Optimal. Leaf size=67 \[ \frac {1}{3} \tanh ^{-1}\left (\frac {(x+1) \sqrt {x^4-2 x^3+3 x^2+4 x+1}}{x^3}\right )+\tanh ^{-1}\left (\frac {\sqrt {x^4-2 x^3+3 x^2+4 x+1}-2 x-1}{x^2}\right ) \]
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Rubi [F] time = 0.06, 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 {x}{\sqrt {1+4 x+3 x^2-2 x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1+4 x+3 x^2-2 x^3+x^4}} \, dx &=\int \frac {x}{\sqrt {1+4 x+3 x^2-2 x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [C] time = 2.25, size = 1105, normalized size = 16.49
result too large to display
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 13.56, size = 67, normalized size = 1.00 \begin {gather*} \frac {1}{3} \tanh ^{-1}\left (\frac {(1+x) \sqrt {1+4 x+3 x^2-2 x^3+x^4}}{x^3}\right )+\tanh ^{-1}\left (\frac {-1-2 x+\sqrt {1+4 x+3 x^2-2 x^3+x^4}}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 70, normalized size = 1.04 \begin {gather*} \frac {1}{6} \, \log \left (2 \, x^{6} - 12 \, x^{5} + 36 \, x^{4} - 56 \, x^{3} + 42 \, x^{2} + 2 \, \sqrt {x^{4} - 2 \, x^{3} + 3 \, x^{2} + 4 \, x + 1} {\left (x^{4} - 5 \, x^{3} + 12 \, x^{2} - 14 \, x + 7\right )} - 13\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}{\sqrt {x^{4} - 2 \, x^{3} + 3 \, x^{2} + 4 \, x + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.68, size = 151, normalized size = 2.25
method | result | size |
trager | \(-\frac {\ln \left (-2 x^{6}+2 \sqrt {x^{4}-2 x^{3}+3 x^{2}+4 x +1}\, x^{4}+12 x^{5}-10 \sqrt {x^{4}-2 x^{3}+3 x^{2}+4 x +1}\, x^{3}-36 x^{4}+24 \sqrt {x^{4}-2 x^{3}+3 x^{2}+4 x +1}\, x^{2}+56 x^{3}-28 x \sqrt {x^{4}-2 x^{3}+3 x^{2}+4 x +1}-42 x^{2}+14 \sqrt {x^{4}-2 x^{3}+3 x^{2}+4 x +1}+13\right )}{6}\) | \(151\) |
default | \(\text {Expression too large to display}\) | \(1609\) |
elliptic | \(\text {Expression too large to display}\) | \(1609\) |
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}{\sqrt {x^{4} - 2 \, x^{3} + 3 \, x^{2} + 4 \, x + 1}}\,{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 = -0.01 \begin {gather*} \int \frac {x}{\sqrt {x^4-2\,x^3+3\,x^2+4\,x+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}{\sqrt {x^{4} - 2 x^{3} + 3 x^{2} + 4 x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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