Optimal. Leaf size=29 \[ 5+e^x+\frac {\log (x)}{\left (e^x+x\right ) \left (25+x+x^2\right )}-\log \left (x^2\right ) \]
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Rubi [F] time = 18.07, 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 {25 x-1249 x^2-99 x^3-102 x^4-4 x^5-2 x^6+e^{3 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^{2 x} \left (-1250-100 x+1148 x^2+96 x^3+100 x^4+4 x^5+2 x^6\right )+e^x \left (25-2499 x-199 x^2+421 x^3+42 x^4+47 x^5+2 x^6+x^7\right )+\left (-25 x-2 x^2-3 x^3+e^x \left (-26 x-3 x^2-x^3\right )\right ) \log (x)}{625 x^3+50 x^4+51 x^5+2 x^6+x^7+e^{2 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^x \left (1250 x^2+100 x^3+102 x^4+4 x^5+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (25+x+x^2\right ) \left (e^{3 x} x \left (25+x+x^2\right )-x \left (-1+50 x+2 x^2+2 x^3\right )+2 e^{2 x} \left (-25-x+24 x^2+x^3+x^4\right )+e^x \left (1-100 x-4 x^2+21 x^3+x^4+x^5\right )\right )-x \left (25+2 x+3 x^2+e^x \left (26+3 x+x^2\right )\right ) \log (x)}{x \left (e^x+x\right )^2 \left (25+x+x^2\right )^2} \, dx\\ &=\int \left (e^x-\frac {2}{x}+\frac {(-1+x) \log (x)}{\left (e^x+x\right )^2 \left (25+x+x^2\right )}-\frac {-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)}{x \left (e^x+x\right ) \left (25+x+x^2\right )^2}\right ) \, dx\\ &=-2 \log (x)+\int e^x \, dx+\int \frac {(-1+x) \log (x)}{\left (e^x+x\right )^2 \left (25+x+x^2\right )} \, dx-\int \frac {-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)}{x \left (e^x+x\right ) \left (25+x+x^2\right )^2} \, dx\\ &=e^x-2 \log (x)-\frac {(2 i \log (x)) \int \frac {1}{\left (-1+3 i \sqrt {11}-2 x\right ) \left (e^x+x\right )^2} \, dx}{3 \sqrt {11}}-\frac {(2 i \log (x)) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx}{3 \sqrt {11}}+\frac {1}{33} \left (\left (33-i \sqrt {11}\right ) \log (x)\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx+\frac {1}{33} \left (\left (33+i \sqrt {11}\right ) \log (x)\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1-3 i \sqrt {11}+2 x\right )} \, dx-\int \left (\frac {-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)}{625 x \left (e^x+x\right )}-\frac {(1+x) \left (-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)\right )}{25 \left (e^x+x\right ) \left (25+x+x^2\right )^2}-\frac {(1+x) \left (-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)\right )}{625 \left (e^x+x\right ) \left (25+x+x^2\right )}\right ) \, dx-\int \frac {-2 i \sqrt {11} \int \frac {1}{\left (-1+3 i \sqrt {11}-2 x\right ) \left (e^x+x\right )^2} \, dx+\left (33+i \sqrt {11}\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1-3 i \sqrt {11}+2 x\right )} \, dx+3 \left (11-i \sqrt {11}\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx}{33 x} \, dx\\ &=e^x-2 \log (x)-\frac {1}{625} \int \frac {-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)}{x \left (e^x+x\right )} \, dx+\frac {1}{625} \int \frac {(1+x) \left (-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)\right )}{\left (e^x+x\right ) \left (25+x+x^2\right )} \, dx-\frac {1}{33} \int \frac {-2 i \sqrt {11} \int \frac {1}{\left (-1+3 i \sqrt {11}-2 x\right ) \left (e^x+x\right )^2} \, dx+\left (33+i \sqrt {11}\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1-3 i \sqrt {11}+2 x\right )} \, dx+3 \left (11-i \sqrt {11}\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx}{x} \, dx+\frac {1}{25} \int \frac {(1+x) \left (-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)\right )}{\left (e^x+x\right ) \left (25+x+x^2\right )^2} \, dx-\frac {(2 i \log (x)) \int \frac {1}{\left (-1+3 i \sqrt {11}-2 x\right ) \left (e^x+x\right )^2} \, dx}{3 \sqrt {11}}-\frac {(2 i \log (x)) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx}{3 \sqrt {11}}+\frac {1}{33} \left (\left (33-i \sqrt {11}\right ) \log (x)\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx+\frac {1}{33} \left (\left (33+i \sqrt {11}\right ) \log (x)\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1-3 i \sqrt {11}+2 x\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 26, normalized size = 0.90 \begin {gather*} e^x-2 \log (x)+\frac {\log (x)}{\left (e^x+x\right ) \left (25+x+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 76, normalized size = 2.62 \begin {gather*} \frac {{\left (x^{2} + x + 25\right )} e^{\left (2 \, x\right )} + {\left (x^{3} + x^{2} + 25 \, x\right )} e^{x} - {\left (2 \, x^{3} + 2 \, x^{2} + 2 \, {\left (x^{2} + x + 25\right )} e^{x} + 50 \, x - 1\right )} \log \relax (x)}{x^{3} + x^{2} + {\left (x^{2} + x + 25\right )} e^{x} + 25 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 108, normalized size = 3.72 \begin {gather*} \frac {x^{3} e^{x} - 2 \, x^{3} \log \relax (x) - 2 \, x^{2} e^{x} \log \relax (x) + x^{2} e^{\left (2 \, x\right )} + x^{2} e^{x} - 2 \, x^{2} \log \relax (x) - 2 \, x e^{x} \log \relax (x) + x e^{\left (2 \, x\right )} + 25 \, x e^{x} - 50 \, x \log \relax (x) - 50 \, e^{x} \log \relax (x) + 25 \, e^{\left (2 \, x\right )} + \log \relax (x)}{x^{3} + x^{2} e^{x} + x^{2} + x e^{x} + 25 \, x + 25 \, e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 0.86
method | result | size |
risch | \(\frac {\ln \relax (x )}{\left (x^{2}+x +25\right ) \left ({\mathrm e}^{x}+x \right )}-2 \ln \relax (x )+{\mathrm e}^{x}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 54, normalized size = 1.86 \begin {gather*} \frac {{\left (x^{2} + x + 25\right )} e^{\left (2 \, x\right )} + {\left (x^{3} + x^{2} + 25 \, x\right )} e^{x} + \log \relax (x)}{x^{3} + x^{2} + {\left (x^{2} + x + 25\right )} e^{x} + 25 \, x} - 2 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 24, normalized size = 0.83 \begin {gather*} {\mathrm {e}}^x-2\,\ln \relax (x)+\frac {\ln \relax (x)}{\left (x+{\mathrm {e}}^x\right )\,\left (x^2+x+25\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 31, normalized size = 1.07 \begin {gather*} e^{x} - 2 \log {\relax (x )} + \frac {\log {\relax (x )}}{x^{3} + x^{2} + 25 x + \left (x^{2} + x + 25\right ) e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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