Optimal. Leaf size=29 \[ -2 \tanh ^{-1}\left (\frac {x}{x^2+\sqrt {x^4-x^2+1}-2 x+1}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 46, normalized size of antiderivative = 1.59, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1741, 12, 1247, 724, 206, 1698} \begin {gather*} -\tanh ^{-1}\left (\frac {x}{\sqrt {x^4-x^2+1}}\right )-\tanh ^{-1}\left (\frac {x^2+1}{2 \sqrt {x^4-x^2+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 1247
Rule 1698
Rule 1741
Rubi steps
\begin {align*} \int \frac {1+x}{(-1+x) \sqrt {1-x^2+x^4}} \, dx &=\int -\frac {2 x}{\left (1-x^2\right ) \sqrt {1-x^2+x^4}} \, dx+\int \frac {-1-x^2}{\left (1-x^2\right ) \sqrt {1-x^2+x^4}} \, dx\\ &=-\left (2 \int \frac {x}{\left (1-x^2\right ) \sqrt {1-x^2+x^4}} \, dx\right )-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1-x^2+x^4}}\right )\\ &=-\tanh ^{-1}\left (\frac {x}{\sqrt {1-x^2+x^4}}\right )-\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {1-x+x^2}} \, dx,x,x^2\right )\\ &=-\tanh ^{-1}\left (\frac {x}{\sqrt {1-x^2+x^4}}\right )+2 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1-x^2}{\sqrt {1-x^2+x^4}}\right )\\ &=-\tanh ^{-1}\left (\frac {x}{\sqrt {1-x^2+x^4}}\right )-\tanh ^{-1}\left (\frac {1+x^2}{2 \sqrt {1-x^2+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 1.72, size = 742, normalized size = 25.59 \begin {gather*} \frac {2 \sqrt [3]{-1} \left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )^2 \sqrt {\frac {\sqrt {1-\sqrt [3]{-1}} \left (\sqrt {1+(-1)^{2/3}}-x\right )}{\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )}} \sqrt {-\frac {\sqrt {1-\sqrt [3]{-1}} \left (x+\sqrt {1+(-1)^{2/3}}\right )}{\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )}} \sqrt {\frac {\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right ) x-\sqrt [3]{-1}}{\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )}} \left (\left (-2+\sqrt [3]{-1}-2 \sqrt {1-\sqrt [3]{-1}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right ) \left (x+\sqrt {1-\sqrt [3]{-1}}\right )}{\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )}}\right )|\frac {\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right )^2}{\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right )^2}\right )+4 \sqrt {1-\sqrt [3]{-1}} \Pi \left (\frac {\left (-1+\sqrt {1-\sqrt [3]{-1}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right )}{\left (1+\sqrt {1-\sqrt [3]{-1}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right ) \left (x+\sqrt {1-\sqrt [3]{-1}}\right )}{\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )}}\right )|\frac {\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right )^2}{\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right )^2}\right )\right )}{\sqrt {x^4-x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.37, size = 29, normalized size = 1.00 \begin {gather*} -2 \tanh ^{-1}\left (\frac {x}{1-2 x+x^2+\sqrt {1-x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 36, normalized size = 1.24 \begin {gather*} \log \left (\frac {2 \, x^{2} - 3 \, x - \sqrt {x^{4} - x^{2} + 1} + 2}{x^{2} - 2 \, 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 + 1}{\sqrt {x^{4} - x^{2} + 1} {\left (x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 31, normalized size = 1.07
method | result | size |
trager | \(\ln \left (-\frac {-2 x^{2}+\sqrt {x^{4}-x^{2}+1}+3 x -2}{\left (-1+x \right )^{2}}\right )\) | \(31\) |
elliptic | \(-\arctanh \left (\frac {x^{2}+1}{2 \sqrt {\left (x^{2}-1\right )^{2}+x^{2}}}\right )-\arctanh \left (\frac {\sqrt {x^{4}-x^{2}+1}}{x}\right )\) | \(44\) |
default | \(\frac {2 \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2-2 i \sqrt {3}}}{2}\right )}{\sqrt {2+2 i \sqrt {3}}\, \sqrt {x^{4}-x^{2}+1}}-\arctanh \left (\frac {x^{2}+1}{2 \sqrt {x^{4}-x^{2}+1}}\right )-\frac {2 \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {1}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}-x^{2}+1}}\) | \(210\) |
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 + 1}{\sqrt {x^{4} - x^{2} + 1} {\left (x - 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 = -0.03 \begin {gather*} \int \frac {x+1}{\left (x-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 {x + 1}{\left (x - 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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