\(\int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx\) [91]

Optimal result

Integrand size = 14, antiderivative size = 224 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {5 a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {5 a \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}+\frac {5 a \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right ) \sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}} \] Output:

-5*a*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)-4*a*(1+1/a/x)^(5/4)/(1-1/a/x)^(1/4)-5 
/2*a*arctan(-1+2^(1/2)*(1-1/a/x)^(1/4)/(1+1/a/x)^(1/4))*2^(1/2)-5/2*a*arct 
an(1+2^(1/2)*(1-1/a/x)^(1/4)/(1+1/a/x)^(1/4))*2^(1/2)+5/2*a*arctanh(2^(1/2 
)*(1-1/a/x)^(1/4)/(1+(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2))/(1+1/a/x)^(1/4))*2^( 
1/2)
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.77 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=a \left (-\frac {10 e^{\frac {1}{2} \coth ^{-1}(a x)}}{1+e^{2 \coth ^{-1}(a x)}}-\frac {8 e^{\frac {5}{2} \coth ^{-1}(a x)}}{1+e^{2 \coth ^{-1}(a x)}}-\frac {5 \arctan \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{\sqrt {2}}+\frac {5 \arctan \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{\sqrt {2}}-\frac {5 \log \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )}{2 \sqrt {2}}+\frac {5 \log \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )}{2 \sqrt {2}}\right ) \] Input:

Integrate[E^((5*ArcCoth[a*x])/2)/x^2,x]
 

Output:

a*((-10*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x])) - (8*E^((5*ArcCoth[a* 
x])/2))/(1 + E^(2*ArcCoth[a*x])) - (5*ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/2 
)])/Sqrt[2] + (5*ArcTan[1 + Sqrt[2]*E^(ArcCoth[a*x]/2)])/Sqrt[2] - (5*Log[ 
1 - Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]])/(2*Sqrt[2]) + (5*Log[1 + 
 Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]])/(2*Sqrt[2]))
 

Rubi [A] (warning: unable to verify)

Time = 0.76 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.13, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6721, 57, 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.

\(\displaystyle \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx\)

\(\Big \downarrow \) 6721

\(\displaystyle -\int \frac {\left (1+\frac {1}{a x}\right )^{5/4}}{\left (1-\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}\)

\(\Big \downarrow \) 57

\(\displaystyle 5 \int \frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}d\frac {1}{x}-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 60

\(\displaystyle 5 \left (\frac {1}{2} \int \frac {1}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 73

\(\displaystyle 5 \left (-2 a \int \frac {1}{\left (2-\frac {1}{x^4}\right )^{3/4} x^2}d\sqrt [4]{1-\frac {1}{a x}}-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 854

\(\displaystyle 5 \left (-2 a \int \frac {1}{\left (1+\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 826

\(\displaystyle 5 \left (-2 a \left (\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 1476

\(\displaystyle 5 \left (-2 a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 1082

\(\displaystyle 5 \left (-2 a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 217

\(\displaystyle 5 \left (-2 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 1479

\(\displaystyle 5 \left (-2 a \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 25

\(\displaystyle 5 \left (-2 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle 5 \left (-2 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 1103

\(\displaystyle 5 \left (-2 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\)

Input:

Int[E^((5*ArcCoth[a*x])/2)/x^2,x]
 

Output:

(-4*a*(1 + 1/(a*x))^(5/4))/(1 - 1/(a*x))^(1/4) + 5*(-(a*(1 - 1/(a*x))^(3/4 
)*(1 + 1/(a*x))^(1/4)) - 2*a*((-(ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/ 
(2 - x^(-4))^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(2 
 - x^(-4))^(1/4)]/Sqrt[2])/2 + (Log[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(2 - 
 x^(-4))^(1/4) + x^(-2)]/(2*Sqrt[2]) - Log[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4 
))/(2 - x^(-4))^(1/4) + x^(-2)]/(2*Sqrt[2]))/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 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, 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 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 6721

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]
 

Maple [F]

\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}} x^{2}}d x\]

Input:

int(1/((a*x-1)/(a*x+1))^(5/4)/x^2,x)
 

Output:

int(1/((a*x-1)/(a*x+1))^(5/4)/x^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {10 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \arctan \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 10 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \arctan \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right ) - 5 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 5 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 4 \, {\left (9 \, a^{2} x^{2} + 8 \, a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{4 \, {\left (a x^{2} - x\right )}} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(5/4)/x^2,x, algorithm="fricas")
 

