Integrand size = 14, antiderivative size = 356 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {17}{24} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {1}{4} a^3 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{1+\frac {1}{a x}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{1+\frac {1}{a x}}}{3 x}-\frac {17 a^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {17 a^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {17 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}-\frac {17 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}} \]
-17/24*a^3*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)-1/4*a^3*(1-1/a/x)^(7/4)*(1+1/a/ x)^(1/4)+1/3*a^2*(1-1/a/x)^(7/4)*(1+1/a/x)^(1/4)/x+17/16*a^3*arctan(-1+(1- 1/a/x)^(1/4)*2^(1/2)/(1+1/a/x)^(1/4))*2^(1/2)+17/16*a^3*arctan(1+(1-1/a/x) ^(1/4)*2^(1/2)/(1+1/a/x)^(1/4))*2^(1/2)+17/32*a^3*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)-17/32*a^3*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 9 vs. order 3 in optimal.
Time = 0.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.26 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{96} a^3 \left (-\frac {8 e^{\frac {1}{2} \coth ^{-1}(a x)} \left (45+30 e^{2 \coth ^{-1}(a x)}+17 e^{4 \coth ^{-1}(a x)}\right )}{\left (1+e^{2 \coth ^{-1}(a x)}\right )^3}+51 \text {RootSum}\left [1+\text {$\#$1}^4\&,\frac {\coth ^{-1}(a x)+2 \log \left (e^{-\frac {1}{2} \coth ^{-1}(a x)}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right ) \]
(a^3*((-8*E^(ArcCoth[a*x]/2)*(45 + 30*E^(2*ArcCoth[a*x]) + 17*E^(4*ArcCoth [a*x])))/(1 + E^(2*ArcCoth[a*x]))^3 + 51*RootSum[1 + #1^4 & , (ArcCoth[a*x ] + 2*Log[E^(-1/2*ArcCoth[a*x]) - #1])/#1 & ]))/96
Time = 0.49 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.85, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6721, 101, 27, 90, 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.
| \(\displaystyle \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^4} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right )^{3/4}}{\left (1+\frac {1}{a x}\right )^{3/4} x^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {1}{3} a^2 \int -\frac {\left (2 a-\frac {3}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/4}}{2 a \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \int \frac {\left (2 a-\frac {3}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/4}}{\left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \int \frac {\left (1-\frac {1}{a x}\right )^{3/4}}{\left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \left (\frac {3}{2} \int \frac {1}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}+a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}\right )+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \left (a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-6 a \int \frac {1}{\left (2-\frac {1}{x^4}\right )^{3/4} x^2}d\sqrt [4]{1-\frac {1}{a x}}\right )+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \left (a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-6 a \int \frac {1}{\left (1+\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \left (a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-6 a \left (\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )\right )+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \left (a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-6 a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )\right )+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \left (a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-6 a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )\right )+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \left (a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-6 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )\right )+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \left (a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-6 a \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )\right )+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \left (a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-6 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )\right )+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {17}{4} a \left (a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-6 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1}{\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1}d\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )\right )\right )+\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}}{3 x}-\frac {1}{6} a \left (\frac {3}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}+\frac {17}{4} a \left (a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-6 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{2-\frac {1}{x^4}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )\right )\right )\) |
