Optimal. Leaf size=22 \[ 3 \left (4+\frac {x}{\left (e^4+e^{4 x/3}-x\right )^2}\right ) \]
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Rubi [F] time = 0.76, 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 {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} \left (3 e^4-3 x\right )-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx\\ &=\int \left (\frac {2 \left (3+4 e^4-4 x\right ) x}{\left (e^4+e^{4 x/3}-x\right )^3}-\frac {-3+8 x}{\left (-e^4-e^{4 x/3}+x\right )^2}\right ) \, dx\\ &=2 \int \frac {\left (3+4 e^4-4 x\right ) x}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx-\int \frac {-3+8 x}{\left (-e^4-e^{4 x/3}+x\right )^2} \, dx\\ &=2 \int \left (\frac {\left (3+4 e^4\right ) x}{\left (e^4+e^{4 x/3}-x\right )^3}-\frac {4 x^2}{\left (e^4+e^{4 x/3}-x\right )^3}\right ) \, dx-\int \left (-\frac {3}{\left (e^4+e^{4 x/3}-x\right )^2}+\frac {8 x}{\left (e^4+e^{4 x/3}-x\right )^2}\right ) \, dx\\ &=3 \int \frac {1}{\left (e^4+e^{4 x/3}-x\right )^2} \, dx-8 \int \frac {x}{\left (e^4+e^{4 x/3}-x\right )^2} \, dx-8 \int \frac {x^2}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx+\left (2 \left (3+4 e^4\right )\right ) \int \frac {x}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx\\ &=-\left (8 \int \frac {x}{\left (e^4+e^{4 x/3}-x\right )^2} \, dx\right )-8 \int \frac {x^2}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx+9 \operatorname {Subst}\left (\int \frac {1}{\left (e^4+e^{4 x}-3 x\right )^2} \, dx,x,\frac {x}{3}\right )+\left (2 \left (3+4 e^4\right )\right ) \int \frac {x}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.21, size = 19, normalized size = 0.86 \begin {gather*} \frac {3 x}{\left (e^4+e^{4 x/3}-x\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 32, normalized size = 1.45 \begin {gather*} \frac {3 \, x}{x^{2} - 2 \, x e^{4} - 2 \, {\left (x - e^{4}\right )} e^{\left (\frac {4}{3} \, x\right )} + e^{8} + e^{\left (\frac {8}{3} \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.69, size = 35, normalized size = 1.59 \begin {gather*} \frac {3 \, x}{x^{2} - 2 \, x e^{4} - 2 \, x e^{\left (\frac {4}{3} \, x\right )} + e^{8} + e^{\left (\frac {8}{3} \, x\right )} + 2 \, e^{\left (\frac {4}{3} \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.91, size = 16, normalized size = 0.73
method | result | size |
norman | \(\frac {3 x}{\left ({\mathrm e}^{\frac {4 x}{3}}-x +{\mathrm e}^{4}\right )^{2}}\) | \(16\) |
risch | \(\frac {3 x}{\left ({\mathrm e}^{\frac {4 x}{3}}-x +{\mathrm e}^{4}\right )^{2}}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 32, normalized size = 1.45 \begin {gather*} \frac {3 \, x}{x^{2} - 2 \, x e^{4} - 2 \, {\left (x - e^{4}\right )} e^{\left (\frac {4}{3} \, x\right )} + e^{8} + e^{\left (\frac {8}{3} \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.91, size = 35, normalized size = 1.59 \begin {gather*} \frac {3\,x}{{\mathrm {e}}^{\frac {8\,x}{3}}+{\mathrm {e}}^8+2\,{\mathrm {e}}^{\frac {4\,x}{3}}\,{\mathrm {e}}^4-2\,x\,{\mathrm {e}}^{\frac {4\,x}{3}}-2\,x\,{\mathrm {e}}^4+x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 37, normalized size = 1.68 \begin {gather*} \frac {3 x}{x^{2} - 2 x e^{4} + \left (- 2 x + 2 e^{4}\right ) e^{\frac {4 x}{3}} + e^{\frac {8 x}{3}} + e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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