Integrand size = 12, antiderivative size = 155 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=-\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )-\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )-\frac {3}{2} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{\frac {-1+x}{x}}\right )-\frac {3}{2} \log \left (1+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{2} \log \left (1+\frac {1}{x}\right )-\frac {\log (x)}{2} \]
-3/2*ln((1+1/x)^(1/3)-((-1+x)/x)^(1/3))-3/2*ln(1+((-1+x)/x)^(1/3)/(1+1/x)^ (1/3))-1/2*ln(1+1/x)-1/2*ln(x)+arctan(-1/3*3^(1/2)+2/3*((-1+x)/x)^(1/3)/(1 +1/x)^(1/3)*3^(1/2))*3^(1/2)-arctan(1/3*3^(1/2)+2/3*((-1+x)/x)^(1/3)/(1+1/ x)^(1/3)*3^(1/2))*3^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.17 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=\frac {3}{2} e^{\frac {8}{3} \coth ^{-1}(x)} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},e^{4 \coth ^{-1}(x)}\right ) \]
Time = 0.25 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6721, 140, 72, 102}
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.
\(\displaystyle \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx\) |
\(\Big \downarrow \) 6721 |
\(\displaystyle -\int \frac {\sqrt [3]{1+\frac {1}{x}} x}{\sqrt [3]{1-\frac {1}{x}}}d\frac {1}{x}\) |
\(\Big \downarrow \) 140 |
\(\displaystyle -\int \frac {1}{\sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}-\int \frac {x}{\sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}\) |
\(\Big \downarrow \) 72 |
\(\displaystyle -\int \frac {x}{\sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}-\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{1-\frac {1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+1\right )-\frac {1}{2} \log \left (\frac {1}{x}+1\right )\) |
\(\Big \downarrow \) 102 |
\(\displaystyle -\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}\right )-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{1-\frac {1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+1\right )-\frac {3}{2} \log \left (\sqrt [3]{1-\frac {1}{x}}-\sqrt [3]{\frac {1}{x}+1}\right )-\frac {1}{2} \log \left (\frac {1}{x}+1\right )+\frac {1}{2} \log \left (\frac {1}{x}\right )\) |
-(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(1 - x^(-1))^(1/3))/(Sqrt[3]*(1 + x^(-1))^ (1/3))]) - Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 - x^(-1))^(1/3))/(Sqrt[3]*(1 + x^(-1))^(1/3))] - (3*Log[1 + (1 - x^(-1))^(1/3)/(1 + x^(-1))^(1/3)])/2 - (3*Log[(1 - x^(-1))^(1/3) - (1 + x^(-1))^(1/3)])/2 - Log[1 + x^(-1)]/2 + L og[x^(-1)]/2
3.2.23.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-d/b, 3]}, Simp[Sqrt[3]*(q/d)*ArcTan[1/Sqrt[3] - 2*q*((a + b* x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))], x] + (Simp[3*(q/(2*d))*Log[q*((a + b* x)^(1/3)/(c + d*x)^(1/3)) + 1], x] + Simp[(q/(2*d))*Log[c + d*x], x])] /; F reeQ[{a, b, c, d}, x] && NegQ[d/b]
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.) *(x_))), x_] :> With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])* q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q *(a + b*x)^(1/3) - (c + d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
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 = 0.66 (sec) , antiderivative size = 1026, normalized size of antiderivative = 6.62
-3*ln((945*(-(1-x)/(1+x))^(2/3)*RootOf(9*_Z^2-3*_Z+1)*x^2+1890*RootOf(9*_Z ^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x-168*(-(1-x)/(1+x))^(2/3)*x^2-504*(-(1-x) /(1+x))^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^2+72*RootOf(9*_Z^2-3*_Z+1)^2*x^2+945 *RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)-336*(-(1-x)/(1+x))^(2/3)*x-147 *(-(1-x)/(1+x))^(1/3)*x^2-180*RootOf(9*_Z^2-3*_Z+1)^2*x-465*RootOf(9*_Z^2- 3*_Z+1)*x^2-168*(-(1-x)/(1+x))^(2/3)+504*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+ x))^(1/3)+72*RootOf(9*_Z^2-3*_Z+1)^2+1026*RootOf(9*_Z^2-3*_Z+1)*x+323*x^2+ 147*(-(1-x)/(1+x))^(1/3)-465*RootOf(9*_Z^2-3*_Z+1)-34*x+323)/x)*RootOf(9*_ Z^2-3*_Z+1)-ln(-(945*(-(1-x)/(1+x))^(2/3)*RootOf(9*_Z^2-3*_Z+1)*x^2+1890*R ootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x-147*(-(1-x)/(1+x))^(2/3)*x^2-4 41*(-(1-x)/(1+x))^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^2-1152*RootOf(9*_Z^2-3*_Z+ 1)^2*x^2+945*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)-294*(-(1-x)/(1+x)) ^(2/3)*x-168*(-(1-x)/(1+x))^(1/3)*x^2+2880*RootOf(9*_Z^2-3*_Z+1)^2*x-120*R ootOf(9*_Z^2-3*_Z+1)*x^2-147*(-(1-x)/(1+x))^(2/3)+441*RootOf(9*_Z^2-3*_Z+1 )*(-(1-x)/(1+x))^(1/3)-1152*RootOf(9*_Z^2-3*_Z+1)^2-1884*RootOf(9*_Z^2-3*_ Z+1)*x+187*x^2+168*(-(1-x)/(1+x))^(1/3)-120*RootOf(9*_Z^2-3*_Z+1)+306*x+18 7)/x)+ln((945*(-(1-x)/(1+x))^(2/3)*RootOf(9*_Z^2-3*_Z+1)*x^2+1890*RootOf(9 *_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x-168*(-(1-x)/(1+x))^(2/3)*x^2-504*(-(1 -x)/(1+x))^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^2+72*RootOf(9*_Z^2-3*_Z+1)^2*x^2+ 945*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)-336*(-(1-x)/(1+x))^(2/3)...
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=\sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - 1\right ) + \frac {1}{2} \, \log \left (\frac {{\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + x + 1}{x + 1}\right ) \]
sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(2/3) + 1/3*sqrt(3)) - log((( x - 1)/(x + 1))^(2/3) - 1) + 1/2*log(((x + 1)*((x - 1)/(x + 1))^(2/3) + (x - 1)*((x - 1)/(x + 1))^(1/3) + x + 1)/(x + 1))
\[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=\int \frac {1}{x \sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=-\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) + \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]
-sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) + sqrt(3)*arc tan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) - 1)) + 1/2*log(((x - 1)/(x + 1 ))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/2*log(((x - 1)/(x + 1))^(2/3) - ((x - 1)/(x + 1))^(1/3) + 1) - log(((x - 1)/(x + 1))^(1/3) + 1) - log((( x - 1)/(x + 1))^(1/3) - 1)
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.51 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + 1\right )}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}}}{x + 1} + 1\right ) - \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - 1 \right |}\right ) \]
sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(2/3) + 1)) + 1/2*log(((x - 1)/(x + 1))^(2/3) + (x - 1)*((x - 1)/(x + 1))^(1/3)/(x + 1) + 1) - log(a bs(((x - 1)/(x + 1))^(2/3) - 1))
Time = 4.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=-\ln \left (1296\,{\left (\frac {x-1}{x+1}\right )}^{2/3}-1296\right )-\ln \left (1296\,{\left (\frac {x-1}{x+1}\right )}^{2/3}+648-\sqrt {3}\,648{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+\ln \left (1296\,{\left (\frac {x-1}{x+1}\right )}^{2/3}+648+\sqrt {3}\,648{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right ) \]