Integrand size = 29, antiderivative size = 187 \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {\left (e+f+\sqrt {3} f\right ) \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\sqrt {2+\sqrt {3}} \left (e+\left (1-\sqrt {3}\right ) f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \]
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Time = 0.19 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2166, 224, 2165, 209} \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\left (1-\sqrt {3}\right ) f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {\arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right ) \left (e+\sqrt {3} f+f\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}} \]
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Rule 209
Rule 224
Rule 2165
Rule 2166
Rubi steps \begin{align*} \text {integral}& = -\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 ) \tan ^{-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*}
Result contains complex when optimal does not.
Time = 20.52 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.56 \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 \sqrt {\frac {2}{3}} \sqrt {-\frac {i (-1+x)}{3 i+\sqrt {3}}} \left (-3 i f \sqrt {-i+\sqrt {3}-2 i x} \left (-i \left ((2+i)+\sqrt {3}\right )+\left ((2-i)+\sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 \left (\sqrt {3} e+\left (3+\sqrt {3}\right ) f\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+(1+2 i) \sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1-x^3}} \]
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Time = 1.47 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.41
method | result | size |
default | \(\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-e -f -f \sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(264\) |
elliptic | \(\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-e -f -f \sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(264\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 735, normalized size of antiderivative = 3.93 \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\left [-\frac {1}{3} \, {\left (\sqrt {3} {\left (i \, e + i \, f\right )} - 3 i \, f\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) + \frac {1}{12} \, \sqrt {3 \, e^{2} - 6 \, e f - 2 \, \sqrt {3} {\left (e^{2} - e f + f^{2}\right )}} \log \left (-\frac {{\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{8} + 16 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{7} + 112 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{6} + 16 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{5} + 112 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{4} - 224 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{3} + 64 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{2} + 4 \, {\left ({\left (2 \, e - f\right )} x^{6} + 18 \, {\left (e - f\right )} x^{5} + 6 \, {\left (7 \, e - 2 \, f\right )} x^{4} + 8 \, {\left (e - 5 \, f\right )} x^{3} + 36 \, f x^{2} - 24 \, {\left (e + f\right )} x + \sqrt {3} {\left ({\left (e - f\right )} x^{6} + 6 \, {\left (2 \, e - f\right )} x^{5} + 6 \, {\left (3 \, e - 4 \, f\right )} x^{4} + 8 \, {\left (2 \, e + f\right )} x^{3} - 12 \, {\left (e + f\right )} x^{2} + 24 \, f x - 8 \, e - 16 \, f\right )} + 8 \, e + 32 \, f\right )} \sqrt {-x^{3} + 1} \sqrt {3 \, e^{2} - 6 \, e f - 2 \, \sqrt {3} {\left (e^{2} - e f + f^{2}\right )}} + 112 \, e^{2} + 224 \, e f - 224 \, f^{2} - 128 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x + 16 \, \sqrt {3} {\left ({\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{7} + 2 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{6} + 6 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{5} - 5 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{4} + 2 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{3} - 6 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{2} - 4 \, e^{2} - 8 \, e f + 8 \, f^{2} + 4 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x\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 (i \, e + i \, f\right )} - 3 i \, f\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) + \frac {1}{6} \, \sqrt {-3 \, e^{2} + 6 \, e f + 2 \, \sqrt {3} {\left (e^{2} - e f + f^{2}\right )}} \arctan \left (\frac {{\left (3 \, f x^{2} - 6 \, {\left (e - f\right )} x - \sqrt {3} {\left ({\left (e + f\right )} x^{2} + 2 \, {\left (2 \, e - f\right )} x - 2 \, e + 4 \, f\right )} + 6 \, e\right )} \sqrt {-x^{3} + 1} \sqrt {-3 \, e^{2} + 6 \, e f + 2 \, \sqrt {3} {\left (e^{2} - e f + f^{2}\right )}}}{6 \, {\left ({\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{3} - e^{2} - 2 \, e f + 2 \, f^{2}\right )}}\right )\right ] \]
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\[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=- \int \frac {e}{x \sqrt {1 - x^{3}} - \sqrt {3} \sqrt {1 - x^{3}} - \sqrt {1 - x^{3}}}\, dx - \int \frac {f x}{x \sqrt {1 - x^{3}} - \sqrt {3} \sqrt {1 - x^{3}} - \sqrt {1 - x^{3}}}\, dx \]
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\[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\int { -\frac {f x + e}{\sqrt {-x^{3} + 1} {\left (x - \sqrt {3} - 1\right )}} \,d x } \]
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Exception generated. \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Hanged} \]
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