Optimal. Leaf size=28 \[ e^{-x} \log ^2\left (\left (3+\frac {\left (-1+\frac {e^2}{3}\right )^2}{x}\right )^2\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(28)=56\).
time = 0.24, antiderivative size = 88, normalized size of antiderivative = 3.14, number of steps
used = 4, number of rules used = 3, integrand size = 154, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6, 1607, 2326}
\begin {gather*} \frac {e^{-x} \left (27 x^2+e^4 x-6 e^2 x+9 x\right ) \log ^2\left (\frac {729 x^2+486 x+54 e^4 (x+1)-108 e^2 (3 x+1)+e^8-12 e^6+81}{81 x^2}\right )}{x \left (27 x+\left (e^2-3\right )^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 1607
Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{e^4 x+\left (9-6 e^2\right ) x+27 x^2} \, dx\\ &=\int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{\left (9-6 e^2+e^4\right ) x+27 x^2} \, dx\\ &=\int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{x \left (9-6 e^2+e^4+27 x\right )} \, dx\\ &=\frac {e^{-x} \left (9 x-6 e^2 x+e^4 x+27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+486 x+729 x^2+54 e^4 (1+x)-108 e^2 (1+3 x)}{81 x^2}\right )}{x \left (\left (-3+e^2\right )^2+27 x\right )}\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs.
\(2(24)=48\).
time = 3.60, size = 52, normalized size = 1.86
method | result | size |
norman | \(\ln \left (\frac {{\mathrm e}^{8}-12 \,{\mathrm e}^{6}+\left (54 x +54\right ) {\mathrm e}^{4}+\left (-324 x -108\right ) {\mathrm e}^{2}+729 x^{2}+486 x +81}{81 x^{2}}\right )^{2} {\mathrm e}^{-x}\) | \(52\) |
risch | \(\text {Expression too large to display}\) | \(3672\) |
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. 58 vs.
\(2 (24) = 48\).
time = 0.55, size = 58, normalized size = 2.07 \begin {gather*} 4 \, {\left (4 \, \log \left (3\right )^{2} - 2 \, {\left (2 \, \log \left (3\right ) + \log \left (x\right )\right )} \log \left (27 \, x + e^{4} - 6 \, e^{2} + 9\right ) + \log \left (27 \, x + e^{4} - 6 \, e^{2} + 9\right )^{2} + 4 \, \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 45, normalized size = 1.61 \begin {gather*} e^{\left (-x\right )} \log \left (\frac {729 \, x^{2} + 54 \, {\left (x + 1\right )} e^{4} - 108 \, {\left (3 \, x + 1\right )} e^{2} + 486 \, x + e^{8} - 12 \, e^{6} + 81}{81 \, x^{2}}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.01, size = 47, normalized size = 1.68 \begin {gather*} e^{\left (-x\right )} \log \left (\frac {729 \, x^{2} + 54 \, x e^{4} - 324 \, x e^{2} + 486 \, x + e^{8} - 12 \, e^{6} + 54 \, e^{4} - 108 \, e^{2} + 81}{81 \, x^{2}}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.80, size = 48, normalized size = 1.71 \begin {gather*} {\mathrm {e}}^{-x}\,{\ln \left (\frac {6\,x-\frac {4\,{\mathrm {e}}^6}{27}+\frac {{\mathrm {e}}^8}{81}+9\,x^2+\frac {{\mathrm {e}}^4\,\left (54\,x+54\right )}{81}-\frac {{\mathrm {e}}^2\,\left (324\,x+108\right )}{81}+1}{x^2}\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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