Integrand size = 15, antiderivative size = 126 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \]
2/3*(1+1/x)^(3/2)*x^2*(1-1/x)^(1/2)/(1-x)^(1/2)+2*x*(1-1/x)^(1/2)*(1+1/x)^ (1/2)/(1-x)^(1/2)-2*arctanh(2^(1/2)*(1/x)^(1/2)/(1+1/x)^(1/2))*2^(1/2)*(1- 1/x)^(1/2)/(1-x)^(1/2)/(1/x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} x \left (\sqrt {1+\frac {1}{x}} (4+x)-3 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )\right )}{3 \sqrt {1-x}} \]
(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 = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.78, 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 |
\(\displaystyle -\frac {\sqrt {1-\frac {1}{x}} \left (2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )-\frac {2 \left (\frac {1}{x}+1\right )^{3/2}}{3 \left (\frac {1}{x}\right )^{3/2}}-\frac {2 \sqrt {\frac {1}{x}+1}}{\sqrt {\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\) |
-((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)]))
3.4.29.3.1 Defintions of rubi rules used
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 = 0.46 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.52
method | result | size |
default | \(\frac {2 \sqrt {1-x}\, \left (3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )-\sqrt {-1-x}\, x -4 \sqrt {-1-x}\right )}{3 \sqrt {\frac {x -1}{1+x}}\, \sqrt {-1-x}}\) | \(66\) |
risch | \(\frac {2 \left (x +4\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{3 \sqrt {-1-x}\, \sqrt {\frac {x -1}{1+x}}\, \sqrt {1-x}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{\sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\) | \(111\) |
2/3/((x-1)/(1+x))^(1/2)*(1-x)^(1/2)*(3*2^(1/2)*arctan(1/2*(-1-x)^(1/2)*2^( 1/2))-(-1-x)^(1/2)*x-4*(-1-x)^(1/2))/(-1-x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.57 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {2} {\left (x - 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{x - 1}\right ) - {\left (x^{2} + 5 \, x + 4\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}\right )}}{3 \, {\left (x - 1\right )}} \]
2/3*(3*sqrt(2)*(x - 1)*arctan(sqrt(2)*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)
\[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \sqrt {1 - x}}\, dx \]
\[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\int { \frac {x}{\sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}} \,d x } \]
Exception generated. \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\text {Exception raised: NotImplementedError} \]
Exception raised: NotImplementedError >> unable to parse Giac output: (-12 *atan(i)+20*i)*1/3/sqrt(2)*sign(sageVARx+1)-(-2/3*sqrt(-sageVARx-1)*(-sage VARx-1)+2*sqrt(-sageVARx-1)+1/3*(12*atan(i)-20*i)/sqrt(2)-4*atan(sqrt(-sag eVARx-1)/sqrt(2))/s
Timed out. \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\int \frac {x}{\sqrt {\frac {x-1}{x+1}}\,\sqrt {1-x}} \,d x \]