Optimal. Leaf size=24 \[ -5+\frac {15 \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2 x} \]
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Rubi [F]
time = 15.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (15 e^5-15 x\right ) \log ^2(x)+\left (\left (30 e^5-30 x\right ) \log (x)+\left (-15 e^5+45 x\right ) \log ^2(x)\right ) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {15 \log (x) \left (-2 \left (-e^5+x\right ) \log (2 x)-\log (x) \left (-e^5+x+\left (e^5-3 x\right ) \log (2 x)\right )\right )}{\left (e^5-x\right )^3 x^2} \, dx\\ &=15 \int \frac {\log (x) \left (-2 \left (-e^5+x\right ) \log (2 x)-\log (x) \left (-e^5+x+\left (e^5-3 x\right ) \log (2 x)\right )\right )}{\left (e^5-x\right )^3 x^2} \, dx\\ &=15 \int \left (\frac {\log ^2(x)}{\left (e^5-x\right )^2 x^2}-\frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{\left (e^5-x\right )^3 x^2}\right ) \, dx\\ &=15 \int \frac {\log ^2(x)}{\left (e^5-x\right )^2 x^2} \, dx-15 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{\left (e^5-x\right )^3 x^2} \, dx\\ &=15 \int \left (\frac {\log ^2(x)}{e^{10} \left (e^5-x\right )^2}+\frac {2 \log ^2(x)}{e^{15} \left (e^5-x\right )}+\frac {\log ^2(x)}{e^{10} x^2}+\frac {2 \log ^2(x)}{e^{15} x}\right ) \, dx-15 \int \left (\frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^{10} \left (e^5-x\right )^3}+\frac {2 \log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^{15} \left (e^5-x\right )^2}+\frac {3 \log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^{20} \left (e^5-x\right )}+\frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^{15} x^2}+\frac {3 \log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^{20} x}\right ) \, dx\\ &=-\frac {45 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^5-x} \, dx}{e^{20}}-\frac {45 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{x} \, dx}{e^{20}}-\frac {15 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{x^2} \, dx}{e^{15}}+\frac {30 \int \frac {\log ^2(x)}{e^5-x} \, dx}{e^{15}}+\frac {30 \int \frac {\log ^2(x)}{x} \, dx}{e^{15}}-\frac {30 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{15}}+\frac {15 \int \frac {\log ^2(x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {15 \int \frac {\log ^2(x)}{x^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^{10}}\\ &=-\frac {15 \log ^2(x)}{e^{10} x}+\frac {15 x \log ^2(x)}{e^{15} \left (e^5-x\right )}-\frac {30 \log ^2(x) \log \left (1-\frac {x}{e^5}\right )}{e^{15}}-\frac {45 \int \left (2 \log (x) \log (2 x)-\frac {2 e^5 \log (x) \log (2 x)}{x}-3 \log ^2(x) \log (2 x)+\frac {e^5 \log ^2(x) \log (2 x)}{x}\right ) \, dx}{e^{20}}-\frac {45 \int \left (-\frac {2 e^5 \log (x) \log (2 x)}{e^5-x}+\frac {2 x \log (x) \log (2 x)}{e^5-x}+\frac {e^5 \log ^2(x) \log (2 x)}{e^5-x}-\frac {3 x \log ^2(x) \log (2 x)}{e^5-x}\right ) \, dx}{e^{20}}-\frac {15 \int \left (-\frac {2 e^5 \log (x) \log (2 x)}{x^2}+\frac {2 \log (x) \log (2 x)}{x}+\frac {e^5 \log ^2(x) \log (2 x)}{x^2}-\frac {3 \log ^2(x) \log (2 x)}{x}\right ) \, dx}{e^{15}}-\frac {30 \int \frac {\log (x)}{e^5-x} \, dx}{e^{15}}-\frac {30 \int \left (-\frac {2 e^5 \log (x) \log (2 x)}{\left (e^5-x\right )^2}+\frac {2 x \log (x) \log (2 x)}{\left (e^5-x\right )^2}+\frac {e^5 \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2}-\frac {3 x \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2}\right ) \, dx}{e^{15}}+\frac {30 \text {Subst}\left (\int x^2 \, dx,x,\log (x)\right )}{e^{15}}+\frac {60 \int \frac {\log (x) \log \left (1-\frac {x}{e^5}\right )}{x} \, dx}{e^{15}}-\frac {15 \int \left (-\frac {2 e^5 \log (x) \log (2 x)}{\left (e^5-x\right )^3}+\frac {2 x \log (x) \log (2 x)}{\left (e^5-x\right )^3}+\frac {e^5 \log ^2(x) \log (2 x)}{\left (e^5-x\right )^3}-\frac {3 