Optimal. Leaf size=183 \[ \frac {\left (e-\left (1+\sqrt {3}\right ) f\right ) \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1+x)}{\sqrt {-1-x^3}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}+\frac {\sqrt {2-\sqrt {3}} \left (e-\left (1-\sqrt {3}\right ) f\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]
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Rubi [A]
time = 0.17, antiderivative size = 183, 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}} (x+1) \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 {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {\left (e-\left (1+\sqrt {3}\right ) f\right ) \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\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 {1}{\sqrt {-1-x^3}} \, dx}{2 \sqrt {3}}+\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}{12 \sqrt {3}}\\ &=\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 (e-\left (1+\sqrt {3}\right ) f\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 )}{\sqrt {3}}\\ &=\frac {\left (e-\left (1+\sqrt {3}\right ) 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 = 20.28, size = 293, normalized size = 1.60 \begin {gather*} \frac {2 \sqrt {\frac {2}{3}} \sqrt {\frac {i (1+x)}{3 i+\sqrt {3}}} \left (3 f \sqrt {-i+\sqrt {3}+2 i x} \left ((-2-i)-\sqrt {3}+\left ((1+2 i)+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 = 258, normalized size = 1.41
method | result | size |
default | \(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i \left (e -f -f \sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(258\) |
elliptic | \(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i \left (e -f -f \sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(258\) |
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 [A]
time = 0.22, size = 708, normalized size = 3.87 \begin {gather*} \left [\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}{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 + f x}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1 + \sqrt {3}\right )}\, 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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