\(\int e^{-\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx\) [97]

Optimal result

Integrand size = 14, antiderivative size = 223 \[ \int e^{-\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx=\frac {17 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 a^3}+\frac {(1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^3}-\frac {x (1-a x)^{7/4} \sqrt [4]{1+a x}}{3 a^2}+\frac {17 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}-\frac {17 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}+\frac {17 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x} \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}\right )}{8 \sqrt {2} a^3} \] Output:

17/24*(-a*x+1)^(3/4)*(a*x+1)^(1/4)/a^3+1/4*(-a*x+1)^(7/4)*(a*x+1)^(1/4)/a^ 
3-1/3*x*(-a*x+1)^(7/4)*(a*x+1)^(1/4)/a^2-17/16*arctan(-1+2^(1/2)*(-a*x+1)^ 
(1/4)/(a*x+1)^(1/4))*2^(1/2)/a^3-17/16*arctan(1+2^(1/2)*(-a*x+1)^(1/4)/(a* 
x+1)^(1/4))*2^(1/2)/a^3+17/16*arctanh(2^(1/2)*(-a*x+1)^(1/4)/(a*x+1)^(1/4) 
/(1+(-a*x+1)^(1/2)/(a*x+1)^(1/2)))*2^(1/2)/a^3
 

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.28 \[ \int e^{-\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx=-\frac {(1-a x)^{7/4} \left (7 \sqrt [4]{1+a x} (-3+4 a x)+17 \sqrt [4]{2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{4},\frac {11}{4},\frac {1}{2} (1-a x)\right )\right )}{84 a^3} \] Input:

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

Output:

-1/84*((1 - a*x)^(7/4)*(7*(1 + a*x)^(1/4)*(-3 + 4*a*x) + 17*2^(1/4)*Hyperg 
eometric2F1[3/4, 7/4, 11/4, (1 - a*x)/2]))/a^3
 

Rubi [A] (warning: unable to verify)

Time = 0.82 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.22, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6676, 101, 27, 90, 60, 73, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 6676

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

\(\Big \downarrow \) 101

\(\displaystyle -\frac {\int -\frac {(2-3 a x) (1-a x)^{3/4}}{2 (a x+1)^{3/4}}dx}{3 a^2}-\frac {x \sqrt [4]{a x+1} (1-a x)^{7/4}}{3 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(2-3 a x) (1-a x)^{3/4}}{(a x+1)^{3/4}}dx}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {17}{4} \int \frac {(1-a x)^{3/4}}{(a x+1)^{3/4}}dx+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {17}{4} \left (\frac {3}{2} \int \frac {1}{\sqrt [4]{1-a x} (a x+1)^{3/4}}dx+\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {17}{4} \left (\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\frac {6 \int \frac {\sqrt {1-a x}}{(a x+1)^{3/4}}d\sqrt [4]{1-a x}}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 854

\(\displaystyle \frac {\frac {17}{4} \left (\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\frac {6 \int \frac {\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {\frac {17}{4} \left (\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\frac {6 \left (\frac {1}{2} \int \frac {\sqrt {1-a x}+1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {17}{4} \left (\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\frac {6 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {17}{4} \left (\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\frac {6 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {17}{4} \left (\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\frac {6 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\frac {17}{4} \left (\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\frac {6 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {17}{4} \left (\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\frac {6 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {17}{4} \left (\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\frac {6 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {17}{4} \left (\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}-\frac {6 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}\right )+\frac {3 \sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a}}{6 a^2}-\frac {x (1-a x)^{7/4} \sqrt [4]{a x+1}}{3 a^2}\)

Input:

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

Output:

-1/3*(x*(1 - a*x)^(7/4)*(1 + a*x)^(1/4))/a^2 + ((3*(1 - a*x)^(7/4)*(1 + a* 
x)^(1/4))/(2*a) + (17*(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/a - (6*((-(ArcTan 
[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqr 
t[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2])/2 + (Log[1 + Sqrt[1 - a*x] 
 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]) - Log[1 + Sqrt[1 
 - a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]))/2))/a))/ 
4)/(6*a^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.14 \[ \int e^{-\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx=-\frac {4 \, {\left (8 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 37 \, a x - 23\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 102 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 102 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 51 \, \sqrt {2} \log \left (\frac {a x + \sqrt {2} {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {-a^{2} x^{2} + 1} - 1}{a x - 1}\right ) + 51 \, \sqrt {2} \log \left (\frac {a x - \sqrt {2} {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {-a^{2} x^{2} + 1} - 1}{a x - 1}\right )}{96 \, a^{3}} \] Input:

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

Output:

-1/96*(4*(8*a^3*x^3 - 22*a^2*x^2 + 37*a*x - 23)*sqrt(-sqrt(-a^2*x^2 + 1)/( 
a*x - 1)) - 102*sqrt(2)*arctan(sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) 
 + 1) - 102*sqrt(2)*arctan(sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1 
) - 51*sqrt(2)*log((a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x 
- 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) + 51*sqrt(2)*log((a*x - sqrt(2) 
*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/( 
a*x - 1)))/a^3
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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