x \log ^2(x) \log (2 x)}{\left (e^5-x\right )^3}\right ) \, dx}{e^{10}}+\frac {30 \int \frac {\log (x)}{x^2} \, dx}{e^{10}}\\ &=-\frac {30}{e^{10} x}+\frac {150 \log \left (e^5-x\right )}{e^{15}}-\frac {30 \log (x)}{e^{10} x}-\frac {15 \log ^2(x)}{e^{10} x}+\frac {15 x \log ^2(x)}{e^{15} \left (e^5-x\right )}+\frac {10 \log ^3(x)}{e^{15}}-\frac {30 \log ^2(x) \log \left (1-\frac {x}{e^5}\right )}{e^{15}}-\frac {60 \log (x) \text {Li}_2\left (\frac {x}{e^5}\right )}{e^{15}}-\frac {90 \int \log (x) \log (2 x) \, dx}{e^{20}}-\frac {90 \int \frac {x \log (x) \log (2 x)}{e^5-x} \, dx}{e^{20}}+\frac {135 \int \log ^2(x) \log (2 x) \, dx}{e^{20}}+\frac {135 \int \frac {x \log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{20}}-\frac {30 \int \frac {\log (x) \log (2 x)}{x} \, dx}{e^{15}}-\frac {30 \int \frac {\log \left (\frac {x}{e^5}\right )}{e^5-x} \, dx}{e^{15}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {60 \int \frac {x \log (x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{15}}+\frac {60 \int \frac {\text {Li}_2\left (\frac {x}{e^5}\right )}{x} \, dx}{e^{15}}+\frac {90 \int \frac {\log (x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {90 \int \frac {\log (x) \log (2 x)}{x} \, dx}{e^{15}}+\frac {90 \int \frac {x \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{15}}-\frac {15 \int \frac {\log ^2(x) \log (2 x)}{x^2} \, dx}{e^{10}}+\frac {30 \int \frac {\log (x) \log (2 x)}{x^2} \, dx}{e^{10}}-\frac {30 \int \frac {x \log (x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^{10}}-\frac {30 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {45 \int \frac {x \log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^{10}}+\frac {60 \int \frac {\log (x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5}+\frac {30 \int \frac {\log (x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.10, size = 22, normalized size = 0.92 \begin {gather*} \frac {15 \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2 x} \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. \(47\) vs.
\(2(23)=46\).
time = 1.08, size = 48, normalized size = 2.00
method | result | size |
risch | \(\frac {15 \ln \left (x \right )^{3}}{x \left ({\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}\right )}+\frac {15 \ln \left (2\right ) \ln \left (x \right )^{2}}{x \left ({\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}\right )}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 31, normalized size = 1.29 \begin {gather*} \frac {15 \, {\left (\log \left (2\right ) \log \left (x\right )^{2} + \log \left (x\right )^{3}\right )}}{x^{3} - 2 \, x^{2} e^{5} + x e^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 31, normalized size = 1.29 \begin {gather*} \frac {15 \, {\left (\log \left (2\right ) \log \left (x\right )^{2} + \log \left (x\right )^{3}\right )}}{x^{3} - 2 \, x^{2} e^{5} + x e^{10}} \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. 49 vs.
\(2 (20) = 40\).
time = 0.17, size = 49, normalized size = 2.04 \begin {gather*} \frac {15 \log {\left (x \right )}^{3}}{x^{3} - 2 x^{2} e^{5} + x e^{10}} + \frac {15 \log {\left (2 \right )} \log {\left (x \right )}^{2}}{x^{3} - 2 x^{2} e^{5} + x e^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 31, normalized size = 1.29 \begin {gather*} \frac {15 \, {\left (\log \left (2\right ) \log \left (x\right )^{2} + \log \left (x\right )^{3}\right )}}{x^{3} - 2 \, x^{2} e^{5} + x e^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.43, size = 22, normalized size = 0.92 \begin {gather*} \frac {15\,{\ln \left (x\right )}^2\,\left (\ln \left (2\right )+\ln \left (x\right )\right )}{x\,{\left (x-{\mathrm {e}}^5\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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