Integrand size = 14, antiderivative size = 90 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx=-\frac {\sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{(1-x)^{3/2}}-\frac {\left (1-\frac {1}{x}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \] Output:
-(1-1/x)^(1/2)*(1+1/x)^(1/2)*x/(1-x)^(3/2)-1/2*(1-1/x)^(3/2)*arctanh(2^(1/ 2)*(1/x)^(1/2)/(1+1/x)^(1/2))*2^(1/2)/(1-x)^(3/2)/(1/x)^(3/2)
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.77 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx=-\frac {\sqrt {\frac {-1+x}{x}} x \left (2 \sqrt {1+\frac {1}{x}}+\sqrt {2} (-1+x) \sqrt {\frac {1}{x}} \text {arctanh}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )\right )}{2 (1-x)^{3/2}} \] Input:
Integrate[E^ArcCoth[x]/(1 - x)^(3/2),x]
Output:
-1/2*(Sqrt[(-1 + x)/x]*x*(2*Sqrt[1 + x^(-1)] + Sqrt[2]*(-1 + x)*Sqrt[x^(-1 )]*ArcTanh[Sqrt[2]*Sqrt[(1 + x)^(-1)]]))/(1 - x)^(3/2)
Time = 0.40 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6727, 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.
\(\displaystyle \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (1-\frac {1}{x}\right )^{3/2} \int \frac {\sqrt {1+\frac {1}{x}}}{\left (1-\frac {1}{x}\right )^2 \sqrt {\frac {1}{x}}}d\frac {1}{x}}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {\left (1-\frac {1}{x}\right )^{3/2} \left (\frac {1}{2} \int \frac {1}{\left (1-\frac {1}{x}\right ) \sqrt {1+\frac {1}{x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}+\frac {\sqrt {\frac {1}{x}+1} \sqrt {\frac {1}{x}}}{1-\frac {1}{x}}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {\left (1-\frac {1}{x}\right )^{3/2} \left (\int \frac {1}{1-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}+\frac {\sqrt {\frac {1}{x}+1} \sqrt {\frac {1}{x}}}{1-\frac {1}{x}}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\left (1-\frac {1}{x}\right )^{3/2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {2}}+\frac {\sqrt {\frac {1}{x}+1} \sqrt {\frac {1}{x}}}{1-\frac {1}{x}}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\) |
Input:
Int[E^ArcCoth[x]/(1 - x)^(3/2),x]
Output:
-(((1 - x^(-1))^(3/2)*((Sqrt[1 + x^(-1)]*Sqrt[x^(-1)])/(1 - x^(-1)) + ArcT anh[(Sqrt[2]*Sqrt[x^(-1)])/Sqrt[1 + x^(-1)]]/Sqrt[2]))/((1 - x)^(3/2)*(x^( -1))^(3/2)))
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_)^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_.))*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Si mp[(-(1/x)^p)*((c + d*x)^p/(1 + c/(d*x))^p) Subst[Int[((1 + c*(x/d))^p*(( 1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a)^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[p]
Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {\left (x -1\right ) \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-x -1}\, \sqrt {2}}{2}\right ) x -\sqrt {2}\, \arctan \left (\frac {\sqrt {-x -1}\, \sqrt {2}}{2}\right )+2 \sqrt {-x -1}\right )}{2 \sqrt {\frac {x -1}{1+x}}\, \left (1-x \right )^{\frac {3}{2}} \sqrt {-x -1}}\) | \(77\) |
risch | \(\frac {\sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}}{\sqrt {-x -1}\, \sqrt {\frac {x -1}{1+x}}\, \sqrt {1-x}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {-x -1}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{2 \sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\) | \(104\) |
Input:
int(1/((x-1)/(1+x))^(1/2)/(1-x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2/((x-1)/(1+x))^(1/2)*(x-1)*(2^(1/2)*arctan(1/2*(-x-1)^(1/2)*2^(1/2))*x -2^(1/2)*arctan(1/2*(-x-1)^(1/2)*2^(1/2))+2*(-x-1)^(1/2))/(1-x)^(3/2)/(-x- 1)^(1/2)
Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx=-\frac {\sqrt {2} {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {2} {\left (x + 1\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{2 \, {\left (x - 1\right )}}\right ) + 2 \, {\left (x + 1\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)/(1-x)^(3/2),x, algorithm="fricas")
Output:
-1/2*(sqrt(2)*(x^2 - 2*x + 1)*arctan(1/2*sqrt(2)*(x + 1)*sqrt(-x + 1)*sqrt ((x - 1)/(x + 1))/(x - 1)) + 2*(x + 1)*sqrt(-x + 1)*sqrt((x - 1)/(x + 1))) /(x^2 - 2*x + 1)
Time = 85.63 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx=- 2 \left (\begin {cases} \frac {\sqrt {2} \left (\frac {\operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}}{2} - \frac {\sqrt {2} \sqrt {1 - \frac {2}{1 - x}}}{2 \sqrt {1 - x}}\right )}{2} & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) \] Input:
integrate(1/((x-1)/(1+x))**(1/2)/(1-x)**(3/2),x)
Output:
-2*Piecewise((sqrt(2)*(acos(sqrt(2)/sqrt(1 - x))/2 - sqrt(2)*sqrt(1 - 2/(1 - x))/(2*sqrt(1 - x)))/2, (sqrt(1 - x) < sqrt(2)) & (sqrt(1 - x) > -sqrt( 2))))
\[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx=\int { \frac {1}{{\left (-x + 1\right )}^{\frac {3}{2}} \sqrt {\frac {x - 1}{x + 1}}} \,d x } \] Input:
integrate(1/((x-1)/(1+x))^(1/2)/(1-x)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-x + 1)^(3/2)*sqrt((x - 1)/(x + 1))), x)
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.44 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-x - 1}\right ) + \frac {2 \, \sqrt {-x - 1}}{x - 1}}{2 \, \mathrm {sgn}\left (x + 1\right )} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)/(1-x)^(3/2),x, algorithm="giac")
Output:
1/2*(sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-x - 1)) + 2*sqrt(-x - 1)/(x - 1))/sg n(x + 1)
Timed out. \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx=\int \frac {1}{\sqrt {\frac {x-1}{x+1}}\,{\left (1-x\right )}^{3/2}} \,d x \] Input:
int(1/(((x - 1)/(x + 1))^(1/2)*(1 - x)^(3/2)),x)
Output:
int(1/(((x - 1)/(x + 1))^(1/2)*(1 - x)^(3/2)), x)
Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx=\frac {i \left (2 \sqrt {x +1}-\sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )\right ) x +\sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )\right )\right )}{2 x -2} \] Input:
int(1/((x-1)/(1+x))^(1/2)/(1-x)^(3/2),x)
Output:
(i*(2*sqrt(x + 1) - sqrt(2)*log(tan(asin(sqrt( - x + 1)/sqrt(2))/2))*x + s qrt(2)*log(tan(asin(sqrt( - x + 1)/sqrt(2))/2))))/(2*(x - 1))