Optimal. Leaf size=32 \[ \left (-\frac {1}{2} e^{\frac {3}{e^5 x^2 \log (x)}} x+\frac {x}{\log \left (2+e^2\right )}\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(137\) vs. \(2(32)=64\).
time = 0.66, antiderivative size = 137, normalized size of antiderivative = 4.28, number of steps
used = 5, number of rules used = 3, integrand size = 114, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {12, 6874,
2326} \begin {gather*} \frac {x^2}{\log ^2\left (2+e^2\right )}+\frac {e^{\frac {6}{e^5 x^2 \log (x)}-5} (2 \log (x)+1)}{4 x \left (\frac {1}{e^5 x^3 \log ^2(x)}+\frac {2}{e^5 x^3 \log (x)}\right ) \log ^2(x)}-\frac {e^{\frac {3}{e^5 x^2 \log (x)}-5} (2 \log (x)+1)}{x \log \left (2+e^2\right ) \left (\frac {1}{e^5 x^3 \log ^2(x)}+\frac {2}{e^5 x^3 \log (x)}\right ) \log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2326
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {4 e^5 x^2 \log ^2(x)+e^{\frac {3}{e^5 x^2 \log (x)}} \log \left (2+e^2\right ) \left (6+12 \log (x)-4 e^5 x^2 \log ^2(x)\right )+e^{\frac {6}{e^5 x^2 \log (x)}} \log ^2\left (2+e^2\right ) \left (-3-6 \log (x)+e^5 x^2 \log ^2(x)\right )}{x \log ^2(x)} \, dx}{2 e^5 \log ^2\left (2+e^2\right )}\\ &=\frac {\int \left (4 e^5 x+\frac {e^{\frac {6}{e^5 x^2 \log (x)}} \log ^2\left (2+e^2\right ) \left (-3-6 \log (x)+e^5 x^2 \log ^2(x)\right )}{x \log ^2(x)}-\frac {2 e^{\frac {3}{e^5 x^2 \log (x)}} \log \left (2+e^2\right ) \left (-3-6 \log (x)+2 e^5 x^2 \log ^2(x)\right )}{x \log ^2(x)}\right ) \, dx}{2 e^5 \log ^2\left (2+e^2\right )}\\ &=\frac {x^2}{\log ^2\left (2+e^2\right )}+\frac {\int \frac {e^{\frac {6}{e^5 x^2 \log (x)}} \left (-3-6 \log (x)+e^5 x^2 \log ^2(x)\right )}{x \log ^2(x)} \, dx}{2 e^5}-\frac {\int \frac {e^{\frac {3}{e^5 x^2 \log (x)}} \left (-3-6 \log (x)+2 e^5 x^2 \log ^2(x)\right )}{x \log ^2(x)} \, dx}{e^5 \log \left (2+e^2\right )}\\ &=\frac {x^2}{\log ^2\left (2+e^2\right )}+\frac {e^{-5+\frac {6}{e^5 x^2 \log (x)}} (1+2 \log (x))}{4 x \left (\frac {1}{e^5 x^3 \log ^2(x)}+\frac {2}{e^5 x^3 \log (x)}\right ) \log ^2(x)}-\frac {e^{-5+\frac {3}{e^5 x^2 \log (x)}} (1+2 \log (x))}{x \log \left (2+e^2\right ) \left (\frac {1}{e^5 x^3 \log ^2(x)}+\frac {2}{e^5 x^3 \log (x)}\right ) \log ^2(x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.13, size = 40, normalized size = 1.25 \begin {gather*} \frac {x^2 \left (-2+e^{\frac {3}{e^5 x^2 \log (x)}} \log \left (2+e^2\right )\right )^2}{4 \log ^2\left (2+e^2\right )} \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. \(71\) vs.
\(2(29)=58\).
time = 1.58, size = 72, normalized size = 2.25
method | result | size |
risch | \(\frac {x^{2}}{\ln \left ({\mathrm e}^{2}+2\right )^{2}}+\frac {x^{2} {\mathrm e}^{\frac {6 \,{\mathrm e}^{-5}}{x^{2} \ln \left (x \right )}}}{4}-\frac {x^{2} {\mathrm e}^{\frac {3 \,{\mathrm e}^{-5}}{x^{2} \ln \left (x \right )}}}{\ln \left ({\mathrm e}^{2}+2\right )}\) | \(54\) |
default | \(\frac {{\mathrm e}^{-5} \left (-2 \,{\mathrm e}^{5} \ln \left ({\mathrm e}^{2}+2\right ) x^{2} {\mathrm e}^{\frac {3 \,{\mathrm e}^{-5}}{x^{2} \ln \left (x \right )}}+\frac {{\mathrm e}^{5} \ln \left ({\mathrm e}^{2}+2\right )^{2} x^{2} {\mathrm e}^{\frac {6 \,{\mathrm e}^{-5}}{x^{2} \ln \left (x \right )}}}{2}+2 x^{2} {\mathrm e}^{5}\right )}{2 \ln \left ({\mathrm e}^{2}+2\right )^{2}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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. 60 vs.
\(2 (29) = 58\).
time = 0.45, size = 60, normalized size = 1.88 \begin {gather*} \frac {x^{2} e^{\left (\frac {6 \, e^{\left (-5\right )}}{x^{2} \log \left (x\right )}\right )} \log \left (e^{2} + 2\right )^{2} - 4 \, x^{2} e^{\left (\frac {3 \, e^{\left (-5\right )}}{x^{2} \log \left (x\right )}\right )} \log \left (e^{2} + 2\right ) + 4 \, x^{2}}{4 \, \log \left (e^{2} + 2\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.89, size = 34, normalized size = 1.06 \begin {gather*} \frac {x^2\,{\left (\ln \left ({\mathrm {e}}^2+2\right )\,{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{-5}}{x^2\,\ln \left (x\right )}}-2\right )}^2}{4\,{\ln \left ({\mathrm {e}}^2+2\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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