Optimal. Leaf size=130 \[ \frac{1}{2} \left (\frac{x-1}{x}\right )^{2/3} \left (\frac{1}{x}+1\right )^{4/3}+\frac{1}{3} \left (\frac{x-1}{x}\right )^{2/3} \sqrt [3]{\frac{1}{x}+1}-\frac{1}{3} \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+1\right )-\frac{1}{9} \log \left (\frac{1}{x}+1\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}\right )}{3 \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0533082, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6171, 80, 50, 60} \[ \frac{1}{2} \left (\frac{x-1}{x}\right )^{2/3} \left (\frac{1}{x}+1\right )^{4/3}+\frac{1}{3} \left (\frac{x-1}{x}\right )^{2/3} \sqrt [3]{\frac{1}{x}+1}-\frac{1}{3} \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+1\right )-\frac{1}{9} \log \left (\frac{1}{x}+1\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6171
Rule 80
Rule 50
Rule 60
Rubi steps
\begin{align*} \int \frac{e^{\frac{2}{3} \coth ^{-1}(x)}}{x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x \sqrt [3]{1+x}}{\sqrt [3]{1-x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \left (1+\frac{1}{x}\right )^{4/3} \left (\frac{-1+x}{x}\right )^{2/3}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3}+\frac{1}{2} \left (1+\frac{1}{x}\right )^{4/3} \left (\frac{-1+x}{x}\right )^{2/3}-\frac{2}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3}+\frac{1}{2} \left (1+\frac{1}{x}\right )^{4/3} \left (\frac{-1+x}{x}\right )^{2/3}-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{-\frac{1-x}{x}}}{\sqrt{3} \sqrt [3]{1+\frac{1}{x}}}\right )}{3 \sqrt{3}}-\frac{1}{3} \log \left (1+\frac{\sqrt [3]{-\frac{1-x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )-\frac{1}{9} \log \left (1+\frac{1}{x}\right )\\ \end{align*}
Mathematica [C] time = 0.275655, size = 134, normalized size = 1.03 \[ -\frac{2}{27} \left (-\text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2+1\& ,\frac{\text{$\#$1}^2 \coth ^{-1}(x)-3 \text{$\#$1}^2 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right )-3 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right )+\coth ^{-1}(x)}{\text{$\#$1}^2-2}\& \right ]-2 \coth ^{-1}(x)-\frac{36 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}+1}+\frac{27 e^{\frac{2}{3} \coth ^{-1}(x)}}{\left (e^{2 \coth ^{-1}(x)}+1\right )^2}+3 \log \left (e^{\frac{2}{3} \coth ^{-1}(x)}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt [3]{{\frac{-1+x}{1+x}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.52712, size = 167, normalized size = 1.28 \begin{align*} \frac{2}{9} \, \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{2 \,{\left (\left (\frac{x - 1}{x + 1}\right )^{\frac{5}{3}} + 4 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}\right )}}{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}{9} \, \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{2}{9} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.58706, size = 321, normalized size = 2.47 \begin{align*} \frac{4 \, \sqrt{3} x^{2} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + 2 \, x^{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 ) - 4 \, x^{2} \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) + 3 \,{\left (5 \, x^{2} + 8 \, x + 3\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{18 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \sqrt [3]{\frac{x - 1}{x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15536, size = 165, normalized size = 1.27 \begin{align*} \frac{2}{9} \, \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{2 \,{\left (\frac{{\left (x - 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{x + 1} + 4 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}\right )}}{3 \,{\left (\frac{x - 1}{x + 1} + 1\right )}^{2}} + \frac{1}{9} \, \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{2}{9} \, \log \left ({\left | \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]