 1/(a^2*x^2)]*x^2)/2 - (-4* a*Sqrt[1 - 1/(a^2*x^2)]*x + 3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a))/(3*a^ 2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
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 = 0.08 (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 | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+3 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -6 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} \sqrt {a^{2}}}\) | \(173\) |
Input:
int(x^2*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/6*(2*a^2*x^2-3*a*x+4)*(a*x+1)/a^3*((a*x-1)/(a*x+1))^(1/2)-1/2/a^2*ln(a^2 *x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)*((a*x-1)/(a*x+1))^(1/2)*((a* x-1)*(a*x+1))^(1/2)/(a*x-1)
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {{\left (2 \, a^{3} x^{3} - a^{2} x^{2} + a x + 4\right )} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{6 \, a^{3}} \] Input:
integrate(x^2*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")
Output:
1/6*((2*a^3*x^3 - a^2*x^2 + a*x + 4)*sqrt((a*x - 1)/(a*x + 1)) - 3*log(sqr t((a*x - 1)/(a*x + 1)) + 1) + 3*log(sqrt((a*x - 1)/(a*x + 1)) - 1))/a^3
\[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\int x^{2} \sqrt {\frac {a x - 1}{a x + 1}}\, dx \] Input:
integrate(x**2*((a*x-1)/(a*x+1))**(1/2),x)
Output:
Integral(x**2*sqrt((a*x - 1)/(a*x + 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (74) = 148\).
Time = 0.03 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.84 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=-\frac {1}{6} \, a {\left (\frac {2 \, {\left (9 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 4 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{4}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{4}} - \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{4}}\right )} \] Input:
integrate(x^2*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")
Output:
-1/6*a*(2*(9*((a*x - 1)/(a*x + 1))^(5/2) - 4*((a*x - 1)/(a*x + 1))^(3/2) + 3*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) + 3*log(sqrt((a*x - 1) /(a*x + 1)) + 1)/a^4 - 3*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^4)
Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {1}{6} \, \sqrt {a^{2} x^{2} - 1} {\left (x {\left (\frac {2 \, x \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {3 \, \mathrm {sgn}\left (a x + 1\right )}{a^{2}}\right )} + \frac {4 \, \mathrm {sgn}\left (a x + 1\right )}{a^{3}}\right )} + \frac {\log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{2 \, a^{2} {\left | a \right |}} \] Input:
integrate(x^2*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")
Output:
1/6*sqrt(a^2*x^2 - 1)*(x*(2*x*sgn(a*x + 1)/a - 3*sgn(a*x + 1)/a^2) + 4*sgn (a*x + 1)/a^3) + 1/2*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/ (a^2*abs(a))
Time = 0.06 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.49 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\left (a\,x-1\right )}{a\,x+1}}-\frac {\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^3} \] Input:
int(x^2*((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
(((a*x - 1)/(a*x + 1))^(1/2) - (4*((a*x - 1)/(a*x + 1))^(3/2))/3 + 3*((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)) - atanh(((a*x - 1)/(a* x + 1))^(1/2))/a^3
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86 \[ \int e^{-\coth ^{-1}(a x)} x^2 \, dx=\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +4 \sqrt {a x +1}\, \sqrt {a x -1}-6 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )}{6 a^{3}} \] Input:
int(x^2*((a*x-1)/(a*x+1))^(1/2),x)
Output:
(2*sqrt(a*x + 1)*sqrt(a*x - 1)*a**2*x**2 - 3*sqrt(a*x + 1)*sqrt(a*x - 1)*a *x + 4*sqrt(a*x + 1)*sqrt(a*x - 1) - 6*log((sqrt(a*x - 1) + sqrt(a*x + 1)) /sqrt(2)))/(6*a**3)