Optimal. Leaf size=190 \[ -\frac {\left (e+f+\sqrt {3} f\right ) \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\sqrt {2-\sqrt {3}} \left (e+\left (1-\sqrt {3}\right ) f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \]
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Rubi [A]
time = 0.17, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2166, 225,
2165, 212} \begin {gather*} -\frac {\sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (e+\left (1-\sqrt {3}\right ) f\right ) F\left (\text {ArcSin}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {\left (e+\sqrt {3} f+f\right ) \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {x^3-1}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 225
Rule 2165
Rule 2166
Rubi steps
\begin {align*} \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {-1+x^3}} \, dx &=-\frac {\left (-e-\left (1+\sqrt {3}\right ) f\right ) \int \frac {\left (1+\sqrt {3}\right ) \left (-22+\left (1+\sqrt {3}\right )^3\right )-6 x}{\left (1+\sqrt {3}-x\right ) \sqrt {-1+x^3}} \, dx}{\left (1+\sqrt {3}\right ) \left (-28+\left (1+\sqrt {3}\right )^3\right )}+\frac {\left (-6 e-\left (1+\sqrt {3}\right ) \left (-22+\left (1+\sqrt {3}\right )^3\right ) f\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx}{\left (1+\sqrt {3}\right ) \left (-28+\left (1+\sqrt {3}\right )^3\right )}\\ &=-\frac {\sqrt {2-\sqrt {3}} \left (e+\left (1-\sqrt {3}\right ) 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^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (12 \left (-e-\left (1+\sqrt {3}\right ) f\right )\right ) \text {Subst}\left (\int \frac {1}{1-\left (3+2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1-x}{\sqrt {-1+x^3}}\right )}{\left (1+\sqrt {3}\right ) \left (-28+\left (1+\sqrt {3}\right )^3\right )}\\ &=-\frac {\left (e+f+\sqrt {3} f\right ) \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\sqrt {2-\sqrt {3}} \left (e+\left (1-\sqrt {3}\right ) 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^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 20.29, size = 289, normalized size = 1.52 \begin {gather*} \frac {2 \sqrt {\frac {2}{3}} \sqrt {-\frac {i (-1+x)}{3 i+\sqrt {3}}} \left (-3 i f \sqrt {-i+\sqrt {3}-2 i x} \left (-i \left ((2+i)+\sqrt {3}\right )+\left ((2-i)+\sqrt {3}\right ) x\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 \left (\sqrt {3} e+\left (3+\sqrt {3}\right ) f\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1+x+x^2} \Pi \left (\frac {2 \sqrt {3}}{3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {-1+x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.28, size = 262, normalized size = 1.38
method | result | size |
default | \(-\frac {2 f \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (-e -f -f \sqrt {3}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}\) | \(262\) |
elliptic | \(-\frac {2 f \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (-e -f -f \sqrt {3}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}\) | \(262\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.17, size = 748, normalized size = 3.94 \begin {gather*} \left [\frac {1}{3} \, {\left (\sqrt {3} {\left (f + e\right )} - 3 \, f\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) + \frac {1}{12} \, \sqrt {6 \, f e + 2 \, \sqrt {3} {\left (f^{2} - f e + e^{2}\right )} - 3 \, e^{2}} \log \left (-\frac {2 \, f^{2} x^{8} + 32 \, f^{2} x^{7} + 224 \, f^{2} x^{6} + 32 \, f^{2} x^{5} + 224 \, f^{2} x^{4} - 448 \, f^{2} x^{3} + 128 \, f^{2} x^{2} - 256 \, f^{2} x + 4 \, {\left (f x^{6} + 18 \, f x^{5} + 12 \, f x^{4} + 40 \, f x^{3} - 36 \, f x^{2} + 24 \, f x - 2 \, {\left (x^{6} + 9 \, x^{5} + 21 \, x^{4} + 4 \, x^{3} - 12 \, x + 4\right )} e + \sqrt {3} {\left (f x^{6} + 6 \, f x^{5} + 24 \, f x^{4} - 8 \, f x^{3} + 12 \, f x^{2} - 24 \, f x - {\left (x^{6} + 12 \, x^{5} + 18 \, x^{4} + 16 \, x^{3} - 12 \, x^{2} - 8\right )} e + 16 \, f\right )} - 32 \, f\right )} \sqrt {x^{3} - 1} \sqrt {6 \, f e + 2 \, \sqrt {3} {\left (f^{2} - f e + e^{2}\right )} - 3 \, e^{2}} + 224 \, f^{2} - {\left (x^{8} + 16 \, x^{7} + 112 \, x^{6} + 16 \, x^{5} + 112 \, x^{4} - 224 \, x^{3} + 64 \, x^{2} - 128 \, x + 112\right )} e^{2} - 2 \, {\left (f x^{8} + 16 \, f x^{7} + 112 \, f x^{6} + 16 \, f x^{5} + 112 \, f x^{4} - 224 \, f x^{3} + 64 \, f x^{2} - 128 \, f x + 112 \, f\right )} e + 16 \, \sqrt {3} {\left (2 \, f^{2} x^{7} + 4 \, f^{2} x^{6} + 12 \, f^{2} x^{5} - 10 \, f^{2} x^{4} + 4 \, f^{2} x^{3} - 12 \, f^{2} x^{2} + 8 \, f^{2} x - 8 \, f^{2} - {\left (x^{7} + 2 \, x^{6} + 6 \, x^{5} - 5 \, x^{4} + 2 \, x^{3} - 6 \, x^{2} + 4 \, x - 4\right )} e^{2} - 2 \, {\left (f x^{7} + 2 \, f x^{6} + 6 \, f x^{5} - 5 \, f x^{4} + 2 \, f x^{3} - 6 \, f x^{2} + 4 \, f x - 4 \, f\right )} e\right )}}{x^{8} - 8 \, x^{7} + 16 \, x^{6} + 16 \, x^{5} - 56 \, x^{4} - 32 \, x^{3} + 64 \, x^{2} + 64 \, x + 16}\right ), \frac {1}{3} \, {\left (\sqrt {3} {\left (f + e\right )} - 3 \, f\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) - \frac {1}{6} \, \sqrt {-6 \, f e - 2 \, \sqrt {3} {\left (f^{2} - f e + e^{2}\right )} + 3 \, e^{2}} \arctan \left (\frac {{\left (3 \, f x^{2} + 6 \, f x - 6 \, {\left (x - 1\right )} e - \sqrt {3} {\left (f x^{2} - 2 \, f x + {\left (x^{2} + 4 \, x - 2\right )} e + 4 \, f\right )}\right )} \sqrt {x^{3} - 1} \sqrt {-6 \, f e - 2 \, \sqrt {3} {\left (f^{2} - f e + e^{2}\right )} + 3 \, e^{2}}}{6 \, {\left (2 \, f^{2} x^{3} - 2 \, f^{2} - {\left (x^{3} - 1\right )} e^{2} - 2 \, {\left (f x^{3} - f\right )} e\right )}}\right )\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {e}{x \sqrt {x^{3} - 1} - \sqrt {3} \sqrt {x^{3} - 1} - \sqrt {x^{3} - 1}}\, dx - \int \frac {f x}{x \sqrt {x^{3} - 1} - \sqrt {3} \sqrt {x^{3} - 1} - \sqrt {x^{3} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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