Optimal. Leaf size=19 \[ x^3+\frac {1}{2+\log \left (x \left (3-e^4+x\right )\right )} \]
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Rubi [A] time = 0.85, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 175, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6, 6688, 6742, 6686} \begin {gather*} x^3+\frac {1}{\log \left (x \left (x-e^4+3\right )\right )+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 6686
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{\left (-12+4 e^4\right ) x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx\\ &=\int \frac {-3-2 x+36 x^3+12 x^4+e^4 \left (1-12 x^3\right )+12 x^3 \left (3-e^4+x\right ) \log \left (x \left (3-e^4+x\right )\right )+3 x^3 \left (3-e^4+x\right ) \log ^2\left (x \left (3-e^4+x\right )\right )}{x \left (3-e^4+x\right ) \left (2+\log \left (x \left (3-e^4+x\right )\right )\right )^2} \, dx\\ &=\int \left (3 x^2+\frac {3-e^4+2 x}{\left (-3+e^4-x\right ) x \left (2+\log \left (x \left (3-e^4+x\right )\right )\right )^2}\right ) \, dx\\ &=x^3+\int \frac {3-e^4+2 x}{\left (-3+e^4-x\right ) x \left (2+\log \left (x \left (3-e^4+x\right )\right )\right )^2} \, dx\\ &=x^3+\frac {1}{2+\log \left (x \left (3-e^4+x\right )\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 19, normalized size = 1.00 \begin {gather*} x^3+\frac {1}{2+\log \left (x \left (3-e^4+x\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 42, normalized size = 2.21 \begin {gather*} \frac {x^{3} \log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2 \, x^{3} + 1}{\log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 42, normalized size = 2.21 \begin {gather*} \frac {x^{3} \log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2 \, x^{3} + 1}{\log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 22, normalized size = 1.16
method | result | size |
risch | \(x^{3}+\frac {1}{\ln \left (-x \,{\mathrm e}^{4}+x^{2}+3 x \right )+2}\) | \(22\) |
norman | \(\frac {x^{3} \ln \left (-x \,{\mathrm e}^{4}+x^{2}+3 x \right )+2 x^{3}+1}{\ln \left (-x \,{\mathrm e}^{4}+x^{2}+3 x \right )+2}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 40, normalized size = 2.11 \begin {gather*} \frac {x^{3} \log \left (x - e^{4} + 3\right ) + x^{3} \log \relax (x) + 2 \, x^{3} + 1}{\log \left (x - e^{4} + 3\right ) + \log \relax (x) + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.06, size = 21, normalized size = 1.11 \begin {gather*} \frac {1}{\ln \left (3\,x-x\,{\mathrm {e}}^4+x^2\right )+2}+x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 19, normalized size = 1.00 \begin {gather*} x^{3} + \frac {1}{\log {\left (x^{2} - x e^{4} + 3 x \right )} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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