Integrand size = 14, antiderivative size = 269 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {3 a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}+\frac {3 a \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}+\frac {3 a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}-\frac {3 a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}} \]
-a*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)+3/2*a*arctan(-1+(1-1/a/x)^(1/4)*2^(1/2) /(1+1/a/x)^(1/4))*2^(1/2)+3/2*a*arctan(1+(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/x) ^(1/4))*2^(1/2)+3/4*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)-3/4*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 = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.55 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^2} \, dx=a \left (-\frac {2 e^{\frac {1}{2} \coth ^{-1}(a x)}}{1+e^{2 \coth ^{-1}(a x)}}+\frac {3 \arctan \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{\sqrt {2}}-\frac {3 \arctan \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{\sqrt {2}}+\frac {3 \log \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )}{2 \sqrt {2}}-\frac {3 \log \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )}{2 \sqrt {2}}\right ) \]
a*((-2*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x])) + (3*ArcTan[1 - Sqrt[2 ]*E^(ArcCoth[a*x]/2)])/Sqrt[2] - (3*ArcTan[1 + Sqrt[2]*E^(ArcCoth[a*x]/2)] )/Sqrt[2] + (3*Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]])/(2*Sq rt[2]) - (3*Log[1 + Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]])/(2*Sqrt[ 2]))
Time = 0.41 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.82, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6721, 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.
| \(\displaystyle \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^2} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right )^{3/4}}{\left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle a \left (-\left (1-\frac {1}{a x}\right )^{3/4}\right ) \sqrt [4]{\frac {1}{a x}+1}-\frac {3}{2} \int \frac {1}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle 6 a \int \frac {1}{\left (2-\frac {1}{x^4}\right )^{3/4} x^2}d\sqrt [4]{1-\frac {1}{a x}}-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle 6 a \int \frac {1}{\left (1+\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 6 a \left (\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 6 a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 6 a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 6 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 6 a \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 6 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 6 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 6 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\) |
-(a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)) + 6*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)
3.2.1.3.1 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 + 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]
\[\int \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{4}}}{x^{2}}d x\]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.71 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^2} \, dx=\frac {3 \, \left (-a^{4}\right )^{\frac {1}{4}} x \log \left (27 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 27 \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) - 3 i \, \left (-a^{4}\right )^{\frac {1}{4}} x \log \left (27 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 27 i \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) + 3 i \, \left (-a^{4}\right )^{\frac {1}{4}} x \log \left (27 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 27 i \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) - 3 \, \left (-a^{4}\right )^{\frac {1}{4}} x \log \left (27 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 27 \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) - 2 \, {\left (a x + 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{2 \, x} \]
1/2*(3*(-a^4)^(1/4)*x*log(27*a^3*((a*x - 1)/(a*x + 1))^(1/4) + 27*(-a^4)^( 3/4)) - 3*I*(-a^4)^(1/4)*x*log(27*a^3*((a*x - 1)/(a*x + 1))^(1/4) + 27*I*( -a^4)^(3/4)) + 3*I*(-a^4)^(1/4)*x*log(27*a^3*((a*x - 1)/(a*x + 1))^(1/4) - 27*I*(-a^4)^(3/4)) - 3*(-a^4)^(1/4)*x*log(27*a^3*((a*x - 1)/(a*x + 1))^(1 /4) - 27*(-a^4)^(3/4)) - 2*(a*x + 1)*((a*x - 1)/(a*x + 1))^(3/4))/x
\[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{x^{2}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^2} \, dx=\frac {1}{4} \, {\left (6 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 6 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 3 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 3 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \frac {8 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{\frac {a x - 1}{a x + 1} + 1}\right )} a \]
1/4*(6*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/4) )) + 6*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/4 ))) - 3*sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/( a*x + 1)) + 1) + 3*sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt ((a*x - 1)/(a*x + 1)) + 1) - 8*((a*x - 1)/(a*x + 1))^(3/4)/((a*x - 1)/(a*x + 1) + 1))*a
Time = 0.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^2} \, dx=\frac {1}{4} \, {\left (6 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 6 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 3 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 3 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \frac {8 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{\frac {a x - 1}{a x + 1} + 1}\right )} a \]
1/4*(6*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/4) )) + 6*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/4 ))) - 3*sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/( a*x + 1)) + 1) + 3*sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt ((a*x - 1)/(a*x + 1)) + 1) - 8*((a*x - 1)/(a*x + 1))^(3/4)/((a*x - 1)/(a*x + 1) + 1))*a
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.33 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^2} \, dx=3\,{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )-3\,{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )-\frac {2\,a\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/4}}{\frac {a\,x-1}{a\,x+1}+1} \]