Integrand size = 12, antiderivative size = 260 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}-\frac {1}{18} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{18} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {\log \left (1-\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{12 \sqrt {3}}-\frac {\log \left (1+\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{12 \sqrt {3}} \]
1/6*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)+1/2*(1+1/x)^(7/6)*((-1+x)/x)^(5/6)+1/9* arctan(((-1+x)/x)^(1/6)/(1+1/x)^(1/6))+1/18*arctan(2*((-1+x)/x)^(1/6)/(1+1 /x)^(1/6)-3^(1/2))+1/18*arctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+3^(1/2))+1 /36*ln(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)-((-1+x)/x)^(1/6)*3^(1/2)/(1+1/x)^( 1/6))*3^(1/2)-1/36*ln(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)+((-1+x)/x)^(1/6)*3^ (1/2)/(1+1/x)^(1/6))*3^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.95 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.48 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {1}{54} \left (\frac {18 e^{\frac {1}{3} \coth ^{-1}(x)} \left (1+7 e^{2 \coth ^{-1}(x)}\right )}{\left (1+e^{2 \coth ^{-1}(x)}\right )^2}-6 \arctan \left (e^{\frac {1}{3} \coth ^{-1}(x)}\right )+\text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {2 \coth ^{-1}(x)-6 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right )-\coth ^{-1}(x) \text {$\#$1}^2+3 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}+2 \text {$\#$1}^3}\&\right ]\right ) \]
((18*E^(ArcCoth[x]/3)*(1 + 7*E^(2*ArcCoth[x])))/(1 + E^(2*ArcCoth[x]))^2 - 6*ArcTan[E^(ArcCoth[x]/3)] + RootSum[1 - #1^2 + #1^4 & , (2*ArcCoth[x] - 6*Log[E^(ArcCoth[x]/3) - #1] - ArcCoth[x]*#1^2 + 3*Log[E^(ArcCoth[x]/3) - #1]*#1^2)/(-#1 + 2*#1^3) & ])/54
Time = 0.41 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.93, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6721, 90, 60, 73, 854, 824, 27, 216, 1142, 25, 1083, 217, 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.
| \(\displaystyle \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}} x}d\frac {1}{x}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}-\frac {1}{6} \int \frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}-\frac {1}{3} \int \frac {1}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{6} \left (2 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {1}{6} \left (2 \int \frac {1}{\left (1+\frac {1}{x^6}\right ) x^4}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {1}{6} \left (2 \left (\frac {1}{3} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{3} \int -\frac {1-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{2 \left (-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{3} \int -\frac {\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+1}{2 \left (\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (2 \left (\frac {1}{3} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{6} \left (2 \left (-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{6} \left (2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{6} \left (2 \left (\frac {1}{6} \left (-\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (-\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (2 \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\frac {1}{6} \left (\arctan \left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{6} \left (2 \left (\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\frac {1}{6} \left (\arctan \left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}\) |
((1 - x^(-1))^(5/6)*(1 + x^(-1))^(7/6))/2 + ((1 - x^(-1))^(5/6)*(1 + x^(-1 ))^(1/6) + 2*(ArcTan[(1 - x^(-1))^(1/6)/(2 - x^(-6))^(1/6)]/3 + (-ArcTan[S qrt[3] - (2*(1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6)] + (Sqrt[3]*Log[1 - (Sq rt[3]*(1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6) + x^(-2)])/2)/6 + (ArcTan[Sqr t[3] + (2*(1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6)] - (Sqrt[3]*Log[1 + (Sqrt [3]*(1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6) + x^(-2)])/2)/6))/6
3.2.18.3.1 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)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 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_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/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] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 15.33 (sec) , antiderivative size = 1262, normalized size of antiderivative = 4.85
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1262\) |
| risch | \(\text {Expression too large to display}\) | \(3478\) |
1/6*(1+x)*(4*x+3)/x^2*(-(1-x)/(1+x))^(5/6)-1/18*RootOf(_Z^2+1)*ln(-(9*Root Of(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(2/3)*x-18*(-(1-x)/(1+x))^ (1/2)*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*x-3*(-(1-x)/(1+x ))^(5/6)*x+6*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+9*RootOf(3*_Z*RootOf(_Z ^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(2/3)-18*(-(1-x)/(1+x))^(1/2)*RootOf(3*_Z*R ootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)-3*(-(1-x)/(1+x))^(5/6)+6*RootOf(_Z^ 2+1)*(-(1-x)/(1+x))^(2/3)-18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/ (1+x))^(1/3)*x+9*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*(-(1- x)/(1+x))^(1/6)*x+6*(-(1-x)/(1+x))^(1/2)*x-3*RootOf(_Z^2+1)*(-(1-x)/(1+x)) ^(1/3)*x-18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(1/3)+9*Ro otOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/6)+6*( -(1-x)/(1+x))^(1/2)-3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)+3*RootOf(3*_Z*Ro otOf(_Z^2+1)+9*_Z^2-1)*x-RootOf(_Z^2+1)*x)/x)-1/18*ln((18*RootOf(3*_Z*Root Of(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)*x+3*Roo tOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1) *(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(5/6)+3*RootOf(_Z^2+1)*(-(1-x)/(1+x ))^(2/3)+18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(1/3)*x+6* (-(1-x)/(1+x))^(1/2)*x+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)*x+18*RootOf(3 *_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(1/3)+6*(-(1-x)/(1+x))^(1/2)+3 *RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)+6*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^...
