### 3.120 $$\int e^{\frac{2}{3} \coth ^{-1}(x)} x^2 \, dx$$

Optimal. Leaf size=157 $\frac{1}{3} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x^3+\frac{4}{9} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x^2+\frac{14}{27} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\frac{11}{27} \log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{11 \log (x)}{81}-\frac{22 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3}}$

[Out]

(14*(1 + x^(-1))^(1/3)*((-1 + x)/x)^(2/3)*x)/27 + (4*(1 + x^(-1))^(1/3)*((-1 + x)/x)^(2/3)*x^2)/9 + ((1 + x^(-
1))^(1/3)*((-1 + x)/x)^(2/3)*x^3)/3 - (22*ArcTan[1/Sqrt[3] + (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + x^(-1))^(1/3
))])/(27*Sqrt[3]) - (11*Log[(1 + x^(-1))^(1/3) - ((-1 + x)/x)^(1/3)])/27 - (11*Log[x])/81

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Rubi [A]  time = 0.0701047, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.417, Rules used = {6171, 99, 151, 12, 91} $\frac{1}{3} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x^3+\frac{4}{9} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x^2+\frac{14}{27} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\frac{11}{27} \log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{11 \log (x)}{81}-\frac{22 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(14*(1 + x^(-1))^(1/3)*((-1 + x)/x)^(2/3)*x)/27 + (4*(1 + x^(-1))^(1/3)*((-1 + x)/x)^(2/3)*x^2)/9 + ((1 + x^(-
1))^(1/3)*((-1 + x)/x)^(2/3)*x^3)/3 - (22*ArcTan[1/Sqrt[3] + (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + x^(-1))^(1/3
))])/(27*Sqrt[3]) - (11*Log[(1 + x^(-1))^(1/3) - ((-1 + x)/x)^(1/3)])/27 - (11*Log[x])/81

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 99

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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rubi steps

\begin{align*} \int e^{\frac{2}{3} \coth ^{-1}(x)} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{2/3} x^3-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{8}{3}+2 x}{\sqrt [3]{1-x} x^3 (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4}{9} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x^2+\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{2/3} x^3+\frac{1}{6} \operatorname{Subst}\left (\int \frac{-\frac{28}{9}-\frac{8 x}{3}}{\sqrt [3]{1-x} x^2 (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{14}{27} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x+\frac{4}{9} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x^2+\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{2/3} x^3-\frac{1}{6} \operatorname{Subst}\left (\int \frac{44}{27 \sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{14}{27} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x+\frac{4}{9} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x^2+\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{2/3} x^3-\frac{22}{81} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{14}{27} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x+\frac{4}{9} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x^2+\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{2/3} x^3-\frac{22 \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{-\frac{1-x}{x}}}{\sqrt{3} \sqrt [3]{1+\frac{1}{x}}}\right )}{27 \sqrt{3}}-\frac{11}{27} \log \left (\sqrt [3]{1+\frac{1}{x}}-\sqrt [3]{-\frac{1-x}{x}}\right )-\frac{11 \log (x)}{81}\\ \end{align*}

