Optimal. Leaf size=99 \[ -\frac {1}{6} \sqrt {15+14 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {x^3-1}}{x^2+x+1}\right )-\frac {1}{6} \sqrt {14 \sqrt {3}-15} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {x^3-1}}{x^2+x+1}\right ) \]
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Rubi [C] time = 0.75, antiderivative size = 457, normalized size of antiderivative = 4.62, number of steps used = 13, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6728, 219, 2135, 2140, 206, 203} \begin {gather*} \frac {1}{6} \sqrt {15+14 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} (1-x)}{\sqrt {x^3-1}}\right )+\frac {1}{6} \sqrt {14 \sqrt {3}-15} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {x^3-1}}\right )-\frac {2 \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 )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {\sqrt {14-5 \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\ 3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {\sqrt {38-21 \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\ 3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 219
Rule 2135
Rule 2140
Rule 6728
Rubi steps
\begin {align*} \int \frac {3-x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx &=\int \left (\frac {1}{\sqrt {-1+x^3}}+\frac {5+x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}}\right ) \, dx\\ &=\int \frac {1}{\sqrt {-1+x^3}} \, dx+\int \frac {5+x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx\\ &=-\frac {2 \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 )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\int \left (\frac {1+2 \sqrt {3}}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}}+\frac {1-2 \sqrt {3}}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}}\right ) \, dx\\ &=-\frac {2 \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 )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\left (1-2 \sqrt {3}\right ) \int \frac {1}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx+\left (1+2 \sqrt {3}\right ) \int \frac {1}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx\\ &=-\frac {2 \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 )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {1}{576} \left (-6+\sqrt {3}\right ) \int \frac {96 \left (1+\sqrt {3}\right )-96 x}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx+\frac {1}{12} \left (-6+\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx-\frac {1}{576} \left (6+\sqrt {3}\right ) \int \frac {96 \left (1-\sqrt {3}\right )-96 x}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx-\frac {1}{12} \left (6+\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx\\ &=\frac {\sqrt {38-21 \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\ 3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {14-5 \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\ 3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {2 \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 )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {1}{6} \left (6-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\left (3-2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1-\sqrt {3}\right ) x}{-2+2 \sqrt {3}}}{\sqrt {-1+x^3}}\right )+\frac {1}{6} \left (6+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\left (3+2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1+\sqrt {3}\right ) x}{-2-2 \sqrt {3}}}{\sqrt {-1+x^3}}\right )\\ &=\frac {1}{6} \sqrt {15+14 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )+\frac {1}{6} \sqrt {-15+14 \sqrt {3}} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )+\frac {\sqrt {38-21 \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\ 3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {14-5 \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\ 3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {2 \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 )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.80, size = 290, normalized size = 2.93 \begin {gather*} \frac {\sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (\sqrt {x^2+x+1} \left (\left ((9+6 i)+(4-i) \sqrt {3}\right ) \Pi \left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )-\left ((-9+6 i)+(4+i) \sqrt {3}\right ) \Pi \left (\frac {2 i \sqrt {3}}{3+(2+i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )\right )+\frac {6 \sqrt {\frac {(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} \left (\left (\sqrt {3}+i\right ) x+2 i\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}\right )}{3 \left (\sqrt {3}+i\right ) \sqrt {x^3-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.61, size = 99, normalized size = 1.00 \begin {gather*} -\frac {1}{6} \sqrt {15+14 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right )-\frac {1}{6} \sqrt {-15+14 \sqrt {3}} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.38, size = 246, normalized size = 2.48 \begin {gather*} -\frac {1}{6} \, \sqrt {14 \, \sqrt {3} + 15} \arctan \left (\frac {\sqrt {x^{3} - 1} \sqrt {14 \, \sqrt {3} + 15} {\left (3 \, \sqrt {3} - 4\right )}}{11 \, {\left (x^{2} + x + 1\right )}}\right ) - \frac {1}{24} \, \sqrt {14 \, \sqrt {3} - 15} \log \left (\frac {11 \, x^{4} + 22 \, x^{3} + 66 \, x^{2} + 2 \, \sqrt {x^{3} - 1} {\left (4 \, x^{2} + \sqrt {3} {\left (3 \, x^{2} + 2 \, x + 4\right )} + 10 \, x - 2\right )} \sqrt {14 \, \sqrt {3} - 15} + 44 \, \sqrt {3} {\left (x^{3} - 1\right )} - 44 \, x + 44}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {14 \, \sqrt {3} - 15} \log \left (\frac {11 \, x^{4} + 22 \, x^{3} + 66 \, x^{2} - 2 \, \sqrt {x^{3} - 1} {\left (4 \, x^{2} + \sqrt {3} {\left (3 \, x^{2} + 2 \, x + 4\right )} + 10 \, x - 2\right )} \sqrt {14 \, \sqrt {3} - 15} + 44 \, \sqrt {3} {\left (x^{3} - 1\right )} - 44 \, x + 44}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - x + 3}{\sqrt {x^{3} - 1} {\left (x^{2} - 2 \, x - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.09, size = 592, normalized size = 5.98
