Integrand size = 14, antiderivative size = 351 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {25}{4} a^2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {25 a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}+\frac {25 a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}-\frac {25 a^2 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {25 a^2 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}} \]
-2*a^2*(1-1/a/x)^(9/4)/(1+1/a/x)^(1/4)-25/4*a^2*(1-1/a/x)^(1/4)*(1+1/a/x)^ (3/4)-5/2*a^2*(1-1/a/x)^(5/4)*(1+1/a/x)^(3/4)+25/8*a^2*arctan(-1+(1-1/a/x) ^(1/4)*2^(1/2)/(1+1/a/x)^(1/4))*2^(1/2)+25/8*a^2*arctan(1+(1-1/a/x)^(1/4)* 2^(1/2)/(1+1/a/x)^(1/4))*2^(1/2)-25/16*a^2*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)+25/16*a^2*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)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.29 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {8}{3} a^2 e^{-\frac {1}{2} \coth ^{-1}(a x)} \left (3+e^{2 \coth ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-e^{2 \coth ^{-1}(a x)}\right )+e^{2 \coth ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},2,\frac {7}{4},-e^{2 \coth ^{-1}(a x)}\right )+2 e^{2 \coth ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},3,\frac {7}{4},-e^{2 \coth ^{-1}(a x)}\right )\right ) \]
(-8*a^2*(3 + E^(2*ArcCoth[a*x])*Hypergeometric2F1[3/4, 1, 7/4, -E^(2*ArcCo th[a*x])] + E^(2*ArcCoth[a*x])*Hypergeometric2F1[3/4, 2, 7/4, -E^(2*ArcCot h[a*x])] + 2*E^(2*ArcCoth[a*x])*Hypergeometric2F1[3/4, 3, 7/4, -E^(2*ArcCo th[a*x])]))/(3*E^(ArcCoth[a*x]/2))
Time = 0.48 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.83, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6721, 87, 60, 60, 73, 770, 755, 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.
| \(\displaystyle \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right )^{5/4}}{\left (1+\frac {1}{a x}\right )^{5/4} x}d\frac {1}{x}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -5 a \int \frac {\left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -5 a \left (\frac {5}{4} \int \frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -5 a \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {1}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}+a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -5 a \left (\frac {5}{4} \left (a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-2 a \int \frac {1}{\sqrt [4]{2-\frac {1}{x^4}}}d\sqrt [4]{1-\frac {1}{a x}}\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle -5 a \left (\frac {5}{4} \left (a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-2 a \int \frac {1}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -5 a \left (\frac {5}{4} \left (a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-2 a \left (\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -5 a \left (\frac {5}{4} \left (a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-2 a \left (\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )\right )\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -5 a \left (\frac {5}{4} \left (a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-2 a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -5 a \left (\frac {5}{4} \left (a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-2 a \left (\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -5 a \left (\frac {5}{4} \left (a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-2 a \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 a \left (\frac {5}{4} \left (a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-2 a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -5 a \left (\frac {5}{4} \left (a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-2 a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}-5 a \left (\frac {5}{4} \left (a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-2 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )\right )+\frac {1}{2} a \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}\right )\) |
(-2*a^2*(1 - 1/(a*x))^(9/4))/(1 + 1/(a*x))^(1/4) - 5*a*((a*(1 - 1/(a*x))^( 5/4)*(1 + 1/(a*x))^(3/4))/2 + (5*(a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4 ) - 2*a*((-(ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(2 - x^(-4))^(1/4)]/S qrt[2]) + ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(2 - x^(-4))^(1/4)]/Sqr t[2])/2 + (-1/2*Log[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(2 - x^(-4))^(1/4) + x^(-2)]/Sqrt[2] + Log[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(2 - x^(-4))^(1/4 ) + x^(-2)]/(2*Sqrt[2]))/2)))/4)
3.2.11.3.1 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 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]
