3.125 $$\int \frac{e^{\frac{2}{3} \coth ^{-1}(x)}}{x^3} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0533082, antiderivative size = 130, normalized size of antiderivative = 1., 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, 80, 50, 60} $\frac{1}{2} \left (\frac{x-1}{x}\right )^{2/3} \left (\frac{1}{x}+1\right )^{4/3}+\frac{1}{3} \left (\frac{x-1}{x}\right )^{2/3} \sqrt [3]{\frac{1}{x}+1}-\frac{1}{3} \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+1\right )-\frac{1}{9} \log \left (\frac{1}{x}+1\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}\right )}{3 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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]

Rubi steps

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

Mathematica [C]  time = 0.275655, size = 134, normalized size = 1.03 $-\frac{2}{27} \left (-\text{RootSum}\left [\text{\#1}^4-\text{\#1}^2+1\& ,\frac{\text{\#1}^2 \coth ^{-1}(x)-3 \text{\#1}^2 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}-\text{\#1}\right )-3 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}-\text{\#1}\right )+\coth ^{-1}(x)}{\text{\#1}^2-2}\& \right ]-2 \coth ^{-1}(x)-\frac{36 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}+1}+\frac{27 e^{\frac{2}{3} \coth ^{-1}(x)}}{\left (e^{2 \coth ^{-1}(x)}+1\right )^2}+3 \log \left (e^{\frac{2}{3} \coth ^{-1}(x)}+1\right )\right )$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*((27*E^((2*ArcCoth[x])/3))/(1 + E^(2*ArcCoth[x]))^2 - (36*E^((2*ArcCoth[x])/3))/(1 + E^(2*ArcCoth[x])) - 2
*ArcCoth[x] + 3*Log[1 + E^((2*ArcCoth[x])/3)] - RootSum[1 - #1^2 + #1^4 & , (ArcCoth[x] - 3*Log[E^(ArcCoth[x]/
3) - #1] + ArcCoth[x]*#1^2 - 3*Log[E^(ArcCoth[x]/3) - #1]*#1^2)/(-2 + #1^2) & ]))/27

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt [3]{{\frac{-1+x}{1+x}}}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.52712, size = 167, normalized size = 1.28 \begin{align*} \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/3)/x^3,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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Fricas [A]  time = 1.58706, size = 321, normalized size = 2.47 \begin{align*} \frac{4 \, \sqrt{3} x^{2} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + 2 \, x^{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 ) - 4 \, x^{2} \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) + 3 \,{\left (5 \, x^{2} + 8 \, x + 3\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{18 \, x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(4*sqrt(3)*x^2*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/3) - 1/3*sqrt(3)) + 2*x^2*log(((x - 1)/(x + 1))^(2
/3) - ((x - 1)/(x + 1))^(1/3) + 1) - 4*x^2*log(((x - 1)/(x + 1))^(1/3) + 1) + 3*(5*x^2 + 8*x + 3)*((x - 1)/(x
+ 1))^(2/3))/x^2

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15536, size = 165, normalized size = 1.27 \begin{align*} \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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