Output:

-1/4*(10*sqrt(2)*(a^2*x^2 - a*x)*arctan(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4 
) + 1) + 10*sqrt(2)*(a^2*x^2 - a*x)*arctan(sqrt(2)*((a*x - 1)/(a*x + 1))^( 
1/4) - 1) - 5*sqrt(2)*(a^2*x^2 - a*x)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1 
/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) + 5*sqrt(2)*(a^2*x^2 - a*x)*log(-sqrt 
(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) + 4*(9*a^ 
2*x^2 + 8*a*x - 1)*((a*x - 1)/(a*x + 1))^(3/4))/(a*x^2 - x)
 

Sympy [F]

\[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {1}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \] Input:

integrate(1/((a*x-1)/(a*x+1))**(5/4)/x**2,x)
 

Output:

Integral(1/(x**2*((a*x - 1)/(a*x + 1))**(5/4)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {1}{4} \, {\left (10 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 10 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 5 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 5 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {8 \, {\left (\frac {5 \, {\left (a x - 1\right )}}{a x + 1} + 4\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}\right )} a \] Input:

integrate(1/((a*x-1)/(a*x+1))^(5/4)/x^2,x, algorithm="maxima")
 

Output:

-1/4*(10*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/ 
4))) + 10*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^( 
1/4))) - 5*sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1 
)/(a*x + 1)) + 1) + 5*sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + s 
qrt((a*x - 1)/(a*x + 1)) + 1) + 8*(5*(a*x - 1)/(a*x + 1) + 4)/(((a*x - 1)/ 
(a*x + 1))^(5/4) + ((a*x - 1)/(a*x + 1))^(1/4)))*a
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {1}{4} \, {\left (10 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 10 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 5 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 5 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {8 \, {\left (\frac {5 \, {\left (a x - 1\right )}}{a x + 1} + 4\right )}}{\frac {{\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}\right )} a \] Input:

integrate(1/((a*x-1)/(a*x+1))^(5/4)/x^2,x, algorithm="giac")
 

Output:

-1/4*(10*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/ 
4))) + 10*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^( 
1/4))) - 5*sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1 
)/(a*x + 1)) + 1) + 5*sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + s 
qrt((a*x - 1)/(a*x + 1)) + 1) + 8*(5*(a*x - 1)/(a*x + 1) + 4)/((a*x - 1)*( 
(a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1) + ((a*x - 1)/(a*x + 1))^(1/4)))*a
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.48 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=5\,{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )-5\,{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )-\frac {8\,a+\frac {10\,a\,\left (a\,x-1\right )}{a\,x+1}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}} \] Input:

int(1/(x^2*((a*x - 1)/(a*x + 1))^(5/4)),x)
 

Output:

5*(-1)^(1/4)*a*atanh((-1)^(1/4)*((a*x - 1)/(a*x + 1))^(1/4)) - 5*(-1)^(1/4 
)*a*atan((-1)^(1/4)*((a*x - 1)/(a*x + 1))^(1/4)) - (8*a + (10*a*(a*x - 1)) 
/(a*x + 1))/(((a*x - 1)/(a*x + 1))^(1/4) + ((a*x - 1)/(a*x + 1))^(5/4))
 

Reduce [F]

\[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {\left (a x +1\right )^{\frac {1}{4}}}{\left (a x -1\right )^{\frac {1}{4}} a \,x^{3}-\left (a x -1\right )^{\frac {1}{4}} x^{2}}d x +\left (\int \frac {\left (a x +1\right )^{\frac {1}{4}}}{\left (a x -1\right )^{\frac {1}{4}} a \,x^{2}-\left (a x -1\right )^{\frac {1}{4}} x}d x \right ) a \] Input:

int(1/((a*x-1)/(a*x+1))^(5/4)/x^2,x)
 

Output:

int((a*x + 1)**(1/4)/((a*x - 1)**(1/4)*a*x**3 - (a*x - 1)**(1/4)*x**2),x) 
+ int((a*x + 1)**(1/4)/((a*x - 1)**(1/4)*a*x**2 - (a*x - 1)**(1/4)*x),x)*a