Optimal. Leaf size=29 \[ x+\frac {e^{2 x} \left (-1+\left (\frac {4}{e^2}-x\right )^2+x\right ) \log ^2(2)}{x} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(29)=58\).
time = 0.13, antiderivative size = 70, normalized size of antiderivative = 2.41, number of steps
used = 11, number of rules used = 7, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.123, Rules used = {12, 14, 2230,
2225, 2208, 2209, 2207} \begin {gather*} x+e^{2 x} x \log ^2(2)-\frac {1}{2} e^{2 x} \log ^2(2)-\frac {1}{2} \left (16-3 e^2\right ) e^{2 x-2} \log ^2(2)+\frac {\left (16-e^4\right ) e^{2 x-4} \log ^2(2)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^4 x^2+e^{2 x} \left (-16+32 x-16 e^2 x^2+e^4 \left (1-2 x+3 x^2+2 x^3\right )\right ) \log ^2(2)}{x^2} \, dx}{e^4}\\ &=\frac {\int \left (e^4+\frac {e^{2 x} \left (-16+e^4+2 \left (16-e^4\right ) x-e^2 \left (16-3 e^2\right ) x^2+2 e^4 x^3\right ) \log ^2(2)}{x^2}\right ) \, dx}{e^4}\\ &=x+\frac {\log ^2(2) \int \frac {e^{2 x} \left (-16+e^4+2 \left (16-e^4\right ) x-e^2 \left (16-3 e^2\right ) x^2+2 e^4 x^3\right )}{x^2} \, dx}{e^4}\\ &=x+\frac {\log ^2(2) \int \left (e^{2+2 x} \left (-16+3 e^2\right )+\frac {e^{2 x} \left (-16+e^4\right )}{x^2}-\frac {2 e^{2 x} \left (-16+e^4\right )}{x}+2 e^{4+2 x} x\right ) \, dx}{e^4}\\ &=x+\frac {\left (2 \log ^2(2)\right ) \int e^{4+2 x} x \, dx}{e^4}-\frac {\left (\left (16-3 e^2\right ) \log ^2(2)\right ) \int e^{2+2 x} \, dx}{e^4}-\frac {\left (\left (16-e^4\right ) \log ^2(2)\right ) \int \frac {e^{2 x}}{x^2} \, dx}{e^4}+\frac {\left (2 \left (16-e^4\right ) \log ^2(2)\right ) \int \frac {e^{2 x}}{x} \, dx}{e^4}\\ &=x-\frac {1}{2} e^{-2+2 x} \left (16-3 e^2\right ) \log ^2(2)+\frac {e^{-4+2 x} \left (16-e^4\right ) \log ^2(2)}{x}+e^{2 x} x \log ^2(2)+\frac {2 \left (16-e^4\right ) \text {Ei}(2 x) \log ^2(2)}{e^4}-\frac {\log ^2(2) \int e^{4+2 x} \, dx}{e^4}-\frac {\left (2 \left (16-e^4\right ) \log ^2(2)\right ) \int \frac {e^{2 x}}{x} \, dx}{e^4}\\ &=x-\frac {1}{2} e^{2 x} \log ^2(2)-\frac {1}{2} e^{-2+2 x} \left (16-3 e^2\right ) \log ^2(2)+\frac {e^{-4+2 x} \left (16-e^4\right ) \log ^2(2)}{x}+e^{2 x} x \log ^2(2)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 37, normalized size = 1.28 \begin {gather*} x+e^{2 x} \left (\frac {-8+e^2}{e^2}+\frac {16-e^4}{e^4 x}+x\right ) \log ^2(2) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs.
\(2(29)=58\).
