Integrand size = 15, antiderivative size = 110 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\frac {8 \sqrt {1-\frac {1}{x^2}} x}{3 \sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x^2}} x^2}{3 \sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \] Output:
8/3*(1-1/x^2)^(1/2)*x/(1-x)^(1/2)+2/3*(1-1/x^2)^(1/2)*x^2/(1-x)^(1/2)-2*2^ (1/2)*(1-1/x)^(1/2)*arctanh(2^(1/2)*(1/x)^(1/2)/(1+1/x)^(1/2))/(1-x)^(1/2) /(1/x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} x \left (\sqrt {1+\frac {1}{x}} (4+x)-3 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )\right )}{3 \sqrt {1-x}} \] Input:
Integrate[(E^ArcCoth[x]*x)/Sqrt[1 - x],x]
Output:
(2*Sqrt[(-1 + x)/x]*x*(Sqrt[1 + x^(-1)]*(4 + x) - 3*Sqrt[2]*Sqrt[x^(-1)]*A rcTanh[Sqrt[2]*Sqrt[(1 + x)^(-1)]]))/(3*Sqrt[1 - x])
Time = 0.43 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6730, 107, 105, 104, 219}
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.
\(\displaystyle \int \frac {x e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\sqrt {1-\frac {1}{x}} \int \frac {\sqrt {1+\frac {1}{x}}}{\left (1-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\) |
\(\Big \downarrow \) 107 |
\(\displaystyle -\frac {\sqrt {1-\frac {1}{x}} \left (\int \frac {\sqrt {1+\frac {1}{x}}}{\left (1-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {2 \left (\frac {1}{x}+1\right )^{3/2}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {\sqrt {1-\frac {1}{x}} \left (2 \int \frac {1}{\left (1-\frac {1}{x}\right ) \sqrt {1+\frac {1}{x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-\frac {2 \left (\frac {1}{x}+1\right )^{3/2}}{3 \left (\frac {1}{x}\right )^{3/2}}-\frac {2 \sqrt {\frac {1}{x}+1}}{\sqrt {\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {\sqrt {1-\frac {1}{x}} \left (4 \int \frac {1}{1-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}-\frac {2 \left (\frac {1}{x}+1\right )^{3/2}}{3 \left (\frac {1}{x}\right )^{3/2}}-\frac {2 \sqrt {\frac {1}{x}+1}}{\sqrt {\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\sqrt {1-\frac {1}{x}} \left (2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )-\frac {2 \left (\frac {1}{x}+1\right )^{3/2}}{3 \left (\frac {1}{x}\right )^{3/2}}-\frac {2 \sqrt {\frac {1}{x}+1}}{\sqrt {\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\) |
Input:
Int[(E^ArcCoth[x]*x)/Sqrt[1 - x],x]
Output:
-((Sqrt[1 - x^(-1)]*((-2*(1 + x^(-1))^(3/2))/(3*(x^(-1))^(3/2)) - (2*Sqrt[ 1 + x^(-1)])/Sqrt[x^(-1)] + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/Sqrt[ 1 + x^(-1)]]))/(Sqrt[1 - x]*Sqrt[x^(-1)]))
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p _), x_Symbol] :> Simp[(-(e*x)^m)*(1/x)^(m + p)*((c + d*x)^p/(1 + c/(d*x))^p ) Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(m + p + 2)))/(1 - x/a)^( n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[a^2*c^2 - d ^2, 0] && !IntegerQ[p]
Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62
method | result | size |
default | \(\frac {2 \left (x -1\right ) \left (-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-x -1}\, \sqrt {2}}{2}\right )+\sqrt {-x -1}\, x +4 \sqrt {-x -1}\right )}{3 \sqrt {\frac {x -1}{1+x}}\, \sqrt {1-x}\, \sqrt {-x -1}}\) | \(68\) |
risch | \(\frac {2 \left (4+x \right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{3 \sqrt {-x -1}\, \sqrt {\frac {x -1}{1+x}}\, \sqrt {1-x}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {-x -1}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{\sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\) | \(111\) |
Input:
int(1/((x-1)/(1+x))^(1/2)*x/(1-x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3*(x-1)*(-3*2^(1/2)*arctan(1/2*(-x-1)^(1/2)*2^(1/2))+(-x-1)^(1/2)*x+4*(- x-1)^(1/2))/((x-1)/(1+x))^(1/2)/(1-x)^(1/2)/(-x-1)^(1/2)
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.69 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {2} {\left (x - 1\right )} \arctan \left (\frac {\sqrt {2} {\left (x + 1\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{2 \, {\left (x - 1\right )}}\right ) - {\left (x^{2} + 5 \, x + 4\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}\right )}}{3 \, {\left (x - 1\right )}} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*x/(1-x)^(1/2),x, algorithm="fricas")
Output:
2/3*(3*sqrt(2)*(x - 1)*arctan(1/2*sqrt(2)*(x + 1)*sqrt(-x + 1)*sqrt((x - 1 )/(x + 1))/(x - 1)) - (x^2 + 5*x + 4)*sqrt(-x + 1)*sqrt((x - 1)/(x + 1)))/ (x - 1)
\[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \sqrt {1 - x}}\, dx \] Input:
integrate(1/((x-1)/(1+x))**(1/2)*x/(1-x)**(1/2),x)
Output:
Integral(x/(sqrt((x - 1)/(x + 1))*sqrt(1 - x)), x)
\[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\int { \frac {x}{\sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}} \,d x } \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*x/(1-x)^(1/2),x, algorithm="maxima")
Output:
integrate(x/(sqrt(-x + 1)*sqrt((x - 1)/(x + 1))), x)
Time = 0.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.45 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=-\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-x - 1}\right )}{\mathrm {sgn}\left (x + 1\right )} - \frac {2 \, {\left ({\left (-x - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {-x - 1}\right )}}{3 \, \mathrm {sgn}\left (x + 1\right )} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*x/(1-x)^(1/2),x, algorithm="giac")
Output:
-2*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-x - 1))/sgn(x + 1) - 2/3*((-x - 1)^(3/ 2) - 3*sqrt(-x - 1))/sgn(x + 1)
Timed out. \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\int \frac {x}{\sqrt {\frac {x-1}{x+1}}\,\sqrt {1-x}} \,d x \] Input:
int(x/(((x - 1)/(x + 1))^(1/2)*(1 - x)^(1/2)),x)
Output:
int(x/(((x - 1)/(x + 1))^(1/2)*(1 - x)^(1/2)), x)
Time = 0.15 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.36 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\frac {2 i \left (\sqrt {x +1}\, x +4 \sqrt {x +1}+3 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )\right )-5 \sqrt {2}\right )}{3} \] Input:
int(1/((x-1)/(1+x))^(1/2)*x/(1-x)^(1/2),x)
Output:
(2*i*(sqrt(x + 1)*x + 4*sqrt(x + 1) + 3*sqrt(2)*log(tan(asin(sqrt( - x + 1 )/sqrt(2))/2)) - 5*sqrt(2)))/3