\[\int \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}{x^{3}}d x\]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.60 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {25 \, \left (-a^{8}\right )^{\frac {1}{4}} x^{2} \log \left (25 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 25 \, \left (-a^{8}\right )^{\frac {1}{4}}\right ) + 25 i \, \left (-a^{8}\right )^{\frac {1}{4}} x^{2} \log \left (25 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 25 i \, \left (-a^{8}\right )^{\frac {1}{4}}\right ) - 25 i \, \left (-a^{8}\right )^{\frac {1}{4}} x^{2} \log \left (25 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 25 i \, \left (-a^{8}\right )^{\frac {1}{4}}\right ) - 25 \, \left (-a^{8}\right )^{\frac {1}{4}} x^{2} \log \left (25 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 25 \, \left (-a^{8}\right )^{\frac {1}{4}}\right ) - 2 \, {\left (43 \, a^{2} x^{2} + 9 \, a x - 2\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{8 \, x^{2}} \]
1/8*(25*(-a^8)^(1/4)*x^2*log(25*a^2*((a*x - 1)/(a*x + 1))^(1/4) + 25*(-a^8 )^(1/4)) + 25*I*(-a^8)^(1/4)*x^2*log(25*a^2*((a*x - 1)/(a*x + 1))^(1/4) + 25*I*(-a^8)^(1/4)) - 25*I*(-a^8)^(1/4)*x^2*log(25*a^2*((a*x - 1)/(a*x + 1) )^(1/4) - 25*I*(-a^8)^(1/4)) - 25*(-a^8)^(1/4)*x^2*log(25*a^2*((a*x - 1)/( a*x + 1))^(1/4) - 25*(-a^8)^(1/4)) - 2*(43*a^2*x^2 + 9*a*x - 2)*((a*x - 1) /(a*x + 1))^(1/4))/x^2
\[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}{x^{3}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{16} \, {\left (50 \, \sqrt {2} a \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 50 \, \sqrt {2} a \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 25 \, \sqrt {2} a \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 25 \, \sqrt {2} a \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 128 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \frac {8 \, {\left (13 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 9 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {2 \, {\left (a x - 1\right )}}{a x + 1} + \frac {{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \]
1/16*(50*sqrt(2)*a*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^( 1/4))) + 50*sqrt(2)*a*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1 ))^(1/4))) + 25*sqrt(2)*a*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt(( a*x - 1)/(a*x + 1)) + 1) - 25*sqrt(2)*a*log(-sqrt(2)*((a*x - 1)/(a*x + 1)) ^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) - 128*a*((a*x - 1)/(a*x + 1))^(1/4 ) - 8*(13*a*((a*x - 1)/(a*x + 1))^(5/4) + 9*a*((a*x - 1)/(a*x + 1))^(1/4)) /(2*(a*x - 1)/(a*x + 1) + (a*x - 1)^2/(a*x + 1)^2 + 1))*a
Time = 0.30 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.69 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{16} \, {\left (50 \, \sqrt {2} a \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 50 \, \sqrt {2} a \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 25 \, \sqrt {2} a \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 25 \, \sqrt {2} a \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 128 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \frac {8 \, {\left (\frac {13 \, {\left (a x - 1\right )} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} + 9 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \]
1/16*(50*sqrt(2)*a*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^( 1/4))) + 50*sqrt(2)*a*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1 ))^(1/4))) + 25*sqrt(2)*a*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt(( a*x - 1)/(a*x + 1)) + 1) - 25*sqrt(2)*a*log(-sqrt(2)*((a*x - 1)/(a*x + 1)) ^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) - 128*a*((a*x - 1)/(a*x + 1))^(1/4 ) - 8*(13*(a*x - 1)*a*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1) + 9*a*((a*x - 1)/(a*x + 1))^(1/4))/((a*x - 1)/(a*x + 1) + 1)^2)*a
Time = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.44 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-8\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-\frac {\frac {9\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{2}+\frac {13\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{2}}{\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2\,\left (a\,x-1\right )}{a\,x+1}+1}-\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )\,25{}\mathrm {i}}{4}-\frac {25\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )}{4} \]
- 8*a^2*((a*x - 1)/(a*x + 1))^(1/4) - ((9*a^2*((a*x - 1)/(a*x + 1))^(1/4)) /2 + (13*a^2*((a*x - 1)/(a*x + 1))^(5/4))/2)/((a*x - 1)^2/(a*x + 1)^2 + (2 *(a*x - 1))/(a*x + 1) + 1) - ((-1)^(1/4)*a^2*atan((-1)^(1/4)*((a*x - 1)/(a *x + 1))^(1/4))*25i)/4 - (25*(-1)^(1/4)*a^2*atan((-1)^(1/4)*((a*x - 1)/(a* x + 1))^(1/4)*1i))/4