Optimal. Leaf size=178 \[ -\frac{2 \left (e+2^{2/3} f\right ) \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{3 \sqrt{3}}-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
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Rubi [A] time = 0.224403, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2139, 219, 2137, 206} \[ -\frac{2 \left (e+2^{2/3} f\right ) \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{3 \sqrt{3}}-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
Antiderivative was successfully verified.
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Rule 2139
Rule 219
Rule 2137
Rule 206
Rubi steps
\begin{align*} \int \frac{e+f x}{\left (2^{2/3}-x\right ) \sqrt{-1+x^3}} \, dx &=-\left (\frac{1}{3} \left (-\sqrt [3]{2} e+f\right ) \int \frac{1}{\sqrt{-1+x^3}} \, dx\right )+\frac{1}{6} \left (\sqrt [3]{2} e+2 f\right ) \int \frac{2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt{-1+x^3}} \, dx\\ &=-\frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{2} e-f\right ) (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 )}{3 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}-\frac{1}{3} \left (2 \left (e+2^{2/3} f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-3 x^2} \, dx,x,\frac{1-\sqrt [3]{2} x}{\sqrt{-1+x^3}}\right )\\ &=-\frac{2 \left (e+2^{2/3} f\right ) \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{-1+x^3}}\right )}{3 \sqrt{3}}-\frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{2} e-f\right ) (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 )}{3 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}\\ \end{align*}
Mathematica [C] time = 0.350988, size = 338, normalized size = 1.9 \[ \frac{2 \sqrt [6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left (2 \sqrt{3} \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \left (\sqrt [3]{2} e+2 f\right ) \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )-i f \sqrt{-2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x-\sqrt [3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt [3]{2}-6 i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (i+2 i 2^{2/3}+\sqrt{3}\right ) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^3-1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.023, size = 270, normalized size = 1.5 \begin{align*} -2\,{\frac{f \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1}}\sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }+2\,{\frac{ \left ( -e-{2}^{2/3}f \right ) \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1} \left ( -{2}^{2/3}+1 \right ) }\sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}},{\frac{3/2+i/2\sqrt{3}}{-{2}^{2/3}+1}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{f x + e}{\sqrt{x^{3} - 1}{\left (x - 2^{\frac{2}{3}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (f x^{3} + e x^{2} + 2^{\frac{2}{3}}{\left (f x^{2} + e x\right )} + 2 \cdot 2^{\frac{1}{3}}{\left (f x + e\right )}\right )} \sqrt{x^{3} - 1}}{x^{6} - 5 \, x^{3} + 4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{e}{x \sqrt{x^{3} - 1} - 2^{\frac{2}{3}} \sqrt{x^{3} - 1}}\, dx - \int \frac{f x}{x \sqrt{x^{3} - 1} - 2^{\frac{2}{3}} \sqrt{x^{3} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{f x + e}{\sqrt{x^{3} - 1}{\left (x - 2^{\frac{2}{3}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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