Optimal. Leaf size=19 \[ -1+e^2+\log ^2\left (3+\frac {1}{x}-x+x^2\right ) \]
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Rubi [F]
time = 16.19, 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 {\left (-2-2 x^2+4 x^3\right ) \log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{x+3 x^2-x^3+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{x}+\frac {2 \left (3-2 x+3 x^2\right ) \log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{1+3 x-x^2+x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{x} \, dx\right )+2 \int \frac {\left (3-2 x+3 x^2\right ) \log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{1+3 x-x^2+x^3} \, dx\\ &=-2 \log (x) \log \left (\frac {1+3 x-x^2+x^3}{x}\right )+2 \int \frac {x \left (\frac {3-2 x+3 x^2}{x}-\frac {1+3 x-x^2+x^3}{x^2}\right ) \log (x)}{1+3 x-x^2+x^3} \, dx+2 \int \left (\frac {3 \log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{1+3 x-x^2+x^3}-\frac {2 x \log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{1+3 x-x^2+x^3}+\frac {3 x^2 \log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{1+3 x-x^2+x^3}\right ) \, dx\\ &=-2 \log (x) \log \left (\frac {1+3 x-x^2+x^3}{x}\right )+2 \int \left (-\frac {\log (x)}{x}+\frac {\left (3-2 x+3 x^2\right ) \log (x)}{1+3 x-x^2+x^3}\right ) \, dx-4 \int \frac {x \log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{1+3 x-x^2+x^3} \, dx+6 \int \frac {\log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{1+3 x-x^2+x^3} \, dx+6 \int \frac {x^2 \log \left (\frac {1+3 x-x^2+x^3}{x}\right )}{1+3 x-x^2+x^3} \, dx\\ &=-2 \log (x) \log \left (\frac {1+3 x-x^2+x^3}{x}\right )-2 \int \frac {\log (x)}{x} \, dx+2 \int \frac {\left (3-2 x+3 x^2\right ) \log (x)}{1+3 x-x^2+x^3} \, dx-4 \int \frac {x \log \left (3+\frac {1}{x}-x+x^2\right )}{1+3 x-x^2+x^3} \, dx+6 \int \frac {\log \left (3+\frac {1}{x}-x+x^2\right )}{1+3 x-x^2+x^3} \, dx+6 \int \frac {x^2 \log \left (3+\frac {1}{x}-x+x^2\right )}{1+3 x-x^2+x^3} \, dx\\ &=-\log ^2(x)-2 \log (x) \log \left (\frac {1+3 x-x^2+x^3}{x}\right )+2 \int \left (\frac {3 \log (x)}{1+3 x-x^2+x^3}-\frac {2 x \log (x)}{1+3 x-x^2+x^3}+\frac {3 x^2 \log (x)}{1+3 x-x^2+x^3}\right ) \, dx-4 \int \frac {x \log \left (3+\frac {1}{x}-x+x^2\right )}{1+3 x-x^2+x^3} \, dx+6 \int \frac {\log \left (3+\frac {1}{x}-x+x^2\right )}{1+3 x-x^2+x^3} \, dx+6 \int \frac {x^2 \log \left (3+\frac {1}{x}-x+x^2\right )}{1+3 x-x^2+x^3} \, dx\\ &=-\log ^2(x)-2 \log (x) \log \left (\frac {1+3 x-x^2+x^3}{x}\right )-4 \int \frac {x \log (x)}{1+3 x-x^2+x^3} \, dx-4 \int \frac {x \log \left (3+\frac {1}{x}-x+x^2\right )}{1+3 x-x^2+x^3} \, dx+6 \int \frac {\log (x)}{1+3 x-x^2+x^3} \, dx+6 \int \frac {x^2 \log (x)}{1+3 x-x^2+x^3} \, dx+6 \int \frac {\log \left (3+\frac {1}{x}-x+x^2\right )}{1+3 x-x^2+x^3} \, dx+6 \int \frac {x^2 \log \left (3+\frac {1}{x}-x+x^2\right )}{1+3 x-x^2+x^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 14, normalized size = 0.74 \begin {gather*} \log ^2\left (3+\frac {1}{x}-x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.55, size = 558, normalized size = 29.37
method | result | size |
norman | \(\ln \left (\frac {x^{3}-x^{2}+3 x +1}{x}\right )^{2}\) | \(21\) |
default | \(\text {Expression too large to display}\) | \(558\) |
risch | \(\text {Expression too large to display}\) | \(5048\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (18) = 36\).
time = 0.28, size = 82, normalized size = 4.32 \begin {gather*} -\log \left (x^{3} - x^{2} + 3 \, x + 1\right )^{2} + 2 \, \log \left (x^{3} - x^{2} + 3 \, x + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2} + 2 \, {\left (\log \left (x^{3} - x^{2} + 3 \, x + 1\right ) - \log \left (x\right )\right )} \log \left (\frac {x^{3} - x^{2} + 3 \, x + 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 20, normalized size = 1.05 \begin {gather*} \log \left (\frac {x^{3} - x^{2} + 3 \, x + 1}{x}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 15, normalized size = 0.79 \begin {gather*} \log {\left (\frac {x^{3} - x^{2} + 3 x + 1}{x} \right )}^{2} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.24, size = 14, normalized size = 0.74 \begin {gather*} {\ln \left (\frac {1}{x}-x+x^2+3\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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