(a^2*(1 - 1/(a*x))^(7/4)*(1 + 1/(a*x))^(1/4))/(3*x) - (a*((3*a^2*(1 - 1/(a *x))^(7/4)*(1 + 1/(a*x))^(1/4))/2 + (17*a*(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)))/4))/6
3.2.3.3.1 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*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
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]
\[\int \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{4}}}{x^{4}}d x\]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.61 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {51 \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (4913 \, a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 4913 \, \left (-a^{12}\right )^{\frac {3}{4}}\right ) - 51 i \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (4913 \, a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 4913 i \, \left (-a^{12}\right )^{\frac {3}{4}}\right ) + 51 i \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (4913 \, a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 4913 i \, \left (-a^{12}\right )^{\frac {3}{4}}\right ) - 51 \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (4913 \, a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 4913 \, \left (-a^{12}\right )^{\frac {3}{4}}\right ) - 2 \, {\left (23 \, a^{3} x^{3} + 9 \, a^{2} x^{2} - 6 \, a x + 8\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{48 \, x^{3}} \]
1/48*(51*(-a^12)^(1/4)*x^3*log(4913*a^9*((a*x - 1)/(a*x + 1))^(1/4) + 4913 *(-a^12)^(3/4)) - 51*I*(-a^12)^(1/4)*x^3*log(4913*a^9*((a*x - 1)/(a*x + 1) )^(1/4) + 4913*I*(-a^12)^(3/4)) + 51*I*(-a^12)^(1/4)*x^3*log(4913*a^9*((a* x - 1)/(a*x + 1))^(1/4) - 4913*I*(-a^12)^(3/4)) - 51*(-a^12)^(1/4)*x^3*log (4913*a^9*((a*x - 1)/(a*x + 1))^(1/4) - 4913*(-a^12)^(3/4)) - 2*(23*a^3*x^ 3 + 9*a^2*x^2 - 6*a*x + 8)*((a*x - 1)/(a*x + 1))^(3/4))/x^3
\[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{x^{4}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{96} \, {\left (51 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right )\right )} a^{2} - \frac {8 \, {\left (45 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{4}} + 30 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{4}} + 17 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \]
1/96*(51*(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) + sqr t((a*x - 1)/(a*x + 1)) + 1))*a^2 - 8*(45*a^2*((a*x - 1)/(a*x + 1))^(11/4) + 30*a^2*((a*x - 1)/(a*x + 1))^(7/4) + 17*a^2*((a*x - 1)/(a*x + 1))^(3/4)) /(3*(a*x - 1)/(a*x + 1) + 3*(a*x - 1)^2/(a*x + 1)^2 + (a*x - 1)^3/(a*x + 1 )^3 + 1))*a
Time = 0.33 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{96} \, {\left (102 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 102 \, \sqrt {2} a^{2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 51 \, \sqrt {2} a^{2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 51 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \frac {8 \, {\left (\frac {30 \, {\left (a x - 1\right )} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} + \frac {45 \, {\left (a x - 1\right )}^{2} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} + 17 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \]
1/96*(102*sqrt(2)*a^2*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1) )^(1/4))) + 102*sqrt(2)*a^2*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a *x + 1))^(1/4))) - 51*sqrt(2)*a^2*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) + 51*sqrt(2)*a^2*log(-sqrt(2)*((a*x - 1)/ (a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) - 8*(30*(a*x - 1)*a^2*(( a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1) + 45*(a*x - 1)^2*a^2*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^2 + 17*a^2*((a*x - 1)/(a*x + 1))^(3/4))/((a*x - 1)/(a *x + 1) + 1)^3)*a
Time = 0.07 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-\frac {3}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {17\,{\left (-1\right )}^{1/4}\,a^3\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8}-\frac {\frac {17\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/4}}{12}+\frac {5\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/4}}{2}+\frac {15\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/4}}{4}}{\frac {3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {3\,\left (a\,x-1\right )}{a\,x+1}+1}-\frac {17\,{\left (-1\right )}^{1/4}\,a^3\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8} \]
(17*(-1)^(1/4)*a^3*atan((-1)^(1/4)*((a*x - 1)/(a*x + 1))^(1/4)))/8 - ((17* a^3*((a*x - 1)/(a*x + 1))^(3/4))/12 + (5*a^3*((a*x - 1)/(a*x + 1))^(7/4))/ 2 + (15*a^3*((a*x - 1)/(a*x + 1))^(11/4))/4)/((3*(a*x - 1)^2)/(a*x + 1)^2 + (a*x - 1)^3/(a*x + 1)^3 + (3*(a*x - 1))/(a*x + 1) + 1) - (17*(-1)^(1/4)* a^3*atanh((-1)^(1/4)*((a*x - 1)/(a*x + 1))^(1/4)))/8