Optimal. Leaf size=18 \[ \frac {\left (8-4 e^x+x\right )^2}{81 \log ^2(3)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(18)=36\).
time = 0.01, antiderivative size = 59, normalized size of antiderivative = 3.28, number of steps
used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 2225, 2207}
\begin {gather*} \frac {x^2}{81 \log ^2(3)}+\frac {16 x}{81 \log ^2(3)}+\frac {8 e^x}{81 \log ^2(3)}+\frac {16 e^{2 x}}{81 \log ^2(3)}-\frac {8 e^x (x+9)}{81 \log ^2(3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2225
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (16+32 e^{2 x}+e^x (-72-8 x)+2 x\right ) \, dx}{81 \log ^2(3)}\\ &=\frac {16 x}{81 \log ^2(3)}+\frac {x^2}{81 \log ^2(3)}+\frac {\int e^x (-72-8 x) \, dx}{81 \log ^2(3)}+\frac {32 \int e^{2 x} \, dx}{81 \log ^2(3)}\\ &=\frac {16 e^{2 x}}{81 \log ^2(3)}+\frac {16 x}{81 \log ^2(3)}+\frac {x^2}{81 \log ^2(3)}-\frac {8 e^x (9+x)}{81 \log ^2(3)}+\frac {8 \int e^x \, dx}{81 \log ^2(3)}\\ &=\frac {8 e^x}{81 \log ^2(3)}+\frac {16 e^{2 x}}{81 \log ^2(3)}+\frac {16 x}{81 \log ^2(3)}+\frac {x^2}{81 \log ^2(3)}-\frac {8 e^x (9+x)}{81 \log ^2(3)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.11 \begin {gather*} \frac {\left (-8+4 e^x-x\right )^2}{81 \log ^2(3)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 29, normalized size = 1.61
method | result | size |
default | \(\frac {16 x -8 \,{\mathrm e}^{x} x -64 \,{\mathrm e}^{x}+x^{2}+16 \,{\mathrm e}^{2 x}}{81 \ln \left (3\right )^{2}}\) | \(29\) |
risch | \(\frac {x^{2}}{81 \ln \left (3\right )^{2}}+\frac {16 \,{\mathrm e}^{2 x}}{81 \ln \left (3\right )^{2}}+\frac {16 x}{81 \ln \left (3\right )^{2}}+\frac {\left (-8 x -64\right ) {\mathrm e}^{x}}{81 \ln \left (3\right )^{2}}\) | \(41\) |
norman | \(\frac {\frac {16 x}{81 \ln \left (3\right )}+\frac {x^{2}}{81 \ln \left (3\right )}-\frac {64 \,{\mathrm e}^{x}}{81 \ln \left (3\right )}+\frac {16 \,{\mathrm e}^{2 x}}{81 \ln \left (3\right )}-\frac {8 x \,{\mathrm e}^{x}}{81 \ln \left (3\right )}}{\ln \left (3\right )}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 26, normalized size = 1.44 \begin {gather*} \frac {x^{2} - 8 \, {\left (x + 8\right )} e^{x} + 16 \, x + 16 \, e^{\left (2 \, x\right )}}{81 \, \log \left (3\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 26, normalized size = 1.44 \begin {gather*} \frac {x^{2} - 8 \, {\left (x + 8\right )} e^{x} + 16 \, x + 16 \, e^{\left (2 \, x\right )}}{81 \, \log \left (3\right )^{2}} \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.
\(2 (15) = 30\).
time = 0.06, size = 58, normalized size = 3.22 \begin {gather*} \frac {x^{2}}{81 \log {\left (3 \right )}^{2}} + \frac {16 x}{81 \log {\left (3 \right )}^{2}} + \frac {\left (- 648 x \log {\left (3 \right )}^{2} - 5184 \log {\left (3 \right )}^{2}\right ) e^{x} + 1296 e^{2 x} \log {\left (3 \right )}^{2}}{6561 \log {\left (3 \right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 26, normalized size = 1.44 \begin {gather*} \frac {x^{2} - 8 \, {\left (x + 8\right )} e^{x} + 16 \, x + 16 \, e^{\left (2 \, x\right )}}{81 \, \log \left (3\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 28, normalized size = 1.56 \begin {gather*} \frac {16\,x+16\,{\mathrm {e}}^{2\,x}-64\,{\mathrm {e}}^x-8\,x\,{\mathrm {e}}^x+x^2}{81\,{\ln \left (3\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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