Optimal. Leaf size=28 \[ \log \left (\left (\frac {2 \left (e^5+x\right )}{5+x}-\frac {e^2}{x^2 \log ^2(x)}\right )^2\right ) \]
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Rubi [F]
time = 10.27, 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 {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-e^2 (5+x)^2-e^2 (5+x)^2 \log (x)+\left (-5+e^5\right ) x^3 \log ^3(x)\right )}{x (5+x) \log (x) \left (e^2 (5+x)-2 x^2 \left (e^5+x\right ) \log ^2(x)\right )} \, dx\\ &=4 \int \frac {-e^2 (5+x)^2-e^2 (5+x)^2 \log (x)+\left (-5+e^5\right ) x^3 \log ^3(x)}{x (5+x) \log (x) \left (e^2 (5+x)-2 x^2 \left (e^5+x\right ) \log ^2(x)\right )} \, dx\\ &=4 \int \left (\frac {5-e^5}{2 (5+x) \left (e^5+x\right )}-\frac {1}{x \log (x)}+\frac {-10 e^7-15 e^2 \left (1+\frac {e^5}{15}\right ) x-2 e^2 x^2-4 e^{10} x^2 \log (x)-8 e^5 x^3 \log (x)-4 x^4 \log (x)}{2 x \left (e^5+x\right ) \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )}\right ) \, dx\\ &=2 \int \frac {-10 e^7-15 e^2 \left (1+\frac {e^5}{15}\right ) x-2 e^2 x^2-4 e^{10} x^2 \log (x)-8 e^5 x^3 \log (x)-4 x^4 \log (x)}{x \left (e^5+x\right ) \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )} \, dx-4 \int \frac {1}{x \log (x)} \, dx+\left (2 \left (5-e^5\right )\right ) \int \frac {1}{(5+x) \left (e^5+x\right )} \, dx\\ &=-\left (2 \int \frac {1}{5+x} \, dx\right )+2 \int \frac {1}{e^5+x} \, dx+2 \int \frac {-e^7 (10+x)-e^2 x (15+2 x)-4 x^2 \left (e^5+x\right )^2 \log (x)}{x \left (e^5+x\right ) \left (e^2 (5+x)-2 x^2 \left (e^5+x\right ) \log ^2(x)\right )} \, dx-4 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-2 \log (5+x)+2 \log \left (e^5+x\right )-4 \log (\log (x))+2 \int \left (\frac {-10 e^7-15 e^2 \left (1+\frac {e^5}{15}\right ) x-2 e^2 x^2-4 e^{10} x^2 \log (x)-8 e^5 x^3 \log (x)-4 x^4 \log (x)}{e^5 x \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )}+\frac {10 e^7+15 e^2 \left (1+\frac {e^5}{15}\right ) x+2 e^2 x^2+4 e^{10} x^2 \log (x)+8 e^5 x^3 \log (x)+4 x^4 \log (x)}{e^5 \left (e^5+x\right ) \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )}\right ) \, dx\\ &=-2 \log (5+x)+2 \log \left (e^5+x\right )-4 \log (\log (x))+\frac {2 \int \frac {-10 e^7-15 e^2 \left (1+\frac {e^5}{15}\right ) x-2 e^2 x^2-4 e^{10} x^2 \log (x)-8 e^5 x^3 \log (x)-4 x^4 \log (x)}{x \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )} \, dx}{e^5}+\frac {2 \int \frac {10 e^7+15 e^2 \left (1+\frac {e^5}{15}\right ) x+2 e^2 x^2+4 e^{10} x^2 \log (x)+8 e^5 x^3 \log (x)+4 x^4 \log (x)}{\left (e^5+x\right ) \left (5 e^2+e^2 x-2 e^5 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )} \, dx}{e^5}\\ &=-2 \log (5+x)+2 \log \left (e^5+x\right )-4 \log (\log (x))+\frac {2 \int \frac {e^7 (10+x)+e^2 x (15+2 x)+4 x^2 \left (e^5+x\right )^2 \log (x)}{\left (e^5+x\right ) \left (e^2 (5+x)-2 x^2 \left (e^5+x\right ) \log ^2(x)\right )} \, dx}{e^5}+\frac {2 \int \frac {-e^7 (10+x)-e^2 x (15+2 x)-4 x^2 \left (e^5+x\right )^2 \log (x)}{e^2 x (5+x)-2 x^3 \left (e^5+x\right ) \log ^2(x)} \, dx}{e^5}\\ &=-2 \log (5+x)+2 \log \left (e^5+x\right )-4 \log (\log (x))+\frac {2 \int \left (\frac {15 e^2 \left (1+\frac {e^5}{15}\right )}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)}+\frac {10 e^7}{x \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}+\frac {2 e^2 x}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)}+\frac {4 e^{10} x \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)}+\frac {8 e^5 x^2 \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)}+\frac {4 x^3 \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)}\right ) \, dx}{e^5}+\frac {2 \int \left (-\frac {10 e^7}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}-\frac {15 e^2 \left (1+\frac {e^5}{15}\right ) x}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}-\frac {2 e^2 x^2}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}-\frac {4 e^{10} x^2 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}-\frac {8 e^5 x^3 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}-\frac {4 x^4 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )}\right ) \, dx}{e^5}\\ &=-2 \log (5+x)+2 \log \left (e^5+x\right )-4 \log (\log (x))+16 \int \frac {x^2 \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)} \, dx-16 \int \frac {x^3 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx+\frac {8 \int \frac {x^3 \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)} \, dx}{e^5}-\frac {8 \int \frac {x^4 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx}{e^5}+\frac {4 \int \frac {x}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)} \, dx}{e^3}-\frac {4 \int \frac {x^2}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx}{e^3}+\left (20 e^2\right ) \int \frac {1}{x \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx-\left (20 e^2\right ) \int \frac {1}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx+\left (8 e^5\right ) \int \frac {x \log (x)}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)} \, dx-\left (8 e^5\right ) \int \frac {x^2 \log (x)}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx+\frac {\left (2 \left (15+e^5\right )\right ) \int \frac {1}{-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)} \, dx}{e^3}-\frac {\left (2 \left (15+e^5\right )\right ) \int \frac {x}{\left (e^5+x\right ) \left (-5 e^2-e^2 x+2 e^5 x^2 \log ^2(x)+2 x^3 \log ^2(x)\right )} \, dx}{e^3}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.97, size = 50, normalized size = 1.79
method | result | size |
risch | \(-2 \ln \left (5+x \right )+2 \ln \left ({\mathrm e}^{5}+x \right )-4 \ln \left (\ln \left (x \right )\right )+2 \ln \left (\ln \left (x \right )^{2}-\frac {{\mathrm e}^{2} \left (5+x \right )}{2 \left ({\mathrm e}^{5}+x \right ) x^{2}}\right )\) | \(44\) |
default | \(-4 \ln \left (x \right )-2 \ln \left (5+x \right )+2 \ln \left (2 x^{2} {\mathrm e}^{5} \ln \left (x \right )^{2}+2 x^{3} \ln \left (x \right )^{2}-{\mathrm e}^{2} x -5 \,{\mathrm e}^{2}\right )-4 \ln \left (\ln \left (x \right )\right )\) | \(50\) |
norman | \(-4 \ln \left (x \right )-2 \ln \left (5+x \right )+2 \ln \left (2 x^{2} {\mathrm e}^{5} \ln \left (x \right )^{2}+2 x^{3} \ln \left (x \right )^{2}-{\mathrm e}^{2} x -5 \,{\mathrm e}^{2}\right )-4 \ln \left (\ln \left (x \right )\right )\) | \(50\) |
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. 62 vs.
\(2 (26) = 52\).
time = 0.31, size = 62, normalized size = 2.21 \begin {gather*} 2 \, \log \left (x + e^{5}\right ) - 2 \, \log \left (x + 5\right ) + 2 \, \log \left (\frac {2 \, {\left (x^{3} + x^{2} e^{5}\right )} \log \left (x\right )^{2} - x e^{2} - 5 \, e^{2}}{2 \, {\left (x^{3} + x^{2} e^{5}\right )}}\right ) - 4 \, \log \left (\log \left (x\right )\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. 59 vs.
\(2 (26) = 52\).
time = 0.33, size = 59, normalized size = 2.11 \begin {gather*} 2 \, \log \left (x + e^{5}\right ) - 2 \, \log \left (x + 5\right ) + 2 \, \log \left (\frac {2 \, {\left (x^{3} + x^{2} e^{5}\right )} \log \left (x\right )^{2} - {\left (x + 5\right )} e^{2}}{x^{3} + x^{2} e^{5}}\right ) - 4 \, \log \left (\log \left (x\right )\right ) \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: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 49, normalized size = 1.75 \begin {gather*} 2 \, \log \left (2 \, x^{3} \log \left (x\right )^{2} + 2 \, x^{2} e^{5} \log \left (x\right )^{2} - x e^{2} - 5 \, e^{2}\right ) - 2 \, \log \left (x + 5\right ) - 4 \, \log \left (x\right ) - 4 \, \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\left (20\,x^3-4\,x^3\,{\mathrm {e}}^5\right )\,{\ln \left (x\right )}^3+{\mathrm {e}}^2\,\left (4\,x^2+40\,x+100\right )\,\ln \left (x\right )+{\mathrm {e}}^2\,\left (4\,x^2+40\,x+100\right )}{{\ln \left (x\right )}^3\,\left ({\mathrm {e}}^5\,\left (2\,x^4+10\,x^3\right )+10\,x^4+2\,x^5\right )-{\mathrm {e}}^2\,\ln \left (x\right )\,\left (x^3+10\,x^2+25\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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