Optimal. Leaf size=65 \[ -\frac {1}{2} i \log \left (\frac {-2 i x^2+\sqrt {x^4-x^2-1} x-i}{-2 i x^2-\sqrt {x^4-x^2-1} x-i}\right ) \]
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Rubi [C] time = 0.41, antiderivative size = 520, normalized size of antiderivative = 8.00, number of steps used = 10, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1710, 1098, 1214, 1456, 540, 421, 419, 538, 537} \begin {gather*} \frac {3 \left (1+\sqrt {5}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {x^4-x^2-1}}-\frac {3 \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {3 \sqrt {2} \left (2+\sqrt {5}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} \left (-1-\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\left (3+\sqrt {5}\right ) \sqrt {x^4-x^2-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 419
Rule 421
Rule 537
Rule 538
Rule 540
Rule 1098
Rule 1214
Rule 1456
Rule 1710
Rubi steps
\begin {align*} \int \frac {-1+2 x^2}{\left (1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx &=2 \int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx-3 \int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx\\ &=\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x^2}{2+\left (1-\sqrt {5}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2-\left (1-\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x^2}} \sqrt {-1-x^2+x^4}}+\frac {3 \int \frac {-1-\sqrt {5}+2 x^2}{\left (1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx}{3+\sqrt {5}}-\frac {6 \int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx}{3+\sqrt {5}}\\ &=\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x^2}{2+\left (1-\sqrt {5}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2-\left (1-\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x^2}} \sqrt {-1-x^2+x^4}}-\frac {3 \sqrt {-2-\left (1-\sqrt {5}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x^2}{2+\left (1-\sqrt {5}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2-\left (1-\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x^2}} \sqrt {-1-x^2+x^4}}+\frac {\left (3 \sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {-1-\sqrt {5}+2 x^2}\right ) \int \frac {\sqrt {-1-\sqrt {5}+2 x^2}}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \left (1+x^2\right )} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1-x^2+x^4}}\\ &=\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x^2}{2+\left (1-\sqrt {5}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2-\left (1-\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x^2}} \sqrt {-1-x^2+x^4}}-\frac {3 \sqrt {-2-\left (1-\sqrt {5}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x^2}{2+\left (1-\sqrt {5}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2-\left (1-\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x^2}} \sqrt {-1-x^2+x^4}}+\frac {\left (6 \sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {-1-\sqrt {5}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {-1-\sqrt {5}+2 x^2}} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1-x^2+x^4}}+\frac {\left (3 \left (-3-\sqrt {5}\right ) \sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {-1-\sqrt {5}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \left (1+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2}} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1-x^2+x^4}}\\ &=\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x^2}{2+\left (1-\sqrt {5}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2-\left (1-\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x^2}} \sqrt {-1-x^2+x^4}}-\frac {3 \sqrt {-2-\left (1-\sqrt {5}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x^2}{2+\left (1-\sqrt {5}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2-\left (1-\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x^2}} \sqrt {-1-x^2+x^4}}+\frac {\left (6 \sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1-x^2+x^4}}+\frac {\left (3 \left (-3-\sqrt {5}\right ) \sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \left (1+x^2\right ) \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1-x^2+x^4}}\\ &=\frac {3 \left (1+\sqrt {5}\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {-1-x^2+x^4}}+\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x^2}{2+\left (1-\sqrt {5}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2-\left (1-\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x^2}} \sqrt {-1-x^2+x^4}}-\frac {3 \sqrt {-2-\left (1-\sqrt {5}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x^2}{2+\left (1-\sqrt {5}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2-\left (1-\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x^2}} \sqrt {-1-x^2+x^4}}-\frac {3 \left (1+\sqrt {5}\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} \left (-1-\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {2} \sqrt {-1-x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 131, normalized size = 2.02 \begin {gather*} -\frac {i \sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {-x^4+x^2+1} \left (2 F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )-3 \Pi \left (\frac {1}{2} \left (-1+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )\right )}{\sqrt {x^4-x^2-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 3.83, size = 65, normalized size = 1.00 \begin {gather*} -\frac {1}{2} i \log \left (\frac {-i-2 i x^2+x \sqrt {-1-x^2+x^4}}{-i-2 i x^2-x \sqrt {-1-x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 41, normalized size = 0.63 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} - 1} {\left (2 \, x^{3} + x\right )}}{x^{6} - 5 \, x^{4} - 5 \, x^{2} - 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 {2 \, x^{2} - 1}{\sqrt {x^{4} - x^{2} - 1} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.48, size = 92, normalized size = 1.42
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-4 \sqrt {x^{4}-x^{2}-1}\, x^{3}-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 x \sqrt {x^{4}-x^{2}-1}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (x^{2}+1\right )^{3}}\right )}{2}\) | \(92\) |
default | \(\frac {4 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2-2 \sqrt {5}}}{2}, \frac {i \sqrt {5}}{2}-\frac {i}{2}\right )}{\sqrt {-2-2 \sqrt {5}}\, \sqrt {x^{4}-x^{2}-1}}-\frac {3 \sqrt {1+\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {\sqrt {5}\, x^{2}}{2}+\frac {x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , -\frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}-x^{2}-1}}\) | \(178\) |
elliptic | \(\frac {4 \sqrt {1+\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {\sqrt {5}\, x^{2}}{2}+\frac {x^{2}}{2}}\, \EllipticF \left (\frac {x \sqrt {-2-2 \sqrt {5}}}{2}, \frac {i \sqrt {5}}{2}-\frac {i}{2}\right )}{\sqrt {-2-2 \sqrt {5}}\, \sqrt {x^{4}-x^{2}-1}}-\frac {3 \sqrt {1+\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {\sqrt {5}\, x^{2}}{2}+\frac {x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , -\frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}-x^{2}-1}}\) | \(180\) |
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 {2 \, x^{2} - 1}{\sqrt {x^{4} - x^{2} - 1} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {2\,x^2-1}{\left (x^2+1\right )\,\sqrt {x^4-x^2-1}} \,d x \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 {2 x^{2} - 1}{\left (x^{2} + 1\right ) \sqrt {x^{4} - x^{2} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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