Optimal. Leaf size=139 \[ \frac{\sqrt{x^3-1}}{2 x^2}-\frac{\sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{2 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0199751, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {325, 219} \[ \frac{\sqrt{x^3-1}}{2 x^2}-\frac{\sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 325
Rule 219
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{-1+x^3}} \, dx &=\frac{\sqrt{-1+x^3}}{2 x^2}+\frac{1}{4} \int \frac{1}{\sqrt{-1+x^3}} \, dx\\ &=\frac{\sqrt{-1+x^3}}{2 x^2}-\frac{\sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{1+x+x^2}{\left (1-\sqrt{3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-x}{1-\sqrt{3}-x}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}\\ \end{align*}
Mathematica [C] time = 0.0053644, size = 40, normalized size = 0.29 \[ -\frac{\sqrt{1-x^3} \, _2F_1\left (-\frac{2}{3},\frac{1}{2};\frac{1}{3};x^3\right )}{2 x^2 \sqrt{x^3-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.016, size = 129, normalized size = 0.9 \begin{align*}{\frac{1}{2\,{x}^{2}}\sqrt{{x}^{3}-1}}+{\frac{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}{2}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{3} - 1} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{3} - 1}}{x^{6} - x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.897636, size = 31, normalized size = 0.22 \begin{align*} - \frac{i \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{2} \\ \frac{1}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac{1}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{3} - 1} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]