Optimal. Leaf size=18 \[ -2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+x+1}}\right ) \]
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Rubi [F] time = 0.46, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2-x+2 x^4}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-2-x+2 x^4}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx &=\int \left (\frac {2}{\sqrt {1+x+x^2+x^4}}-\frac {4+3 x}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {1+x+x^2+x^4}} \, dx-\int \frac {4+3 x}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx\\ &=2 \int \frac {1}{\sqrt {1+x+x^2+x^4}} \, dx-\int \left (\frac {4}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}}+\frac {3 x}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {1+x+x^2+x^4}} \, dx-3 \int \frac {x}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx-4 \int \frac {1}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx\\ \end {align*}
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Mathematica [C] time = 6.21, size = 17638, normalized size = 979.89 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.16, size = 18, normalized size = 1.00 \begin {gather*} -2 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x+x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 35, normalized size = 1.94 \begin {gather*} \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} + x + 1} x + x + 1}{x^{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 {2 \, x^{4} - x - 2}{\sqrt {x^{4} + x^{2} + x + 1} {\left (x^{4} + x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 6.08, size = 39, normalized size = 2.17
method | result | size |
trager | \(-\ln \left (-\frac {x^{4}+2 \sqrt {x^{4}+x^{2}+x +1}\, x +2 x^{2}+x +1}{x^{4}+x +1}\right )\) | \(39\) |
default | \(\text {Expression too large to display}\) | \(4880\) |
elliptic | \(\text {Expression too large to display}\) | \(4880\) |
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^{4} - x - 2}{\sqrt {x^{4} + x^{2} + x + 1} {\left (x^{4} + 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 = -0.06 \begin {gather*} \int -\frac {-2\,x^4+x+2}{\left (x^4+x+1\right )\,\sqrt {x^4+x^2+x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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