Integrand size = 8, antiderivative size = 96 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx=\left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{1+\frac {1}{x}} x-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{\sqrt {3}}-\log \left (\sqrt [3]{1-\frac {1}{x}}-\sqrt [3]{1+\frac {1}{x}}\right )-\frac {\log (x)}{3} \] Output:
(1-1/x)^(2/3)*(1+1/x)^(1/3)*x-2/3*arctan(1/3*3^(1/2)+2/3*(1-1/x)^(1/3)*3^( 1/2)/(1+1/x)^(1/3))*3^(1/2)-ln((1-1/x)^(1/3)-(1+1/x)^(1/3))-1/3*ln(x)
Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx=\frac {1}{3} \left (\frac {6 e^{\frac {2}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+2 \sqrt {3} \arctan \left (\frac {1+2 e^{\frac {2}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-2 \log \left (1-e^{\frac {2}{3} \coth ^{-1}(x)}\right )+\log \left (1+e^{\frac {2}{3} \coth ^{-1}(x)}+e^{\frac {4}{3} \coth ^{-1}(x)}\right )\right ) \] Input:
Integrate[E^((2*ArcCoth[x])/3),x]
Output:
((6*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x])) + 2*Sqrt[3]*ArcTan[(1 + 2*E^((2*ArcCoth[x])/3))/Sqrt[3]] - 2*Log[1 - E^((2*ArcCoth[x])/3)] + Log[1 + E^((2*ArcCoth[x])/3) + E^((4*ArcCoth[x])/3)])/3
Time = 0.37 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6720, 105, 102}
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.
| \(\displaystyle \int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx\) |
| \(\Big \downarrow \) 6720 |
| \(\displaystyle -\int \frac {\sqrt [3]{1+\frac {1}{x}} x^2}{\sqrt [3]{1-\frac {1}{x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x-\frac {2}{3} \int \frac {x}{\sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x-\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )+\frac {3}{2} \log \left (\sqrt [3]{1-\frac {1}{x}}-\sqrt [3]{\frac {1}{x}+1}\right )-\frac {1}{2} \log \left (\frac {1}{x}\right )\right )\) |
Input:
Int[E^((2*ArcCoth[x])/3),x]
Output:
(1 - x^(-1))^(2/3)*(1 + x^(-1))^(1/3)*x - (2*(Sqrt[3]*ArcTan[1/Sqrt[3] + ( 2*(1 - x^(-1))^(1/3))/(Sqrt[3]*(1 + x^(-1))^(1/3))] + (3*Log[(1 - x^(-1))^ (1/3) - (1 + x^(-1))^(1/3)])/2 - Log[x^(-1)]/2))/3
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.) *(x_))), x_] :> With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])* q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q *(a + b*x)^(1/3) - (c + d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, 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[E^(ArcCoth[(a_.)*(x_)]*(n_)), x_Symbol] :> -Subst[Int[(1 + x/a)^(n/2)/( x^2*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && !IntegerQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.63 (sec) , antiderivative size = 397, normalized size of antiderivative = 4.14
| method | result | size |
| risch | \(\frac {x -1}{\left (\frac {x -1}{1+x}\right )^{\frac {1}{3}}}+\frac {\left (-\frac {2 \ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-1}{1+x}\right )}{3}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+4 x +2}{1+x}\right )}{3}\right ) \left (\left (x -1\right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}{\left (\frac {x -1}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )}\) | \(397\) |
| trager | \(\left (1+x \right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-\frac {2 \ln \left (9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x -36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}+12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-x -1\right )}{3}+\frac {2 \ln \left (-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-x +1\right )}{3}-2 \ln \left (-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-x +1\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )\) | \(503\) |
Input:
int(1/((x-1)/(1+x))^(1/3),x,method=_RETURNVERBOSE)