Mathematica [C]  time = 7.16166, size = 340, normalized size = 2.17 $-\frac{e^{-\frac{10}{3} \coth ^{-1}(x)} \left (54 e^{8 \coth ^{-1}(x)} \left (782 e^{2 \coth ^{-1}(x)}+325 e^{4 \coth ^{-1}(x)}+475\right ) \text{HypergeometricPFQ}\left (\left \{2,2,2,\frac{7}{3}\right \},\left \{1,1,\frac{16}{3}\right \},e^{2 \coth ^{-1}(x)}\right )+162 e^{8 \coth ^{-1}(x)} \left (64 e^{2 \coth ^{-1}(x)}+29 e^{4 \coth ^{-1}(x)}+35\right ) \text{HypergeometricPFQ}\left (\left \{2,2,2,2,\frac{7}{3}\right \},\left \{1,1,1,\frac{16}{3}\right \},e^{2 \coth ^{-1}(x)}\right )+486 e^{8 \coth ^{-1}(x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{7}{3}\right \},\left \{1,1,1,1,\frac{16}{3}\right \},e^{2 \coth ^{-1}(x)}\right )+972 e^{10 \coth ^{-1}(x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{7}{3}\right \},\left \{1,1,1,1,\frac{16}{3}\right \},e^{2 \coth ^{-1}(x)}\right )+486 e^{12 \coth ^{-1}(x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{7}{3}\right \},\left \{1,1,1,1,\frac{16}{3}\right \},e^{2 \coth ^{-1}(x)}\right )+15227940 e^{2 \coth ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{1}{3},1,\frac{4}{3},e^{2 \coth ^{-1}(x)}\right )-14083160 e^{4 \coth ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{1}{3},1,\frac{4}{3},e^{2 \coth ^{-1}(x)}\right )-8250060 e^{6 \coth ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{1}{3},1,\frac{4}{3},e^{2 \coth ^{-1}(x)}\right )+1456000 e^{8 \coth ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{1}{3},1,\frac{4}{3},e^{2 \coth ^{-1}(x)}\right )+22750000 \text{Hypergeometric2F1}\left (\frac{1}{3},1,\frac{4}{3},e^{2 \coth ^{-1}(x)}\right )-20915440 e^{2 \coth ^{-1}(x)}+7026175 e^{4 \coth ^{-1}(x)}+7394140 e^{6 \coth ^{-1}(x)}-433485 e^{8 \coth ^{-1}(x)}-22750000\right )}{49140}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-22750000 - 20915440*E^(2*ArcCoth[x]) + 7026175*E^(4*ArcCoth[x]) + 7394140*E^(6*ArcCoth[x]) - 433485*E^(8*Ar
cCoth[x]) + 22750000*Hypergeometric2F1[1/3, 1, 4/3, E^(2*ArcCoth[x])] + 15227940*E^(2*ArcCoth[x])*Hypergeometr
ic2F1[1/3, 1, 4/3, E^(2*ArcCoth[x])] - 14083160*E^(4*ArcCoth[x])*Hypergeometric2F1[1/3, 1, 4/3, E^(2*ArcCoth[x
])] - 8250060*E^(6*ArcCoth[x])*Hypergeometric2F1[1/3, 1, 4/3, E^(2*ArcCoth[x])] + 1456000*E^(8*ArcCoth[x])*Hyp
ergeometric2F1[1/3, 1, 4/3, E^(2*ArcCoth[x])] + 54*E^(8*ArcCoth[x])*(475 + 782*E^(2*ArcCoth[x]) + 325*E^(4*Arc
Coth[x]))*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 16/3}, E^(2*ArcCoth[x])] + 162*E^(8*ArcCoth[x])*(35 + 64*E^
(2*ArcCoth[x]) + 29*E^(4*ArcCoth[x]))*HypergeometricPFQ[{2, 2, 2, 2, 7/3}, {1, 1, 1, 16/3}, E^(2*ArcCoth[x])]
+ 486*E^(8*ArcCoth[x])*HypergeometricPFQ[{2, 2, 2, 2, 2, 7/3}, {1, 1, 1, 1, 16/3}, E^(2*ArcCoth[x])] + 972*E^(
10*ArcCoth[x])*HypergeometricPFQ[{2, 2, 2, 2, 2, 7/3}, {1, 1, 1, 1, 16/3}, E^(2*ArcCoth[x])] + 486*E^(12*ArcCo
th[x])*HypergeometricPFQ[{2, 2, 2, 2, 2, 7/3}, {1, 1, 1, 1, 16/3}, E^(2*ArcCoth[x])])/(49140*E^((10*ArcCoth[x]
)/3))

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Maple [F]  time = 0.119, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\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^2,x)

[Out]

int(1/((-1+x)/(1+x))^(1/3)*x^2,x)

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Maxima [A]  time = 1.54007, size = 201, normalized size = 1.28 \begin{align*} -\frac{22}{81} \, \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 (11 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{8}{3}} - 10 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{3}} + 35 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}\right )}}{27 \,{\left (\frac{3 \,{\left (x - 1\right )}}{x + 1} - \frac{3 \,{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \frac{11}{81} \, \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{22}{81} \, \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^2,x, algorithm="maxima")

[Out]

-22/81*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) - 2/27*(11*((x - 1)/(x + 1))^(8/3) - 10*((x
- 1)/(x + 1))^(5/3) + 35*((x - 1)/(x + 1))^(2/3))/(3*(x - 1)/(x + 1) - 3*(x - 1)^2/(x + 1)^2 + (x - 1)^3/(x +
1)^3 - 1) + 11/81*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) - 22/81*log(((x - 1)/(x + 1))^(1
/3) - 1)

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Fricas [A]  time = 1.46335, size = 325, normalized size = 2.07 \begin{align*} \frac{1}{27} \,{\left (9 \, x^{3} + 21 \, x^{2} + 26 \, x + 14\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \frac{22}{81} \, \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{11}{81} \, \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{22}{81} \, \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^2,x, algorithm="fricas")

[Out]

1/27*(9*x^3 + 21*x^2 + 26*x + 14)*((x - 1)/(x + 1))^(2/3) - 22/81*sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))
^(1/3) + 1/3*sqrt(3)) + 11/81*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) - 22/81*log(((x - 1)/
(x + 1))^(1/3) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\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**2,x)

[Out]

Integral(x**2/((x - 1)/(x + 1))**(1/3), x)

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Giac [A]  time = 1.18165, size = 194, normalized size = 1.24 \begin{align*} -\frac{22}{81} \, \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{10 \,{\left (x - 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{x + 1} - \frac{11 \,{\left (x - 1\right )}^{2} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{{\left (x + 1\right )}^{2}} - 35 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}\right )}}{27 \,{\left (\frac{x - 1}{x + 1} - 1\right )}^{3}} + \frac{11}{81} \, \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{22}{81} \, \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^2,x, algorithm="giac")

[Out]

-22/81*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) + 2/27*(10*(x - 1)*((x - 1)/(x + 1))^(2/3)/
(x + 1) - 11*(x - 1)^2*((x - 1)/(x + 1))^(2/3)/(x + 1)^2 - 35*((x - 1)/(x + 1))^(2/3))/((x - 1)/(x + 1) - 1)^3
+ 11/81*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) - 22/81*log(abs(((x - 1)/(x + 1))^(1/3) -
1))