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) \ln \left (-\frac {3888 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{4} x^{2}+7776 x \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{4} \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right )+8280 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2} \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) x^{2}+10512 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2} \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) x +44352 \sqrt {x^{3}-1}\, \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+6048 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2} \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right )+4147 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) x^{2}+286 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) x +44352 \sqrt {x^{3}-1}+8008 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right )}{\left (36 x \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+29 x +28\right )^{2}}\right )}{12}+\frac {\RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right ) \ln \left (\frac {3888 x^{2} \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{5}+7776 x \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{5}-1800 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{3} x^{2}+2448 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{3} x -6048 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{3}+7392 \sqrt {x^{3}-1}\, \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}-53 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right ) x^{2}-3074 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right ) x +2968 \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )-1232 \sqrt {x^{3}-1}}{\left (36 x \RootOf \left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+x -28\right )^{2}}\right )}{2}\) | \(592\) |
default | \(\frac {2 \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 {3 \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 )}{\sqrt {x^{3}-1}}+\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 )}{2 \sqrt {x^{3}-1}}+\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 ) \sqrt {3}}{\sqrt {x^{3}-1}}+\frac {\sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 ) \sqrt {3}}{2 \sqrt {x^{3}-1}}+\frac {3 \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 )}{\sqrt {x^{3}-1}}-\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 )}{2 \sqrt {x^{3}-1}}+\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 ) \sqrt {3}}{\sqrt {x^{3}-1}}-\frac {\sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 ) \sqrt {3}}{2 \sqrt {x^{3}-1}}\) | \(1517\) |
elliptic | \(-\frac {3 \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \sqrt {3}}}\, \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 {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \sqrt {3}}}\, \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 {3}}{\sqrt {x^{3}-1}}+\frac {3 \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 )}{\sqrt {x^{3}-1}}+\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 )}{2 \sqrt {x^{3}-1}}+\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 ) \sqrt {3}}{\sqrt {x^{3}-1}}+\frac {\sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 ) \sqrt {3}}{2 \sqrt {x^{3}-1}}+\frac {3 \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 )}{\sqrt {x^{3}-1}}-\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 )}{2 \sqrt {x^{3}-1}}+\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 ) \sqrt {3}}{\sqrt {x^{3}-1}}-\frac {\sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \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 ) \sqrt {3}}{2 \sqrt {x^{3}-1}}\) | \(1726\) |
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*} \int \frac {x^{2} - x + 3}{\sqrt {x^{3} - 1} {\left (x^{2} - 2 \, x - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 505, normalized size = 5.10 \begin {gather*} -\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}-\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}-6\right )\,\Pi \left (\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}+\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+6\right )\,\Pi \left (-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\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 {x^{2} - x + 3}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - 2 x - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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