Output:
(x-1)/((x-1)/(1+x))^(1/3)+(-2/3*ln(-(4*RootOf(_Z^2-_Z+1)^2*x^2+4*RootOf(_Z ^2-_Z+1)^2*x+3*RootOf(_Z^2-_Z+1)*(x^3+x^2-x-1)^(2/3)-3*RootOf(_Z^2-_Z+1)*( x^3+x^2-x-1)^(1/3)*x-4*RootOf(_Z^2-_Z+1)*x^2-3*RootOf(_Z^2-_Z+1)*(x^3+x^2- x-1)^(1/3)-2*RootOf(_Z^2-_Z+1)*x+3*(x^3+x^2-x-1)^(1/3)*x+x^2+2*RootOf(_Z^2 -_Z+1)+3*(x^3+x^2-x-1)^(1/3)-1)/(1+x))+2/3*RootOf(_Z^2-_Z+1)*ln((2*RootOf( _Z^2-_Z+1)^2*x^2+2*RootOf(_Z^2-_Z+1)^2*x+3*RootOf(_Z^2-_Z+1)*(x^3+x^2-x-1) ^(2/3)-5*RootOf(_Z^2-_Z+1)*x^2-6*RootOf(_Z^2-_Z+1)*x-3*(x^3+x^2-x-1)^(2/3) +3*(x^3+x^2-x-1)^(1/3)*x+2*x^2-RootOf(_Z^2-_Z+1)+3*(x^3+x^2-x-1)^(1/3)+4*x +2)/(1+x)))/((x-1)/(1+x))^(1/3)*((x-1)*(1+x)^2)^(1/3)/(1+x)
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx={\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \] Input:
integrate(1/((x-1)/(1+x))^(1/3),x, algorithm="fricas")
Output:
(x + 1)*((x - 1)/(x + 1))^(2/3) - 2/3*sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/ (x + 1))^(1/3) + 1/3*sqrt(3)) + 1/3*log(((x - 1)/(x + 1))^(2/3) + ((x - 1) /(x + 1))^(1/3) + 1) - 2/3*log(((x - 1)/(x + 1))^(1/3) - 1)
\[ \int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx=\int \frac {1}{\sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \] Input:
integrate(1/((x-1)/(1+x))**(1/3),x)
Output:
Integral(((x - 1)/(x + 1))**(-1/3), x)
Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx=-\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) - \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{\frac {x - 1}{x + 1} - 1} + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \] Input:
integrate(1/((x-1)/(1+x))^(1/3),x, algorithm="maxima")
Output:
-2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) - 2*((x - 1)/(x + 1))^(2/3)/((x - 1)/(x + 1) - 1) + 1/3*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) - 2/3*log(((x - 1)/(x + 1))^(1/3) - 1)
Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx=-\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) - \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{\frac {x - 1}{x + 1} - 1} + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{3} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1 \right |}\right ) \] Input:
integrate(1/((x-1)/(1+x))^(1/3),x, algorithm="giac")
Output:
-2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) - 2*((x - 1)/(x + 1))^(2/3)/((x - 1)/(x + 1) - 1) + 1/3*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) - 2/3*log(abs(((x - 1)/(x + 1))^(1/3) - 1) )
Time = 23.39 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.23 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx=-\frac {2\,\ln \left (4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}-4\right )}{3}-\ln \left (4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}-9\,{\left (-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )}^2\right )\,\left (-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )+\ln \left (4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}-9\,{\left (\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )}^2\right )\,\left (\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )-\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{\frac {x-1}{x+1}-1} \] Input:
int(1/((x - 1)/(x + 1))^(1/3),x)
Output:
log(4*((x - 1)/(x + 1))^(1/3) - 9*((3^(1/2)*1i)/3 + 1/3)^2)*((3^(1/2)*1i)/ 3 + 1/3) - log(4*((x - 1)/(x + 1))^(1/3) - 9*((3^(1/2)*1i)/3 - 1/3)^2)*((3 ^(1/2)*1i)/3 - 1/3) - (2*log(4*((x - 1)/(x + 1))^(1/3) - 4))/3 - (2*((x - 1)/(x + 1))^(2/3))/((x - 1)/(x + 1) - 1)
\[ \int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx=\int \frac {\left (x +1\right )^{\frac {1}{3}}}{\left (x -1\right )^{\frac {1}{3}}}d x \] Input:
int(1/((x-1)/(1+x))^(1/3),x)
Output:
int((x + 1)**(1/3)/(x - 1)**(1/3),x)