Integrand size = 14, antiderivative size = 299 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {5 a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {5 a \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {5 a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}+\frac {5 a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}} \]
-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+(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/x)^(1/4))*2^(1/2)-5/2*a*arct an(1+(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/x)^(1/4))*2^(1/2)-5/4*a*ln(1-(1-1/a/x) ^(1/4)*2^(1/2)/(1+1/a/x)^(1/4)+(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2))*2^(1/2)+5/ 4*a*ln(1+(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/x)^(1/4)+(1-1/a/x)^(1/2)/(1+1/a/x) ^(1/2))*2^(1/2)
Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.58 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=a \left (-\frac {10 e^{\frac {1}{2} \coth ^{-1}(a x)}}{1+e^{2 \coth ^{-1}(a x)}}-\frac {8 e^{\frac {5}{2} \coth ^{-1}(a x)}}{1+e^{2 \coth ^{-1}(a x)}}-\frac {5 \arctan \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{\sqrt {2}}+\frac {5 \arctan \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{\sqrt {2}}-\frac {5 \log \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )}{2 \sqrt {2}}+\frac {5 \log \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )}{2 \sqrt {2}}\right ) \]
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]))
Time = 0.45 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.85, 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 |
\(\displaystyle 5 \left (-2 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
(-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))
3.1.83.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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])
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]]))
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]
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]
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]
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]
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]
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]
\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}} x^{2}}d x\]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.81 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {5 \, \left (-a^{4}\right )^{\frac {1}{4}} {\left (a x^{2} - x\right )} \log \left (125 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 125 \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) - 5 \, \left (-a^{4}\right )^{\frac {1}{4}} {\left (i \, a x^{2} - i \, x\right )} \log \left (125 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 125 i \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) - 5 \, \left (-a^{4}\right )^{\frac {1}{4}} {\left (-i \, a x^{2} + i \, x\right )} \log \left (125 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 125 i \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) - 5 \, \left (-a^{4}\right )^{\frac {1}{4}} {\left (a x^{2} - x\right )} \log \left (125 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 125 \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) + 2 \, {\left (9 \, a^{2} x^{2} + 8 \, a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{2 \, {\left (a x^{2} - x\right )}} \]
-1/2*(5*(-a^4)^(1/4)*(a*x^2 - x)*log(125*a^3*((a*x - 1)/(a*x + 1))^(1/4) + 125*(-a^4)^(3/4)) - 5*(-a^4)^(1/4)*(I*a*x^2 - I*x)*log(125*a^3*((a*x - 1) /(a*x + 1))^(1/4) + 125*I*(-a^4)^(3/4)) - 5*(-a^4)^(1/4)*(-I*a*x^2 + I*x)* log(125*a^3*((a*x - 1)/(a*x + 1))^(1/4) - 125*I*(-a^4)^(3/4)) - 5*(-a^4)^( 1/4)*(a*x^2 - x)*log(125*a^3*((a*x - 1)/(a*x + 1))^(1/4) - 125*(-a^4)^(3/4 )) + 2*(9*a^2*x^2 + 8*a*x - 1)*((a*x - 1)/(a*x + 1))^(3/4))/(a*x^2 - x)
\[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {1}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {1}{4} \, {\left (10 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 10 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 5 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 5 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {8 \, {\left (\frac {5 \, {\left (a x - 1\right )}}{a x + 1} + 4\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}\right )} a \]
-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
Time = 0.29 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {1}{4} \, {\left (10 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 10 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 5 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 5 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {8 \, {\left (\frac {5 \, {\left (a x - 1\right )}}{a x + 1} + 4\right )}}{\frac {{\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}\right )} a \]
-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
Time = 4.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.36 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=5\,{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )-5\,{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )-\frac {8\,a+\frac {10\,a\,\left (a\,x-1\right )}{a\,x+1}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}} \]