3.2.22 \(\int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx\) [122]

Optimal. Leaf size=96 \[ \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x-\frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{\sqrt {3}}-\log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{\frac {-1+x}{x}}\right )-\frac {\log (x)}{3} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6305, 96, 93} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}}+\sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x-\log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 93

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 6305

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]

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 85, normalized size = 0.89 \begin {gather*} \frac {1}{3} \left (\frac {6 e^{\frac {2}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+2 \sqrt {3} \text {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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.97, size = 397, normalized size = 4.14

method	result	size
risch	\(\frac {-1+x}{\left (\frac {-1+x}{1+x}\right )^{\frac {1}{3}}}+\frac {\left (-\frac {2 \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{1+x}\right )}{3}+\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -5 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +2 x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+4 x +2}{1+x}\right )}{3}\right ) \left (\left (1+x \right )^{2} \left (-1+x \right )\right )^{\frac {1}{3}}}{\left (\frac {-1+x}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )}\)	\(397\)
trager	\(\left (1+x \right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+\frac {2 \ln \left (9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-x -1\right )}{3}-2 \ln \left (9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-x -1\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x -9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-15 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2 x -2\right )\)	\(657\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((-1+x)/(1+x))^(1/3),x,method=_RETURNVERBOSE)

[Out]

(-1+x)/((-1+x)/(1+x))^(1/3)+(-2/3*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*Roo
tOf(_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))+2/3*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)^2*(-1+x))^(1/3)/(1+x)

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Maxima [A]
time = 0.47, size = 96, normalized size = 1.00 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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Fricas [A]
time = 0.35, size = 87, normalized size = 0.91 \begin {gather*} {\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.42, size = 97, normalized size = 1.01 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))

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Mupad [B]
time = 0.05, size = 118, normalized size = 1.23 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x - 1)/(x + 1))^(1/3),x)

[Out]

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)

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