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

Optimal. Leaf size=155 \[ -\frac {3}{2} \log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+1\right )-\frac {1}{2} \log \left (\frac {1}{x}+1\right )-\frac {\log (x)}{2}-\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right ) \]

[Out]

-3/2*ln((1/x+1)^(1/3)-((-1+x)/x)^(1/3))-3/2*ln(1+((-1+x)/x)^(1/3)/(1/x+1)^(1/3))-1/2*ln(1/x+1)-1/2*ln(x)+arcta
n(-1/3*3^(1/2)+2/3*((-1+x)/x)^(1/3)/(1/x+1)^(1/3)*3^(1/2))*3^(1/2)-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 = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6171, 105, 60, 91} \[ -\frac {3}{2} \log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+1\right )-\frac {1}{2} \log \left (\frac {1}{x}+1\right )-\frac {\log (x)}{2}-\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 60

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-(d/b), 3]}, Simp[(Sq
rt[3]*q*ArcTan[1/Sqrt[3] - (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3))])/d, x] + (Simp[(3*q*Log[(q*(a + b*
x)^(1/3))/(c + d*x)^(1/3) + 1])/(2*d), x] + Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ
[b*c - a*d, 0] && NegQ[d/b]

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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -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 \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right )-\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right )\\ &=-\sqrt {3} \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} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )-\frac {3}{2} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{\frac {-1+x}{x}}\right )-\frac {3}{2} \log \left (1+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{2} \log \left (1+\frac {1}{x}\right )-\frac {\log (x)}{2}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 26, normalized size = 0.17 \[ \frac {3}{2} e^{\frac {8}{3} \coth ^{-1}(x)} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};e^{4 \coth ^{-1}(x)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*E^((8*ArcCoth[x])/3)*Hypergeometric2F1[2/3, 1, 5/3, E^(4*ArcCoth[x])])/2

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fricas [A]  time = 0.65, size = 86, normalized size = 0.55 \[ \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - 1\right ) + \frac {1}{2} \, \log \left (\frac {{\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + x + 1}{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 79, normalized size = 0.51 \[ \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + 1\right )}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}}}{x + 1} + 1\right ) - \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.92, size = 1038, normalized size = 6.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*RootOf(9*_Z^2-3*_Z+1)*ln((945*(-(1-x)/(1+x))^(2/3)*RootOf(9*_Z^2-3*_Z+1)*x^2+1890*RootOf(9*_Z^2-3*_Z+1)*(-(1
-x)/(1+x))^(2/3)*x-168*(-(1-x)/(1+x))^(2/3)*x^2+945*(-(1-x)/(1+x))^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^2-1224*RootOf
(9*_Z^2-3*_Z+1)^2*x^2+945*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)-336*(-(1-x)/(1+x))^(2/3)*x-168*(-(1-x)/(1
+x))^(1/3)*x^2+3060*RootOf(9*_Z^2-3*_Z+1)^2*x+1353*RootOf(9*_Z^2-3*_Z+1)*x^2-168*(-(1-x)/(1+x))^(2/3)-945*Root
Of(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)-1224*RootOf(9*_Z^2-3*_Z+1)^2-1062*RootOf(9*_Z^2-3*_Z+1)*x-304*x^2+168*(
-(1-x)/(1+x))^(1/3)+1353*RootOf(9*_Z^2-3*_Z+1)+32*x-304)/x)-3*ln(-(945*(-(1-x)/(1+x))^(2/3)*RootOf(9*_Z^2-3*_Z
+1)*x^2+1890*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x-147*(-(1-x)/(1+x))^(2/3)*x^2+945*(-(1-x)/(1+x))^(1/3
)*RootOf(9*_Z^2-3*_Z+1)*x^2+1224*RootOf(9*_Z^2-3*_Z+1)^2*x^2+945*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)-29
4*(-(1-x)/(1+x))^(2/3)*x-147*(-(1-x)/(1+x))^(1/3)*x^2-3060*RootOf(9*_Z^2-3*_Z+1)^2*x+537*RootOf(9*_Z^2-3*_Z+1)
*x^2-147*(-(1-x)/(1+x))^(2/3)-945*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)+1224*RootOf(9*_Z^2-3*_Z+1)^2+978*
RootOf(9*_Z^2-3*_Z+1)*x-11*x^2+147*(-(1-x)/(1+x))^(1/3)+537*RootOf(9*_Z^2-3*_Z+1)-18*x-11)/x)*RootOf(9*_Z^2-3*
_Z+1)+ln(-(945*(-(1-x)/(1+x))^(2/3)*RootOf(9*_Z^2-3*_Z+1)*x^2+1890*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*
x-147*(-(1-x)/(1+x))^(2/3)*x^2+945*(-(1-x)/(1+x))^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^2+1224*RootOf(9*_Z^2-3*_Z+1)^2
*x^2+945*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)-294*(-(1-x)/(1+x))^(2/3)*x-147*(-(1-x)/(1+x))^(1/3)*x^2-30
60*RootOf(9*_Z^2-3*_Z+1)^2*x+537*RootOf(9*_Z^2-3*_Z+1)*x^2-147*(-(1-x)/(1+x))^(2/3)-945*RootOf(9*_Z^2-3*_Z+1)*
(-(1-x)/(1+x))^(1/3)+1224*RootOf(9*_Z^2-3*_Z+1)^2+978*RootOf(9*_Z^2-3*_Z+1)*x-11*x^2+147*(-(1-x)/(1+x))^(1/3)+
537*RootOf(9*_Z^2-3*_Z+1)-18*x-11)/x)

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maxima [A]  time = 0.42, size = 140, normalized size = 0.90 \[ -\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) + \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.42, size = 82, normalized size = 0.53 \[ -\ln \left (1296\,{\left (\frac {x-1}{x+1}\right )}^{2/3}-1296\right )-\ln \left (1296\,{\left (\frac {x-1}{x+1}\right )}^{2/3}+648-\sqrt {3}\,648{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+\ln \left (1296\,{\left (\frac {x-1}{x+1}\right )}^{2/3}+648+\sqrt {3}\,648{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*((x - 1)/(x + 1))^(1/3)),x)

[Out]

log(3^(1/2)*648i + 1296*((x - 1)/(x + 1))^(2/3) + 648)*((3^(1/2)*1i)/2 + 1/2) - log(1296*((x - 1)/(x + 1))^(2/
3) - 3^(1/2)*648i + 648)*((3^(1/2)*1i)/2 - 1/2) - log(1296*((x - 1)/(x + 1))^(2/3) - 1296)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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