time = 0.29, size = 82, normalized size = 2.83
method | result | size |
risch | \(x +\frac {\left (x^{2} {\mathrm e}^{4}+x \,{\mathrm e}^{4}-{\mathrm e}^{4}-8 \,{\mathrm e}^{2} x +16\right ) \ln \left (2\right )^{2} {\mathrm e}^{2 x -4}}{x}\) | \(38\) |
norman | \(\frac {\left (x^{2} {\mathrm e}^{2}+{\mathrm e}^{2} \ln \left (2\right )^{2} {\mathrm e}^{2 x} x^{2}+\ln \left (2\right )^{2} \left ({\mathrm e}^{2}-8\right ) x \,{\mathrm e}^{2 x}-\ln \left (2\right )^{2} \left ({\mathrm e}^{4}-16\right ) {\mathrm e}^{-2} {\mathrm e}^{2 x}\right ) {\mathrm e}^{-2}}{x}\) | \(64\) |
default | \({\mathrm e}^{-4} \left (x \,{\mathrm e}^{4}+{\mathrm e}^{4} \ln \left (2\right )^{2} {\mathrm e}^{2 x} x +{\mathrm e}^{4} \ln \left (2\right )^{2} {\mathrm e}^{2 x}+\frac {16 \ln \left (2\right )^{2} {\mathrm e}^{2 x}}{x}-8 \ln \left (2\right )^{2} {\mathrm e}^{2 x} {\mathrm e}^{2}-\frac {\ln \left (2\right )^{2} {\mathrm e}^{2 x} {\mathrm e}^{4}}{x}\right )\) | \(82\) |
derivativedivides | \(\frac {{\mathrm e}^{-4} \left (2 x \,{\mathrm e}^{4}+2 \,{\mathrm e}^{4} \ln \left (2\right )^{2} {\mathrm e}^{2 x} x +2 \,{\mathrm e}^{4} \ln \left (2\right )^{2} {\mathrm e}^{2 x}+\frac {32 \ln \left (2\right )^{2} {\mathrm e}^{2 x}}{x}-16 \ln \left (2\right )^{2} {\mathrm e}^{2 x} {\mathrm e}^{2}-\frac {2 \ln \left (2\right )^{2} {\mathrm e}^{2 x} {\mathrm e}^{4}}{x}\right )}{2}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.29, size = 100, normalized size = 3.45 \begin {gather*} -\frac {1}{2} \, {\left (4 \, {\rm Ei}\left (2 \, x\right ) e^{4} \log \left (2\right )^{2} - {\left (2 \, x e^{4} - e^{4}\right )} e^{\left (2 \, x\right )} \log \left (2\right )^{2} - 4 \, e^{4} \Gamma \left (-1, -2 \, x\right ) \log \left (2\right )^{2} - 64 \, {\rm Ei}\left (2 \, x\right ) \log \left (2\right )^{2} - 3 \, e^{\left (2 \, x + 4\right )} \log \left (2\right )^{2} + 16 \, e^{\left (2 \, x + 2\right )} \log \left (2\right )^{2} + 64 \, \Gamma \left (-1, -2 \, x\right ) \log \left (2\right )^{2} - 2 \, x e^{4}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 38, normalized size = 1.31 \begin {gather*} \frac {{\left ({\left ({\left (x^{2} + x - 1\right )} e^{4} - 8 \, x e^{2} + 16\right )} e^{\left (2 \, x\right )} \log \left (2\right )^{2} + x^{2} e^{4}\right )} e^{\left (-4\right )}}{x} \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. 60 vs.
\(2 (24) = 48\).
time = 0.07, size = 60, normalized size = 2.07 \begin {gather*} x + \frac {\left (x^{2} e^{4} \log {\left (2 \right )}^{2} - 8 x e^{2} \log {\left (2 \right )}^{2} + x e^{4} \log {\left (2 \right )}^{2} - e^{4} \log {\left (2 \right )}^{2} + 16 \log {\left (2 \right )}^{2}\right ) e^{2 x}}{x e^{4}} \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. 74 vs.
\(2 (25) = 50\).
time = 0.40, size = 74, normalized size = 2.55 \begin {gather*} \frac {{\left (x^{2} e^{\left (2 \, x + 4\right )} \log \left (2\right )^{2} + x e^{\left (2 \, x + 4\right )} \log \left (2\right )^{2} - 8 \, x e^{\left (2 \, x + 2\right )} \log \left (2\right )^{2} + x^{2} e^{4} + 16 \, e^{\left (2 \, x\right )} \log \left (2\right )^{2} - e^{\left (2 \, x + 4\right )} \log \left (2\right )^{2}\right )} e^{\left (-4\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 58, normalized size = 2.00 \begin {gather*} x\,\left ({\mathrm {e}}^{2\,x}\,{\ln \left (2\right )}^2+1\right )-{\mathrm {e}}^{2\,x}\,\left (8\,{\mathrm {e}}^{-2}\,{\ln \left (2\right )}^2-{\ln \left (2\right )}^2\right )+\frac {{\mathrm {e}}^{2\,x}\,\left (16\,{\mathrm {e}}^{-4}\,{\ln \left (2\right )}^2-{\ln \left (2\right )}^2\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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