Optimal. Leaf size=18 \[ -6+\frac {2}{25 \left (5-\frac {5 e^x}{x}\right )^2} \]
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Rubi [A] time = 0.39, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1593, 6688, 12, 6712, 32} \begin {gather*} \frac {2}{625 \left (1-\frac {e^x}{x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 1593
Rule 6688
Rule 6712
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x (4-4 x) x}{625 e^{3 x}-1875 e^{2 x} x+1875 e^x x^2-625 x^3} \, dx\\ &=\int \frac {4 e^x (1-x) x}{625 \left (e^x-x\right )^3} \, dx\\ &=\frac {4}{625} \int \frac {e^x (1-x) x}{\left (e^x-x\right )^3} \, dx\\ &=-\left (\frac {4}{625} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^3} \, dx,x,\frac {e^x}{x}\right )\right )\\ &=\frac {2}{625 \left (1-\frac {e^x}{x}\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 16, normalized size = 0.89 \begin {gather*} \frac {2 x^2}{625 \left (e^x-x\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 20, normalized size = 1.11 \begin {gather*} \frac {2 \, x^{2}}{625 \, {\left (x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 20, normalized size = 1.11 \begin {gather*} \frac {2 \, x^{2}}{625 \, {\left (x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 14, normalized size = 0.78
method | result | size |
risch | \(\frac {2 x^{2}}{625 \left (x -{\mathrm e}^{x}\right )^{2}}\) | \(14\) |
norman | \(\frac {-\frac {2 \,{\mathrm e}^{2 x}}{625}+\frac {4 \,{\mathrm e}^{x} x}{625}}{\left (x -{\mathrm e}^{x}\right )^{2}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 20, normalized size = 1.11 \begin {gather*} \frac {2 \, x^{2}}{625 \, {\left (x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 13, normalized size = 0.72 \begin {gather*} \frac {2\,x^2}{625\,{\left (x-{\mathrm {e}}^x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 22, normalized size = 1.22 \begin {gather*} \frac {2 x^{2}}{625 x^{2} - 1250 x e^{x} + 625 e^{2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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