Optimal. Leaf size=31 \[ 4+\frac {3 x^2 \left (-x+5 \left (\frac {1}{x}+x\right )\right ) (3-\log (x))^2}{e^3}+\log (x) \]
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Rubi [A]
time = 0.06, antiderivative size = 57, normalized size of antiderivative = 1.84, number of steps
used = 13, number of rules used = 8, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {12, 14, 2350,
2367, 2333, 2332, 2342, 2341} \begin {gather*} \frac {108 x^3}{e^3}+\frac {12 x^3 \log ^2(x)}{e^3}-\frac {72 x^3 \log (x)}{e^3}+\frac {135 x}{e^3}+\frac {15 x \log ^2(x)}{e^3}-\frac {90 x \log (x)}{e^3}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2350
Rule 2367
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^3+45 x+252 x^3+\left (-60 x-192 x^3\right ) \log (x)+\left (15 x+36 x^3\right ) \log ^2(x)}{x} \, dx}{e^3}\\ &=\frac {\int \left (\frac {e^3+45 x+252 x^3}{x}-12 \left (5+16 x^2\right ) \log (x)+3 \left (5+12 x^2\right ) \log ^2(x)\right ) \, dx}{e^3}\\ &=\frac {\int \frac {e^3+45 x+252 x^3}{x} \, dx}{e^3}+\frac {3 \int \left (5+12 x^2\right ) \log ^2(x) \, dx}{e^3}-\frac {12 \int \left (5+16 x^2\right ) \log (x) \, dx}{e^3}\\ &=-\frac {4 \left (15 x+16 x^3\right ) \log (x)}{e^3}+\frac {\int \left (45+\frac {e^3}{x}+252 x^2\right ) \, dx}{e^3}+\frac {3 \int \left (5 \log ^2(x)+12 x^2 \log ^2(x)\right ) \, dx}{e^3}+\frac {12 \int \left (5+\frac {16 x^2}{3}\right ) \, dx}{e^3}\\ &=\frac {105 x}{e^3}+\frac {316 x^3}{3 e^3}+\log (x)-\frac {4 \left (15 x+16 x^3\right ) \log (x)}{e^3}+\frac {15 \int \log ^2(x) \, dx}{e^3}+\frac {36 \int x^2 \log ^2(x) \, dx}{e^3}\\ &=\frac {105 x}{e^3}+\frac {316 x^3}{3 e^3}+\log (x)-\frac {4 \left (15 x+16 x^3\right ) \log (x)}{e^3}+\frac {15 x \log ^2(x)}{e^3}+\frac {12 x^3 \log ^2(x)}{e^3}-\frac {24 \int x^2 \log (x) \, dx}{e^3}-\frac {30 \int \log (x) \, dx}{e^3}\\ &=\frac {135 x}{e^3}+\frac {108 x^3}{e^3}+\log (x)-\frac {30 x \log (x)}{e^3}-\frac {8 x^3 \log (x)}{e^3}-\frac {4 \left (15 x+16 x^3\right ) \log (x)}{e^3}+\frac {15 x \log ^2(x)}{e^3}+\frac {12 x^3 \log ^2(x)}{e^3}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 1.52 \begin {gather*} \frac {135 x+108 x^3+e^3 \log (x)-90 x \log (x)-72 x^3 \log (x)+15 x \log ^2(x)+12 x^3 \log ^2(x)}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 48, normalized size = 1.55
method | result | size |
risch | \({\mathrm e}^{-3} \left (12 x^{3}+15 x \right ) \ln \left (x \right )^{2}+{\mathrm e}^{-3} \left (-72 x^{3}-90 x \right ) \ln \left (x \right )+108 \,{\mathrm e}^{-3} x^{3}+135 \,{\mathrm e}^{-3} x +\ln \left (x \right )\) | \(46\) |
default | \({\mathrm e}^{-3} \left (12 x^{3} \ln \left (x \right )^{2}-72 x^{3} \ln \left (x \right )+108 x^{3}+15 x \ln \left (x \right )^{2}-90 x \ln \left (x \right )+135 x +\ln \left (x \right ) {\mathrm e}^{3}\right )\) | \(48\) |
norman | \(\ln \left (x \right )+135 \,{\mathrm e}^{-3} x +108 \,{\mathrm e}^{-3} x^{3}-90 x \,{\mathrm e}^{-3} \ln \left (x \right )+15 x \,{\mathrm e}^{-3} \ln \left (x \right )^{2}-72 \,{\mathrm e}^{-3} x^{3} \ln \left (x \right )+12 \,{\mathrm e}^{-3} x^{3} \ln \left (x \right )^{2}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (26) = 52\).
time = 0.26, size = 61, normalized size = 1.97 \begin {gather*} \frac {1}{3} \, {\left (4 \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} - 192 \, x^{3} \log \left (x\right ) + 316 \, x^{3} + 45 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - 180 \, x \log \left (x\right ) + 3 \, e^{3} \log \left (x\right ) + 315 \, x\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 44, normalized size = 1.42 \begin {gather*} {\left (108 \, x^{3} + 3 \, {\left (4 \, x^{3} + 5 \, x\right )} \log \left (x\right )^{2} - {\left (72 \, x^{3} + 90 \, x - e^{3}\right )} \log \left (x\right ) + 135 \, x\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 51, normalized size = 1.65 \begin {gather*} \frac {\left (- 72 x^{3} - 90 x\right ) \log {\left (x \right )}}{e^{3}} + \frac {\left (12 x^{3} + 15 x\right ) \log {\left (x \right )}^{2}}{e^{3}} + \frac {108 x^{3} + 135 x + e^{3} \log {\left (x \right )}}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 45, normalized size = 1.45 \begin {gather*} {\left (12 \, x^{3} \log \left (x\right )^{2} - 72 \, x^{3} \log \left (x\right ) + 108 \, x^{3} + 15 \, x \log \left (x\right )^{2} - 90 \, x \log \left (x\right ) + e^{3} \log \left (x\right ) + 135 \, x\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.78, size = 51, normalized size = 1.65 \begin {gather*} 12\,{\mathrm {e}}^{-3}\,x^3\,{\ln \left (x\right )}^2-72\,{\mathrm {e}}^{-3}\,x^3\,\ln \left (x\right )+108\,{\mathrm {e}}^{-3}\,x^3+15\,{\mathrm {e}}^{-3}\,x\,{\ln \left (x\right )}^2-90\,{\mathrm {e}}^{-3}\,x\,\ln \left (x\right )+135\,{\mathrm {e}}^{-3}\,x+\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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