Optimal. Leaf size=27 \[ \log \left (25+e^x-\frac {1}{5} x (-x+x (2 x-\log (x)))^2\right ) \]
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Rubi [F]
time = 3.84, 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 {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {-125-5 x^2+21 x^3-24 x^4+4 x^5-8 x^2 \log (x)+18 x^3 \log (x)-4 x^4 \log (x)-3 x^2 \log ^2(x)+x^3 \log ^2(x)}{125+5 e^x-x^3+4 x^4-4 x^5-2 x^3 \log (x)+4 x^4 \log (x)-x^3 \log ^2(x)}\right ) \, dx\\ &=x+\int \frac {-125-5 x^2+21 x^3-24 x^4+4 x^5-8 x^2 \log (x)+18 x^3 \log (x)-4 x^4 \log (x)-3 x^2 \log ^2(x)+x^3 \log ^2(x)}{125+5 e^x-x^3+4 x^4-4 x^5-2 x^3 \log (x)+4 x^4 \log (x)-x^3 \log ^2(x)} \, dx\\ &=x+\int \left (-\frac {125}{125+5 e^x-x^3+4 x^4-4 x^5-2 x^3 \log (x)+4 x^4 \log (x)-x^3 \log ^2(x)}+\frac {5 x^2}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}-\frac {21 x^3}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}+\frac {24 x^4}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}-\frac {4 x^5}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}+\frac {8 x^2 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}-\frac {18 x^3 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}+\frac {4 x^4 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}+\frac {3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}-\frac {x^3 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)}\right ) \, dx\\ &=x+3 \int \frac {x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx-4 \int \frac {x^5}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx+4 \int \frac {x^4 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx+5 \int \frac {x^2}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx+8 \int \frac {x^2 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx-18 \int \frac {x^3 \log (x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx-21 \int \frac {x^3}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx+24 \int \frac {x^4}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx-125 \int \frac {1}{125+5 e^x-x^3+4 x^4-4 x^5-2 x^3 \log (x)+4 x^4 \log (x)-x^3 \log ^2(x)} \, dx-\int \frac {x^3 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+2 x^3 \log (x)-4 x^4 \log (x)+x^3 \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.04, size = 43, normalized size = 1.59
method | result | size |
risch | \(3 \ln \left (x \right )+\ln \left (\ln \left (x \right )^{2}+\left (-4 x +2\right ) \ln \left (x \right )+\frac {4 x^{5}-4 x^{4}+x^{3}-5 \,{\mathrm e}^{x}-125}{x^{3}}\right )\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 44, normalized size = 1.63 \begin {gather*} \log \left (-\frac {4}{5} \, x^{5} - \frac {1}{5} \, x^{3} \log \left (x\right )^{2} + \frac {4}{5} \, x^{4} - \frac {1}{5} \, x^{3} + \frac {2}{5} \, {\left (2 \, x^{4} - x^{3}\right )} \log \left (x\right ) + e^{x} + 25\right ) \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. 52 vs.
\(2 (24) = 48\).
time = 0.40, size = 52, normalized size = 1.93 \begin {gather*} 3 \, \log \left (x\right ) + \log \left (\frac {4 \, x^{5} + x^{3} \log \left (x\right )^{2} - 4 \, x^{4} + x^{3} - 2 \, {\left (2 \, x^{4} - x^{3}\right )} \log \left (x\right ) - 5 \, e^{x} - 125}{x^{3}}\right ) \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. 54 vs.
\(2 (20) = 40\).
time = 0.16, size = 54, normalized size = 2.00 \begin {gather*} \log {\left (- \frac {4 x^{5}}{5} + \frac {4 x^{4} \log {\left (x \right )}}{5} + \frac {4 x^{4}}{5} - \frac {x^{3} \log {\left (x \right )}^{2}}{5} - \frac {2 x^{3} \log {\left (x \right )}}{5} - \frac {x^{3}}{5} + e^{x} + 25 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 45, normalized size = 1.67 \begin {gather*} \log \left (-4 \, x^{5} + 4 \, x^{4} \log \left (x\right ) - x^{3} \log \left (x\right )^{2} + 4 \, x^{4} - 2 \, x^{3} \log \left (x\right ) - x^{3} + 5 \, e^{x} + 125\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.31, size = 42, normalized size = 1.56 \begin {gather*} \ln \left (\ln \left (x\right )\,\left (2\,x^3-4\,x^4\right )-5\,{\mathrm {e}}^x+x^3\,{\ln \left (x\right )}^2+x^3-4\,x^4+4\,x^5-125\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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