Optimal. Leaf size=18 \[ \frac {1}{3 \left (\frac {3}{4}+x+e^x x+\log (3)\right )^2} \]
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Rubi [A]
time = 0.18, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 129, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6820, 12,
6818} \begin {gather*} \frac {16}{3 \left (4 \left (e^x+1\right ) x+3+\log (81)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6818
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {128 \left (-1-e^x (1+x)\right )}{3 \left (4 \left (1+e^x\right ) x+3 \left (1+\frac {\log (81)}{3}\right )\right )^3} \, dx\\ &=\frac {128}{3} \int \frac {-1-e^x (1+x)}{\left (4 \left (1+e^x\right ) x+3 \left (1+\frac {\log (81)}{3}\right )\right )^3} \, dx\\ &=\frac {16}{3 \left (3+4 \left (1+e^x\right ) x+\log (81)\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 18, normalized size = 1.00 \begin {gather*} \frac {16}{3 \left (3+4 \left (1+e^x\right ) x+\log (81)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 19, normalized size = 1.06
method | result | size |
norman | \(\frac {16}{3 \left (4 \,{\mathrm e}^{x} x +4 \ln \left (3\right )+4 x +3\right )^{2}}\) | \(19\) |
risch | \(\frac {16}{3 \left (4 \,{\mathrm e}^{x} x +4 \ln \left (3\right )+4 x +3\right )^{2}}\) | \(19\) |
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. 57 vs.
\(2 (18) = 36\).
time = 0.56, size = 57, normalized size = 3.17 \begin {gather*} \frac {16}{3 \, {\left (16 \, x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 8 \, x {\left (4 \, \log \left (3\right ) + 3\right )} + 8 \, {\left (4 \, x^{2} + x {\left (4 \, \log \left (3\right ) + 3\right )}\right )} e^{x} + 16 \, \log \left (3\right )^{2} + 24 \, \log \left (3\right ) + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (18) = 36\).
time = 0.36, size = 56, normalized size = 3.11 \begin {gather*} \frac {16}{3 \, {\left (16 \, x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 8 \, {\left (4 \, x^{2} + 4 \, x \log \left (3\right ) + 3 \, x\right )} e^{x} + 8 \, {\left (4 \, x + 3\right )} \log \left (3\right ) + 16 \, \log \left (3\right )^{2} + 24 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (17) = 34\).
time = 0.15, size = 58, normalized size = 3.22 \begin {gather*} \frac {16}{48 x^{2} e^{2 x} + 48 x^{2} + 72 x + 96 x \log {\left (3 \right )} + \left (96 x^{2} + 72 x + 96 x \log {\left (3 \right )}\right ) e^{x} + 27 + 48 \log {\left (3 \right )}^{2} + 72 \log {\left (3 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (18) = 36\).
time = 0.42, size = 57, normalized size = 3.17 \begin {gather*} \frac {16}{3 \, {\left (16 \, x^{2} e^{\left (2 \, x\right )} + 32 \, x^{2} e^{x} + 32 \, x e^{x} \log \left (3\right ) + 16 \, x^{2} + 24 \, x e^{x} + 32 \, x \log \left (3\right ) + 16 \, \log \left (3\right )^{2} + 24 \, x + 24 \, \log \left (3\right ) + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.29, size = 123, normalized size = 6.83 \begin {gather*} -\frac {\frac {128\,x}{4\,\ln \left (3\right )+3}+\frac {256\,x^2}{{\left (4\,\ln \left (3\right )+3\right )}^2}+\frac {512\,x^2\,{\mathrm {e}}^x}{{\left (4\,\ln \left (3\right )+3\right )}^2}+\frac {256\,x^2\,{\mathrm {e}}^{2\,x}}{{\left (\ln \left (81\right )+3\right )}^2}+\frac {128\,x\,{\mathrm {e}}^x}{\ln \left (81\right )+3}}{72\,x+72\,\ln \left (3\right )+96\,x^2\,{\mathrm {e}}^x+96\,x\,\ln \left (3\right )+48\,x^2\,{\mathrm {e}}^{2\,x}+72\,x\,{\mathrm {e}}^x+48\,{\ln \left (3\right )}^2+48\,x^2+96\,x\,{\mathrm {e}}^x\,\ln \left (3\right )+27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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