3.121 \(\int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx\)

Optimal. Leaf size=130 \[ \frac {1}{2} \left (\frac {1}{x}+1\right )^{4/3} \left (\frac {x-1}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x-\frac {1}{3} \log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {\log (x)}{9}-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]

[Out]

1/3*(1/x+1)^(1/3)*((-1+x)/x)^(2/3)*x+1/2*(1/x+1)^(4/3)*((-1+x)/x)^(2/3)*x^2-1/3*ln((1/x+1)^(1/3)-((-1+x)/x)^(1
/3))-1/9*ln(x)-2/9*arctan(1/3*3^(1/2)+2/3*((-1+x)/x)^(1/3)/(1/x+1)^(1/3)*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6171, 96, 94, 91} \[ \frac {1}{2} \left (\frac {1}{x}+1\right )^{4/3} \left (\frac {x-1}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x-\frac {1}{3} \log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {\log (x)}{9}-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[E^((2*ArcCoth[x])/3)*x,x]

[Out]

((1 + x^(-1))^(1/3)*((-1 + x)/x)^(2/3)*x)/3 + ((1 + x^(-1))^(4/3)*((-1 + x)/x)^(2/3)*x^2)/2 - (2*ArcTan[1/Sqrt
[3] + (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + x^(-1))^(1/3))])/(3*Sqrt[3]) - Log[(1 + x^(-1))^(1/3) - ((-1 + x)/x
)^(1/3)]/3 - Log[x]/9

Rule 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> 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]

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(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[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[(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] || Sum
SimplerQ[m, 1])

Rule 6171

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2
)), x], x, 1/x] /; FreeQ[{a, n}, x] &&  !IntegerQ[n] && IntegerQ[m]

Rubi steps

\begin {align*} \int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3} x^2-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3} x^2-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3} x^2-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-\frac {1-x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{-\frac {1-x}{x}}\right )-\frac {\log (x)}{9}\\ \end {align*}

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Mathematica [A]  time = 0.49, size = 165, normalized size = 1.27 \[ \frac {1}{9} \left (\frac {24 e^{\frac {2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}+\frac {18 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (e^{2 \coth ^{-1}(x)}-1\right )^2}-2 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )-2 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}+1\right )+\log \left (-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}+1\right )+\log \left (e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 e^{\frac {1}{3} \coth ^{-1}(x)}-1}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 e^{\frac {1}{3} \coth ^{-1}(x)}+1}{\sqrt {3}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^((2*ArcCoth[x])/3)*x,x]

[Out]

((18*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x]))^2 + (24*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x])) + 2*S
qrt[3]*ArcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Log[
1 - E^(ArcCoth[x]/3)] - 2*Log[1 + E^(ArcCoth[x]/3)] + Log[1 - E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)] + Log[1
+ E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)])/9

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fricas [A]  time = 0.58, size = 95, normalized size = 0.73 \[ \frac {1}{6} \, {\left (3 \, x^{2} + 8 \, x + 5\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/3)*x,x, algorithm="fricas")

[Out]

1/6*(3*x^2 + 8*x + 5)*((x - 1)/(x + 1))^(2/3) - 2/9*sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/3) + 1/3*s
qrt(3)) + 1/9*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) - 2/9*log(((x - 1)/(x + 1))^(1/3) - 1
)

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giac [A]  time = 0.17, size = 120, normalized size = 0.92 \[ -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) - \frac {2 \, {\left (\frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{x + 1} - 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{3 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{2}} + \frac {1}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/3)*x,x, algorithm="giac")

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) - 2/3*((x - 1)*((x - 1)/(x + 1))^(2/3)/(x + 1
) - 4*((x - 1)/(x + 1))^(2/3))/((x - 1)/(x + 1) - 1)^2 + 1/9*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(
1/3) + 1) - 2/9*log(abs(((x - 1)/(x + 1))^(1/3) - 1))

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maple [C]  time = 0.58, size = 403, normalized size = 3.10 \[ \frac {\left (5+3 x \right ) \left (-1+x \right )}{6 \left (\frac {-1+x}{1+x}\right )^{\frac {1}{3}}}+\frac {\left (-\frac {2 \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{1+x}\right )}{9}+\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -5 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +2 x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+4 x +2}{1+x}\right )}{9}\right ) \left (\left (-1+x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}{\left (\frac {-1+x}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((-1+x)/(1+x))^(1/3)*x,x)

[Out]

1/6*(5+3*x)*(-1+x)/((-1+x)/(1+x))^(1/3)+(-2/9*ln(-(4*RootOf(_Z^2-_Z+1)^2*x^2+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)^2*x-4*RootOf(_Z^2-_Z+1)*x^2-3*RootOf(_Z^2
-_Z+1)*(x^3+x^2-x-1)^(1/3)+3*(x^3+x^2-x-1)^(1/3)*x-2*RootOf(_Z^2-_Z+1)*x+x^2+3*(x^3+x^2-x-1)^(1/3)+2*RootOf(_Z
^2-_Z+1)-1)/(1+x))+2/9*RootOf(_Z^2-_Z+1)*ln((2*RootOf(_Z^2-_Z+1)^2*x^2+3*RootOf(_Z^2-_Z+1)*(x^3+x^2-x-1)^(2/3)
+2*RootOf(_Z^2-_Z+1)^2*x-5*RootOf(_Z^2-_Z+1)*x^2-3*(x^3+x^2-x-1)^(2/3)+3*(x^3+x^2-x-1)^(1/3)*x-6*RootOf(_Z^2-_
Z+1)*x+2*x^2+3*(x^3+x^2-x-1)^(1/3)-RootOf(_Z^2-_Z+1)+4*x+2)/(1+x)))/((-1+x)/(1+x))^(1/3)*((-1+x)*(1+x)^2)^(1/3
)/(1+x)

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maxima [A]  time = 0.40, size = 123, normalized size = 0.95 \[ -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) + \frac {2 \, {\left (\left (\frac {x - 1}{x + 1}\right )^{\frac {5}{3}} - 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{3 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} - \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1\right )}} + \frac {1}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/3)*x,x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) + 2/3*(((x - 1)/(x + 1))^(5/3) - 4*((x - 1)/(
x + 1))^(2/3))/(2*(x - 1)/(x + 1) - (x - 1)^2/(x + 1)^2 - 1) + 1/9*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x +
1))^(1/3) + 1) - 2/9*log(((x - 1)/(x + 1))^(1/3) - 1)

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mupad [B]  time = 0.05, size = 145, normalized size = 1.12 \[ \frac {\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{3}-\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{5/3}}{3}}{\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}-\frac {2\,\left (x-1\right )}{x+1}+1}-\frac {2\,\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}-\frac {4}{9}\right )}{9}-\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}-9\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}-9\,{\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((x - 1)/(x + 1))^(1/3),x)

[Out]

((8*((x - 1)/(x + 1))^(2/3))/3 - (2*((x - 1)/(x + 1))^(5/3))/3)/((x - 1)^2/(x + 1)^2 - (2*(x - 1))/(x + 1) + 1
) - (2*log((4*((x - 1)/(x + 1))^(1/3))/9 - 4/9))/9 - log((4*((x - 1)/(x + 1))^(1/3))/9 - 9*((3^(1/2)*1i)/9 - 1
/9)^2)*((3^(1/2)*1i)/9 - 1/9) + log((4*((x - 1)/(x + 1))^(1/3))/9 - 9*((3^(1/2)*1i)/9 + 1/9)^2)*((3^(1/2)*1i)/
9 + 1/9)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))**(1/3)*x,x)

[Out]

Integral(x/((x - 1)/(x + 1))**(1/3), x)

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