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.02 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {\sqrt {2} x^{2} \sqrt {i \, \sqrt {3} + 1} \log \left ({\left (i \, \sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {i \, \sqrt {3} + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - \sqrt {2} x^{2} \sqrt {i \, \sqrt {3} + 1} \log \left ({\left (-i \, \sqrt {3} \sqrt {2} + \sqrt {2}\right )} \sqrt {i \, \sqrt {3} + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - \sqrt {2} x^{2} \sqrt {-i \, \sqrt {3} + 1} \log \left ({\left (i \, \sqrt {3} \sqrt {2} + \sqrt {2}\right )} \sqrt {-i \, \sqrt {3} + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \sqrt {2} x^{2} \sqrt {-i \, \sqrt {3} + 1} \log \left ({\left (-i \, \sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {-i \, \sqrt {3} + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 4 \, x^{2} \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 6 \, {\left (4 \, x^{2} + 7 \, x + 3\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{36 \, x^{2}} \]
1/36*(sqrt(2)*x^2*sqrt(I*sqrt(3) + 1)*log((I*sqrt(3)*sqrt(2) - sqrt(2))*sq rt(I*sqrt(3) + 1) + 4*((x - 1)/(x + 1))^(1/6)) - sqrt(2)*x^2*sqrt(I*sqrt(3 ) + 1)*log((-I*sqrt(3)*sqrt(2) + sqrt(2))*sqrt(I*sqrt(3) + 1) + 4*((x - 1) /(x + 1))^(1/6)) - sqrt(2)*x^2*sqrt(-I*sqrt(3) + 1)*log((I*sqrt(3)*sqrt(2) + sqrt(2))*sqrt(-I*sqrt(3) + 1) + 4*((x - 1)/(x + 1))^(1/6)) + sqrt(2)*x^ 2*sqrt(-I*sqrt(3) + 1)*log((-I*sqrt(3)*sqrt(2) - sqrt(2))*sqrt(-I*sqrt(3) + 1) + 4*((x - 1)/(x + 1))^(1/6)) + 4*x^2*arctan(((x - 1)/(x + 1))^(1/6)) + 6*(4*x^2 + 7*x + 3)*((x - 1)/(x + 1))^(5/6))/x^2
\[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=\int \frac {1}{x^{3} \sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=-\frac {1}{36} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{36} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {\left (\frac {x - 1}{x + 1}\right )^{\frac {11}{6}} + 7 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{3 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} + \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + 1\right )}} + \frac {1}{18} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{18} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{9} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]
-1/36*sqrt(3)*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3 ) + 1) + 1/36*sqrt(3)*log(-sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/3*(((x - 1)/(x + 1))^(11/6) + 7*((x - 1)/(x + 1))^(5/6 ))/(2*(x - 1)/(x + 1) + (x - 1)^2/(x + 1)^2 + 1) + 1/18*arctan(sqrt(3) + 2 *((x - 1)/(x + 1))^(1/6)) + 1/18*arctan(-sqrt(3) + 2*((x - 1)/(x + 1))^(1/ 6)) + 1/9*arctan(((x - 1)/(x + 1))^(1/6))
Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.67 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=-\frac {1}{36} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{36} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {\frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x + 1} + 7 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{3 \, {\left (\frac {x - 1}{x + 1} + 1\right )}^{2}} + \frac {1}{18} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{18} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{9} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]
-1/36*sqrt(3)*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3 ) + 1) + 1/36*sqrt(3)*log(-sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/3*((x - 1)*((x - 1)/(x + 1))^(5/6)/(x + 1) + 7*((x - 1 )/(x + 1))^(5/6))/((x - 1)/(x + 1) + 1)^2 + 1/18*arctan(sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 1/18*arctan(-sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 1 /9*arctan(((x - 1)/(x + 1))^(1/6))
Time = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )}{9}+\frac {\frac {7\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{3}+\frac {{\left (\frac {x-1}{x+1}\right )}^{11/6}}{3}}{\frac {2\,\left (x-1\right )}{x+1}+\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+1}-\mathrm {atan}\left (\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{243\,\left (-\frac {1}{243}+\frac {\sqrt {3}\,1{}\mathrm {i}}{243}\right )}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\mathrm {atan}\left (\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{243\,\left (\frac {1}{243}+\frac {\sqrt {3}\,1{}\mathrm {i}}{243}\right )}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right ) \]
atan(((x - 1)/(x + 1))^(1/6))/9 + ((7*((x - 1)/(x + 1))^(5/6))/3 + ((x - 1 )/(x + 1))^(11/6)/3)/((2*(x - 1))/(x + 1) + (x - 1)^2/(x + 1)^2 + 1) - ata n((2*((x - 1)/(x + 1))^(1/6))/(243*((3^(1/2)*1i)/243 - 1/243)))*((3^(1/2)* 1i)/18 + 1/18) - atan((2*((x - 1)/(x + 1))^(1/6))/(243*((3^(1/2)*1i)/243 + 1/243)))*((3^(1/2)*1i)/18 - 1/18)