Integrand size = 13, antiderivative size = 87 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1+x)^{3/2}} \, dx=\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{(1+x)^{3/2}}+\frac {\sqrt {2} \left (1+\frac {1}{x}\right )^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1-\frac {1}{x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}} \] Output:
2*(1-1/x)^(1/2)*(1+1/x)^(3/2)*x^2/(1+x)^(3/2)+2^(1/2)*(1+1/x)^(3/2)*arctan (2^(1/2)*(1/x)^(1/2)/(1-1/x)^(1/2))/(1/x)^(3/2)/(1+x)^(3/2)
Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1+x)^{3/2}} \, dx=\frac {\sqrt {1+\frac {1}{x}} x \left (2 \sqrt {\frac {-1+x}{x}}-\sqrt {2} \sqrt {\frac {1}{x}} \arctan \left (\frac {\sqrt {\frac {-1+x}{x^2}} x}{\sqrt {2}}\right )\right )}{\sqrt {1+x}} \] Input:
Integrate[(E^ArcCoth[x]*x)/(1 + x)^(3/2),x]
Output:
(Sqrt[1 + x^(-1)]*x*(2*Sqrt[(-1 + x)/x] - Sqrt[2]*Sqrt[x^(-1)]*ArcTan[(Sqr t[(-1 + x)/x^2]*x)/Sqrt[2]]))/Sqrt[1 + x]
Time = 0.41 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6730, 107, 104, 216}
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.
\(\displaystyle \int \frac {x e^{\coth ^{-1}(x)}}{(x+1)^{3/2}} \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\left (\frac {1}{x}+1\right )^{3/2} \int \frac {1}{\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 107 |
\(\displaystyle -\frac {\left (\frac {1}{x}+1\right )^{3/2} \left (-\int \frac {1}{\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right ) \sqrt {\frac {1}{x}}}d\frac {1}{x}-\frac {2 \sqrt {1-\frac {1}{x}}}{\sqrt {\frac {1}{x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {\left (\frac {1}{x}+1\right )^{3/2} \left (-2 \int \frac {1}{1+\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1-\frac {1}{x}}}-\frac {2 \sqrt {1-\frac {1}{x}}}{\sqrt {\frac {1}{x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {\left (\frac {1}{x}+1\right )^{3/2} \left (-\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1-\frac {1}{x}}}\right )-\frac {2 \sqrt {1-\frac {1}{x}}}{\sqrt {\frac {1}{x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2}}\) |
Input:
Int[(E^ArcCoth[x]*x)/(1 + x)^(3/2),x]
Output:
-(((1 + x^(-1))^(3/2)*((-2*Sqrt[1 - x^(-1)])/Sqrt[x^(-1)] - Sqrt[2]*ArcTan [(Sqrt[2]*Sqrt[x^(-1)])/Sqrt[1 - x^(-1)]]))/((x^(-1))^(3/2)*(1 + x)^(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[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]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[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 = 0.14 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.54
method | result | size |
default | \(-\frac {\sqrt {x -1}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {x -1}\, \sqrt {2}}{2}\right )-2 \sqrt {x -1}\right )}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\) | \(47\) |
risch | \(\frac {2 x -2}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x -1}\, \sqrt {2}}{2}\right ) \sqrt {x -1}}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\) | \(60\) |
Input:
int(1/((x-1)/(1+x))^(1/2)*x/(1+x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-(x-1)^(1/2)*(2^(1/2)*arctan(1/2*(x-1)^(1/2)*2^(1/2))-2*(x-1)^(1/2))/((x-1 )/(1+x))^(1/2)/(1+x)^(1/2)
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1+x)^{3/2}} \, dx=-\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}}\right ) + 2 \, \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*x/(1+x)^(3/2),x, algorithm="fricas")
Output:
-sqrt(2)*arctan(1/2*sqrt(2)*sqrt(x + 1)*sqrt((x - 1)/(x + 1))) + 2*sqrt(x + 1)*sqrt((x - 1)/(x + 1))
\[ \int \frac {e^{\coth ^{-1}(x)} x}{(1+x)^{3/2}} \, dx=\int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \left (x + 1\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/((x-1)/(1+x))**(1/2)*x/(1+x)**(3/2),x)
Output:
Integral(x/(sqrt((x - 1)/(x + 1))*(x + 1)**(3/2)), x)
\[ \int \frac {e^{\coth ^{-1}(x)} x}{(1+x)^{3/2}} \, dx=\int { \frac {x}{{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {\frac {x - 1}{x + 1}}} \,d x } \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*x/(1+x)^(3/2),x, algorithm="maxima")
Output:
integrate(x/((x + 1)^(3/2)*sqrt((x - 1)/(x + 1))), x)
Exception generated. \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1+x)^{3/2}} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*x/(1+x)^(3/2),x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: (2*a tan(i)-4*i)/sqrt(2)*sign(sageVARx+1)+2*(sqrt(sageVARx-1)/sign(sageVARx+1)- atan(sqrt(sageVARx-1)/sqrt(2))/sqrt(2)/sign(sageVARx+1))
Timed out. \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1+x)^{3/2}} \, dx=\int \frac {x}{\sqrt {\frac {x-1}{x+1}}\,{\left (x+1\right )}^{3/2}} \,d x \] Input:
int(x/(((x - 1)/(x + 1))^(1/2)*(x + 1)^(3/2)),x)
Output:
int(x/(((x - 1)/(x + 1))^(1/2)*(x + 1)^(3/2)), x)
Time = 0.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.24 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1+x)^{3/2}} \, dx=-\sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x -1}}{\sqrt {2}}\right )+2 \sqrt {x -1} \] Input:
int(1/((x-1)/(1+x))^(1/2)*x/(1+x)^(3/2),x)
Output:
- sqrt(2)*atan(sqrt(x - 1)/sqrt(2)) + 2*sqrt(x - 1)