Optimal. Leaf size=35 \[ -\frac {2 \tanh ^{-1}\left (\frac {\frac {x}{\sqrt {3}}-\frac {1}{\sqrt {3}}}{\sqrt {x^3-x}}\right )}{\sqrt {3}} \]
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Rubi [C] time = 0.92, antiderivative size = 229, normalized size of antiderivative = 6.54, number of steps used = 13, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2056, 6728, 329, 222, 933, 168, 537} \begin {gather*} \frac {\sqrt {2} \sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{3 \sqrt {x^3-x}}+\frac {2 \left (1+2 i \sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {3}{4-i \sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{3 \left (\sqrt {2}+4 i\right ) \sqrt {x^3-x}}-\frac {2 \left (1-2 i \sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {3}{4+i \sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{3 \left (-\sqrt {2}+4 i\right ) \sqrt {x^3-x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 168
Rule 222
Rule 329
Rule 537
Rule 933
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {-1-2 x+x^2}{\left (1+2 x+3 x^2\right ) \sqrt {-x+x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {-1-2 x+x^2}{\sqrt {x} \sqrt {-1+x^2} \left (1+2 x+3 x^2\right )} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \left (\frac {1}{3 \sqrt {x} \sqrt {-1+x^2}}-\frac {4 (1+2 x)}{3 \sqrt {x} \sqrt {-1+x^2} \left (1+2 x+3 x^2\right )}\right ) \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1+x^2}} \, dx}{3 \sqrt {-x+x^3}}-\frac {\left (4 \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1+2 x}{\sqrt {x} \sqrt {-1+x^2} \left (1+2 x+3 x^2\right )} \, dx}{3 \sqrt {-x+x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^3}}-\frac {\left (4 \sqrt {x} \sqrt {-1+x^2}\right ) \int \left (\frac {2-\frac {i}{\sqrt {2}}}{\sqrt {x} \left (2-2 i \sqrt {2}+6 x\right ) \sqrt {-1+x^2}}+\frac {2+\frac {i}{\sqrt {2}}}{\sqrt {x} \left (2+2 i \sqrt {2}+6 x\right ) \sqrt {-1+x^2}}\right ) \, dx}{3 \sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{3 \sqrt {-x+x^3}}-\frac {\left (2 \left (4-i \sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (2-2 i \sqrt {2}+6 x\right ) \sqrt {-1+x^2}} \, dx}{3 \sqrt {-x+x^3}}-\frac {\left (2 \left (4+i \sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (2+2 i \sqrt {2}+6 x\right ) \sqrt {-1+x^2}} \, dx}{3 \sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{3 \sqrt {-x+x^3}}-\frac {\left (2 \left (4-i \sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x} \left (2-2 i \sqrt {2}+6 x\right )} \, dx}{3 \sqrt {-x+x^3}}-\frac {\left (2 \left (4+i \sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x} \left (2+2 i \sqrt {2}+6 x\right )} \, dx}{3 \sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{3 \sqrt {-x+x^3}}+\frac {\left (4 \left (4-i \sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \left (4-i \sqrt {2}\right )-6 x^2\right ) \sqrt {1-x^2} \sqrt {2-x^2}} \, dx,x,\sqrt {1-x}\right )}{3 \sqrt {-x+x^3}}+\frac {\left (4 \left (4+i \sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \left (4+i \sqrt {2}\right )-6 x^2\right ) \sqrt {1-x^2} \sqrt {2-x^2}} \, dx,x,\sqrt {1-x}\right )}{3 \sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{3 \sqrt {-x+x^3}}+\frac {\sqrt {2} \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {3}{4-i \sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{3 \sqrt {-x+x^3}}+\frac {\sqrt {2} \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {3}{4+i \sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{3 \sqrt {-x+x^3}}\\ \end {align*}
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Mathematica [C] time = 1.04, size = 112, normalized size = 3.20 \begin {gather*} -\frac {2 \sqrt {1-\frac {1}{x^2}} x^{3/2} \left (-3 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )+\left (2+i \sqrt {2}\right ) \Pi \left (\frac {i}{-i+\sqrt {2}};\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )+\left (2-i \sqrt {2}\right ) \Pi \left (-\frac {i}{i+\sqrt {2}};\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )\right )}{3 \sqrt {x \left (x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 35, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {3}}+\frac {x}{\sqrt {3}}}{\sqrt {-x+x^3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 73, normalized size = 2.09 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (\frac {9 \, x^{4} + 36 \, x^{3} - 4 \, \sqrt {3} \sqrt {x^{3} - x} {\left (3 \, x^{2} + 4 \, x - 1\right )} + 10 \, x^{2} - 20 \, x + 1}{9 \, x^{4} + 12 \, x^{3} + 10 \, x^{2} + 4 \, x + 1}\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} - 2 \, x - 1}{\sqrt {x^{3} - x} {\left (3 \, x^{2} + 2 \, x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.57, size = 61, normalized size = 1.74
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {x^{3}-x}+\RootOf \left (\textit {\_Z}^{2}-3\right )}{3 x^{2}+2 x +1}\right )}{3}\) | \(61\) |
elliptic | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {x^{3}-x}}+\frac {\left (-\frac {4}{9}+\frac {i \sqrt {2}}{9}\right ) \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \left (-1+\frac {i \sqrt {2}}{2}\right ) \EllipticPi \left (\sqrt {1+x}, 1-\frac {i \sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}+\frac {\left (-\frac {4}{9}-\frac {i \sqrt {2}}{9}\right ) \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \left (-1-\frac {i \sqrt {2}}{2}\right ) \EllipticPi \left (\sqrt {1+x}, 1+\frac {i \sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}\) | \(165\) |
default | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {x^{3}-x}}-\frac {4 \left (\frac {1}{3}-\frac {i \sqrt {2}}{12}\right ) \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \left (-1+\frac {i \sqrt {2}}{2}\right ) \EllipticPi \left (\sqrt {1+x}, 1-\frac {i \sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {x^{3}-x}}-\frac {4 \left (\frac {1}{3}+\frac {i \sqrt {2}}{12}\right ) \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \left (-1-\frac {i \sqrt {2}}{2}\right ) \EllipticPi \left (\sqrt {1+x}, 1+\frac {i \sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {x^{3}-x}}\) | \(167\) |
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} - 2 \, x - 1}{\sqrt {x^{3} - x} {\left (3 \, x^{2} + 2 \, x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 175, normalized size = 5.00 \begin {gather*} -\frac {2\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{3\,\sqrt {x^3-x}}-\frac {\sqrt {2}\,\sqrt {-x}\,\left (-\frac {4}{9}+\frac {\sqrt {2}\,8{}\mathrm {i}}{9}\right )\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\frac {1}{\frac {1}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )\,1{}\mathrm {i}}{2\,\sqrt {x^3-x}\,\left (\frac {1}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {2}\,\sqrt {-x}\,\left (\frac {4}{9}+\frac {\sqrt {2}\,8{}\mathrm {i}}{9}\right )\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\frac {1}{-\frac {1}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )\,1{}\mathrm {i}}{2\,\sqrt {x^3-x}\,\left (-\frac {1}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}\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} - 2 x - 1}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (3 x^{2} + 2 x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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