Integrand size = 14, antiderivative size = 351 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {25}{4} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {25 a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}-\frac {25 a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}-\frac {25 a^2 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {25 a^2 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}} \]
-25/4*a^2*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)-5/2*a^2*(1-1/a/x)^(3/4)*(1+1/a/x )^(5/4)-2*a^2*(1+1/a/x)^(9/4)/(1-1/a/x)^(1/4)-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)
Time = 0.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{16} a^2 \left (-128 e^{\frac {1}{2} \coth ^{-1}(a x)}+\frac {32 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (1+e^{2 \coth ^{-1}(a x)}\right )^2}-\frac {104 e^{\frac {1}{2} \coth ^{-1}(a x)}}{1+e^{2 \coth ^{-1}(a x)}}-50 \sqrt {2} \arctan \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+50 \sqrt {2} \arctan \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )-25 \sqrt {2} \log \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )+25 \sqrt {2} \log \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )\right ) \]
(a^2*(-128*E^(ArcCoth[a*x]/2) + (32*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[ a*x]))^2 - (104*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x])) - 50*Sqrt[2]* ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/2)] + 50*Sqrt[2]*ArcTan[1 + Sqrt[2]*E^( ArcCoth[a*x]/2)] - 25*Sqrt[2]*Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCo th[a*x]] + 25*Sqrt[2]*Log[1 + Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]] ))/16
Time = 0.51 (sec) , antiderivative size = 293, 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, 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.
| \(\displaystyle \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle 5 a \int \frac {\left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}d\frac {1}{x}-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle 5 a \left (\frac {5}{4} \int \frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}d\frac {1}{x}-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (-2 a \int \frac {1}{\left (2-\frac {1}{x^4}\right )^{3/4} x^2}d\sqrt [4]{1-\frac {1}{a x}}-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (-2 a \int \frac {1}{\left (1+\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (-2 a \left (\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (-2 a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (-2 a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (-2 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (-2 a \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (-2 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (-2 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 5 a \left (\frac {5}{4} \left (-2 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )-a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )-\frac {1}{2} a \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}\) |
(-2*a^2*(1 + 1/(a*x))^(9/4))/(1 - 1/(a*x))^(1/4) + 5*a*(-1/2*(a*(1 - 1/(a* x))^(3/4)*(1 + 1/(a*x))^(5/4)) + (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)))/4)
3.1.84.3.1 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[(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]
\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}} x^{3}}d x\]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {25 \, \left (-a^{8}\right )^{\frac {1}{4}} {\left (a x^{3} - x^{2}\right )} \log \left (15625 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 15625 \, \left (-a^{8}\right )^{\frac {3}{4}}\right ) - 25 \, \left (-a^{8}\right )^{\frac {1}{4}} {\left (i \, a x^{3} - i \, x^{2}\right )} \log \left (15625 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 15625 i \, \left (-a^{8}\right )^{\frac {3}{4}}\right ) - 25 \, \left (-a^{8}\right )^{\frac {1}{4}} {\left (-i \, a x^{3} + i \, x^{2}\right )} \log \left (15625 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 15625 i \, \left (-a^{8}\right )^{\frac {3}{4}}\right ) - 25 \, \left (-a^{8}\right )^{\frac {1}{4}} {\left (a x^{3} - x^{2}\right )} \log \left (15625 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 15625 \, \left (-a^{8}\right )^{\frac {3}{4}}\right ) + 2 \, {\left (43 \, a^{3} x^{3} + 34 \, a^{2} x^{2} - 11 \, a x - 2\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{8 \, {\left (a x^{3} - x^{2}\right )}} \]
-1/8*(25*(-a^8)^(1/4)*(a*x^3 - x^2)*log(15625*a^6*((a*x - 1)/(a*x + 1))^(1 /4) + 15625*(-a^8)^(3/4)) - 25*(-a^8)^(1/4)*(I*a*x^3 - I*x^2)*log(15625*a^ 6*((a*x - 1)/(a*x + 1))^(1/4) + 15625*I*(-a^8)^(3/4)) - 25*(-a^8)^(1/4)*(- I*a*x^3 + I*x^2)*log(15625*a^6*((a*x - 1)/(a*x + 1))^(1/4) - 15625*I*(-a^8 )^(3/4)) - 25*(-a^8)^(1/4)*(a*x^3 - x^2)*log(15625*a^6*((a*x - 1)/(a*x + 1 ))^(1/4) - 15625*(-a^8)^(3/4)) + 2*(43*a^3*x^3 + 34*a^2*x^2 - 11*a*x - 2)* ((a*x - 1)/(a*x + 1))^(3/4))/(a*x^3 - x^2)
\[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {1}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {1}{16} \, {\left (25 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right )\right )} a + \frac {8 \, {\left (\frac {45 \, {\left (a x - 1\right )} a}{a x + 1} + \frac {25 \, {\left (a x - 1\right )}^{2} a}{{\left (a x + 1\right )}^{2}} + 16 \, a\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}\right )} a \]
-1/16*(25*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1)) ^(1/4))) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1) )^(1/4))) - sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) + sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sq rt((a*x - 1)/(a*x + 1)) + 1))*a + 8*(45*(a*x - 1)*a/(a*x + 1) + 25*(a*x - 1)^2*a/(a*x + 1)^2 + 16*a)/(((a*x - 1)/(a*x + 1))^(9/4) + 2*((a*x - 1)/(a* x + 1))^(5/4) + ((a*x - 1)/(a*x + 1))^(1/4)))*a
Time = 0.32 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.69 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {1}{16} \, {\left (50 \, \sqrt {2} a \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 50 \, \sqrt {2} a \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 25 \, \sqrt {2} a \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 25 \, \sqrt {2} a \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {128 \, a}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {8 \, {\left (\frac {9 \, {\left (a x - 1\right )} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} + 13 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \]
-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*(9*(a*x - 1)*a*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1) + 13*a*((a*x - 1)/(a*x + 1))^(3/4))/((a*x - 1)/(a*x + 1) + 1)^2)*a
Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.43 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {25\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4}-\frac {25\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4}-\frac {8\,a^2+\frac {25\,a^2\,{\left (a\,x-1\right )}^2}{2\,{\left (a\,x+1\right )}^2}+\frac {45\,a^2\,\left (a\,x-1\right )}{2\,\left (a\,x+1\right )}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}+2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}} \]
(25*(-1)^(1/4)*a^2*atanh((-1)^(1/4)*((a*x - 1)/(a*x + 1))^(1/4)))/4 - (25* (-1)^(1/4)*a^2*atan((-1)^(1/4)*((a*x - 1)/(a*x + 1))^(1/4)))/4 - (8*a^2 + (25*a^2*(a*x - 1)^2)/(2*(a*x + 1)^2) + (45*a^2*(a*x - 1))/(2*(a*x + 1)))/( ((a*x - 1)/(a*x + 1))^(1/4) + 2*((a*x - 1)/(a*x + 1))^(5/4) + ((a*x - 1)/( a*x + 1))^(9/4))