Optimal. Leaf size=29 \[ e+\frac {1+\left (\frac {5}{e^3}+x\right )^2+\frac {\log ^2(x)}{x}}{2 \log (2)} \]
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Rubi [A]
time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.24, number of steps
used = 6, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 14, 2341,
2342} \begin {gather*} \frac {\log ^2(x)}{2 x \log (2)}+\frac {\left (e^3 x+5\right )^2}{2 e^6 \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2341
Rule 2342
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {10 x^2+2 e^3 x^3+2 e^3 \log (x)-e^3 \log ^2(x)}{x^2} \, dx}{2 e^3 \log (2)}\\ &=\frac {\int \left (2 \left (5+e^3 x\right )+\frac {2 e^3 \log (x)}{x^2}-\frac {e^3 \log ^2(x)}{x^2}\right ) \, dx}{2 e^3 \log (2)}\\ &=\frac {\left (5+e^3 x\right )^2}{2 e^6 \log (2)}-\frac {\int \frac {\log ^2(x)}{x^2} \, dx}{2 \log (2)}+\frac {\int \frac {\log (x)}{x^2} \, dx}{\log (2)}\\ &=-\frac {1}{x \log (2)}+\frac {\left (5+e^3 x\right )^2}{2 e^6 \log (2)}-\frac {\log (x)}{x \log (2)}+\frac {\log ^2(x)}{2 x \log (2)}-\frac {\int \frac {\log (x)}{x^2} \, dx}{\log (2)}\\ &=\frac {\left (5+e^3 x\right )^2}{2 e^6 \log (2)}+\frac {\log ^2(x)}{2 x \log (2)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 1.03 \begin {gather*} \frac {10 x+e^3 x^2+\frac {e^3 \log ^2(x)}{x}}{e^3 \log (4)} \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. \(63\) vs.
\(2(29)=58\).
time = 0.03, size = 64, normalized size = 2.21
method | result | size |
risch | \(\frac {\ln \left (x \right )^{2}}{2 \ln \left (2\right ) x}+\frac {x^{2}}{2 \ln \left (2\right )}+\frac {5 \,{\mathrm e}^{-3} x}{\ln \left (2\right )}\) | \(33\) |
norman | \(\frac {\frac {x^{3}}{2 \ln \left (2\right )}+\frac {\ln \left (x \right )^{2}}{2 \ln \left (2\right )}+\frac {5 \,{\mathrm e}^{-3} x^{2}}{\ln \left (2\right )}}{x}\) | \(38\) |
default | \(\frac {{\mathrm e}^{-3} \left (x^{2} {\mathrm e}^{3}-{\mathrm e}^{3} \left (-\frac {\ln \left (x \right )^{2}}{x}-\frac {2 \ln \left (x \right )}{x}-\frac {2}{x}\right )+2 \,{\mathrm e}^{3} \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )+10 x \right )}{2 \ln \left (2\right )}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 48, normalized size = 1.66 \begin {gather*} \frac {{\left (x^{2} e^{3} - 2 \, {\left (\frac {\log \left (x\right )}{x} + \frac {1}{x}\right )} e^{3} + 10 \, x + \frac {{\left (\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 2\right )} e^{3}}{x}\right )} e^{\left (-3\right )}}{2 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 30, normalized size = 1.03 \begin {gather*} \frac {{\left (x^{3} e^{3} + e^{3} \log \left (x\right )^{2} + 10 \, x^{2}\right )} e^{\left (-3\right )}}{2 \, x \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 29, normalized size = 1.00 \begin {gather*} \frac {x^{2}}{2 \log {\left (2 \right )}} + \frac {5 x}{e^{3} \log {\left (2 \right )}} + \frac {\log {\left (x \right )}^{2}}{2 x \log {\left (2 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 30, normalized size = 1.03 \begin {gather*} \frac {{\left (x^{3} e^{3} + e^{3} \log \left (x\right )^{2} + 10 \, x^{2}\right )} e^{\left (-3\right )}}{2 \, x \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.67, size = 29, normalized size = 1.00 \begin {gather*} \frac {{\ln \left (x\right )}^2}{2\,x\,\ln \left (2\right )}+\frac {x\,{\mathrm {e}}^{-3}\,\left (x\,{\mathrm {e}}^3+10\right )}{2\,\ln \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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