Optimal. Leaf size=43 \[ -\frac {\sqrt {x^4+x^2+x+1} x}{x^4+x+1}-\tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+x+1}}\right ) \]
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Rubi [F] time = 0.59, 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 {\sqrt {1+x+x^2+x^4} \left (-2-x+2 x^4\right )}{\left (1+x+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+x+x^2+x^4} \left (-2-x+2 x^4\right )}{\left (1+x+x^4\right )^2} \, dx &=\int \left (\frac {(-4-3 x) \sqrt {1+x+x^2+x^4}}{\left (1+x+x^4\right )^2}+\frac {2 \sqrt {1+x+x^2+x^4}}{1+x+x^4}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+x+x^2+x^4}}{1+x+x^4} \, dx+\int \frac {(-4-3 x) \sqrt {1+x+x^2+x^4}}{\left (1+x+x^4\right )^2} \, dx\\ &=2 \int \frac {\sqrt {1+x+x^2+x^4}}{1+x+x^4} \, dx+\int \left (-\frac {4 \sqrt {1+x+x^2+x^4}}{\left (1+x+x^4\right )^2}-\frac {3 x \sqrt {1+x+x^2+x^4}}{\left (1+x+x^4\right )^2}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+x+x^2+x^4}}{1+x+x^4} \, dx-3 \int \frac {x \sqrt {1+x+x^2+x^4}}{\left (1+x+x^4\right )^2} \, dx-4 \int \frac {\sqrt {1+x+x^2+x^4}}{\left (1+x+x^4\right )^2} \, dx\\ \end {align*}
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Mathematica [C] time = 6.21, size = 17667, normalized size = 410.86 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.14, size = 43, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x^4+x^2+x+1} x}{x^4+x+1}-\tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 67, normalized size = 1.56 \begin {gather*} \frac {{\left (x^{4} + x + 1\right )} \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} + x + 1} x + x + 1}{x^{4} + x + 1}\right ) - 2 \, \sqrt {x^{4} + x^{2} + x + 1} x}{2 \, {\left (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 {{\left (2 \, x^{4} - x - 2\right )} \sqrt {x^{4} + x^{2} + x + 1}}{{\left (x^{4} + x + 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.48, size = 9105, normalized size = 211.74 \begin {gather*} \text {output too large to display} \end {gather*}
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 {{\left (2 \, x^{4} - x - 2\right )} \sqrt {x^{4} + x^{2} + x + 1}}{{\left (x^{4} + x + 1\right )}^{2}}\,{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 {\left (-2\,x^4+x+2\right )\,\sqrt {x^4+x^2+x+1}}{{\left (x^4+x+1\right )}^2} \,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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