Optimal. Leaf size=19 \[ x^2-\left (19-e^x+\log ^2(x)\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(19)=38\).
time = 0.10, antiderivative size = 41, normalized size of antiderivative = 2.16, number of steps
used = 9, number of rules used = 6, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {14, 2225, 2326,
2338, 2339, 30} \begin {gather*} x^2-e^{2 x}-\log ^4(x)-38 \log ^2(x)+\frac {2 e^x \left (19 x+x \log ^2(x)\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2225
Rule 2326
Rule 2338
Rule 2339
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-2 e^{2 x}+\frac {2 e^x \left (19 x+2 \log (x)+x \log ^2(x)\right )}{x}+\frac {2 \left (x^2-38 \log (x)-2 \log ^3(x)\right )}{x}\right ) \, dx\\ &=-\left (2 \int e^{2 x} \, dx\right )+2 \int \frac {e^x \left (19 x+2 \log (x)+x \log ^2(x)\right )}{x} \, dx+2 \int \frac {x^2-38 \log (x)-2 \log ^3(x)}{x} \, dx\\ &=-e^{2 x}+\frac {2 e^x \left (19 x+x \log ^2(x)\right )}{x}+2 \int \left (x-\frac {38 \log (x)}{x}-\frac {2 \log ^3(x)}{x}\right ) \, dx\\ &=-e^{2 x}+x^2+\frac {2 e^x \left (19 x+x \log ^2(x)\right )}{x}-4 \int \frac {\log ^3(x)}{x} \, dx-76 \int \frac {\log (x)}{x} \, dx\\ &=-e^{2 x}+x^2-38 \log ^2(x)+\frac {2 e^x \left (19 x+x \log ^2(x)\right )}{x}-4 \text {Subst}\left (\int x^3 \, dx,x,\log (x)\right )\\ &=-e^{2 x}+x^2-38 \log ^2(x)-\log ^4(x)+\frac {2 e^x \left (19 x+x \log ^2(x)\right )}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.14, size = 34, normalized size = 1.79 \begin {gather*} -361+38 e^x-e^{2 x}+x^2+2 \left (-19+e^x\right ) \log ^2(x)-\log ^4(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 35, normalized size = 1.84
method | result | size |
risch | \(-\ln \left (x \right )^{4}+\left (2 \,{\mathrm e}^{x}-38\right ) \ln \left (x \right )^{2}+x^{2}-{\mathrm e}^{2 x}+38 \,{\mathrm e}^{x}\) | \(32\) |
default | \(2 \,{\mathrm e}^{x} \ln \left (x \right )^{2}+38 \,{\mathrm e}^{x}+x^{2}-{\mathrm e}^{2 x}-38 \ln \left (x \right )^{2}-\ln \left (x \right )^{4}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 34, normalized size = 1.79 \begin {gather*} -\log \left (x\right )^{4} + 2 \, e^{x} \log \left (x\right )^{2} + x^{2} - 38 \, \log \left (x\right )^{2} - e^{\left (2 \, x\right )} + 38 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 30, normalized size = 1.58 \begin {gather*} -\log \left (x\right )^{4} + 2 \, {\left (e^{x} - 19\right )} \log \left (x\right )^{2} + x^{2} - e^{\left (2 \, x\right )} + 38 \, e^{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. 31 vs.
\(2 (14) = 28\).
time = 0.13, size = 31, normalized size = 1.63 \begin {gather*} x^{2} + \left (2 \log {\left (x \right )}^{2} + 38\right ) e^{x} - e^{2 x} - \log {\left (x \right )}^{4} - 38 \log {\left (x \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 34, normalized size = 1.79 \begin {gather*} -\log \left (x\right )^{4} + 2 \, e^{x} \log \left (x\right )^{2} + x^{2} - 38 \, \log \left (x\right )^{2} - e^{\left (2 \, x\right )} + 38 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.67, size = 34, normalized size = 1.79 \begin {gather*} 38\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}-38\,{\ln \left (x\right )}^2-{\ln \left (x\right )}^4+2\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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