\(\int e^{\frac {2 \text {arctanh}(x)}{3}} x^2 \, dx\) [120]

Optimal result

Integrand size = 12, antiderivative size = 133 \[ \int e^{\frac {2 \text {arctanh}(x)}{3}} x^2 \, dx=-\frac {11}{27} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{9} (1-x)^{2/3} (1+x)^{4/3}-\frac {1}{3} (1-x)^{2/3} x (1+x)^{4/3}+\frac {22 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{27 \sqrt {3}}+\frac {11}{81} \log (1+x)+\frac {11}{27} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right ) \] Output:

-11/27*(1-x)^(2/3)*(1+x)^(1/3)-1/9*(1-x)^(2/3)*(1+x)^(4/3)-1/3*(1-x)^(2/3) 
*x*(1+x)^(4/3)-22/81*arctan(-1/3*3^(1/2)+2/3*(1-x)^(1/3)*3^(1/2)/(1+x)^(1/ 
3))*3^(1/2)+11/81*ln(1+x)+11/27*ln(1+(1-x)^(1/3)/(1+x)^(1/3))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.44 \[ \int e^{\frac {2 \text {arctanh}(x)}{3}} x^2 \, dx=-\frac {1}{18} (1-x)^{2/3} \left (2 \sqrt [3]{1+x} \left (1+4 x+3 x^2\right )+11 \sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {1-x}{2}\right )\right ) \] Input:

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

Output:

-1/18*((1 - x)^(2/3)*(2*(1 + x)^(1/3)*(1 + 4*x + 3*x^2) + 11*2^(1/3)*Hyper 
geometric2F1[-1/3, 2/3, 5/3, (1 - x)/2]))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6676, 101, 27, 90, 60, 72}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 e^{\frac {2 \text {arctanh}(x)}{3}} \, dx\)

\(\Big \downarrow \) 6676

\(\displaystyle \int \frac {x^2 \sqrt [3]{x+1}}{\sqrt [3]{1-x}}dx\)

\(\Big \downarrow \) 101

\(\displaystyle -\frac {1}{3} \int -\frac {\sqrt [3]{x+1} (2 x+3)}{3 \sqrt [3]{1-x}}dx-\frac {1}{3} (1-x)^{2/3} x (x+1)^{4/3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int \frac {\sqrt [3]{x+1} (2 x+3)}{\sqrt [3]{1-x}}dx-\frac {1}{3} (1-x)^{2/3} x (x+1)^{4/3}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{9} \left (\frac {11}{3} \int \frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}dx-(1-x)^{2/3} (x+1)^{4/3}\right )-\frac {1}{3} (1-x)^{2/3} x (x+1)^{4/3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{9} \left (\frac {11}{3} \left (\frac {2}{3} \int \frac {1}{\sqrt [3]{1-x} (x+1)^{2/3}}dx-(1-x)^{2/3} \sqrt [3]{x+1}\right )-(1-x)^{2/3} (x+1)^{4/3}\right )-\frac {1}{3} (1-x)^{2/3} x (x+1)^{4/3}\)

\(\Big \downarrow \) 72

\(\displaystyle \frac {1}{9} \left (\frac {11}{3} \left (\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}\right )+\frac {1}{2} \log (x+1)+\frac {3}{2} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right )\right )-(1-x)^{2/3} \sqrt [3]{x+1}\right )-(1-x)^{2/3} (x+1)^{4/3}\right )-\frac {1}{3} (1-x)^{2/3} x (x+1)^{4/3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 72

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

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(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 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 6676

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

Maple [F]

\[\int {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )}^{\frac {2}{3}} x^{2}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.20 \[ \int e^{\frac {2 \text {arctanh}(x)}{3}} x^2 \, dx=\frac {1}{27} \, {\left (9 \, x^{3} + 3 \, x^{2} + 2 \, x - 14\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + \frac {22}{81} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {22}{81} \, \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 1\right ) - \frac {11}{81} \, \log \left (-\frac {{\left (x - 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - x + \sqrt {-x^{2} + 1} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1}{x - 1}\right ) \] Input:

integrate(((1+x)/(-x^2+1)^(1/2))^(2/3)*x^2,x, algorithm="fricas")
 

Output:

1/27*(9*x^3 + 3*x^2 + 2*x - 14)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 22/81*sq 
rt(3)*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) - 1/3*sqrt(3)) + 
22/81*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1) - 11/81*log(-((x - 1)*(-sqr 
t(-x^2 + 1)/(x - 1))^(2/3) - x + sqrt(-x^2 + 1)*(-sqrt(-x^2 + 1)/(x - 1))^ 
(1/3) + 1)/(x - 1))
 

Sympy [F(-1)]

Timed out. \[ \int e^{\frac {2 \text {arctanh}(x)}{3}} x^2 \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int e^{\frac {2 \text {arctanh}(x)}{3}} x^2 \, dx=\int { x^{2} \left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {2}{3}} \,d x } \] Input:

integrate(((1+x)/(-x^2+1)^(1/2))^(2/3)*x^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int e^{\frac {2 \text {arctanh}(x)}{3}} x^2 \, dx=\int { x^{2} \left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {2}{3}} \,d x } \] Input:

integrate(((1+x)/(-x^2+1)^(1/2))^(2/3)*x^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {2 \text {arctanh}(x)}{3}} x^2 \, dx=\int x^2\,{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{2/3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int e^{\frac {2 \text {arctanh}(x)}{3}} x^2 \, dx=\int \frac {\left (x +1\right )^{\frac {2}{3}} x^{2}}{\left (-x^{2}+1\right )^{\frac {1}{3}}}d x \] Input:

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

Output:

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