Optimal. Leaf size=28 \[ \frac {e^5 \left (x-\log \left (x+\log \left (5-4 \left (x+\frac {1}{-5+x^2}\right )\right )\right )\right )}{x} \]
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Rubi [F] time = 3.92, 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^5 \left (45 x-92 x^2-14 x^3+40 x^4+x^5-4 x^6\right )+\left (e^5 \left (-145 x+100 x^2+54 x^3-40 x^4-5 x^5+4 x^6\right )+e^5 \left (-145+100 x+54 x^2-40 x^3-5 x^4+4 x^5\right ) \log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right ) \log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{-145 x^3+100 x^4+54 x^5-40 x^6-5 x^7+4 x^8+\left (-145 x^2+100 x^3+54 x^4-40 x^5-5 x^6+4 x^7\right ) \log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\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 {e^5 \left (\frac {x \left (45-92 x-14 x^2+40 x^3+x^4-4 x^5\right )}{\left (-145+100 x+54 x^2-40 x^3-5 x^4+4 x^5\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )\right )}{x^2} \, dx\\ &=e^5 \int \frac {\frac {x \left (45-92 x-14 x^2+40 x^3+x^4-4 x^5\right )}{\left (-145+100 x+54 x^2-40 x^3-5 x^4+4 x^5\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx\\ &=e^5 \int \left (\frac {45-92 x-14 x^2+40 x^3+x^4-4 x^5}{x \left (-5+x^2\right ) \left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2}\right ) \, dx\\ &=e^5 \int \frac {45-92 x-14 x^2+40 x^3+x^4-4 x^5}{x \left (-5+x^2\right ) \left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx+e^5 \int \frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx\\ &=e^5 \int \left (-\frac {9}{29 x \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\frac {2}{\left (-5+x^2\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}-\frac {2 \left (-345+124 x+40 x^2\right )}{29 \left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}\right ) \, dx+e^5 \int \frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx\\ &=-\left (\frac {1}{29} \left (2 e^5\right ) \int \frac {-345+124 x+40 x^2}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx\right )-\frac {1}{29} \left (9 e^5\right ) \int \frac {1}{x \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx+e^5 \int \frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx+\left (2 e^5\right ) \int \frac {1}{\left (-5+x^2\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx\\ &=-\left (\frac {1}{29} \left (2 e^5\right ) \int \left (-\frac {345}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\frac {124 x}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\frac {40 x^2}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}\right ) \, dx\right )-\frac {1}{29} \left (9 e^5\right ) \int \frac {1}{x \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx+e^5 \int \frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx+\left (2 e^5\right ) \int \left (-\frac {1}{2 \sqrt {5} \left (\sqrt {5}-x\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}-\frac {1}{2 \sqrt {5} \left (\sqrt {5}+x\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}\right ) \, dx\\ &=-\left (\frac {1}{29} \left (9 e^5\right ) \int \frac {1}{x \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx\right )+e^5 \int \frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx-\frac {1}{29} \left (80 e^5\right ) \int \frac {x^2}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx-\frac {1}{29} \left (248 e^5\right ) \int \frac {x}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx+\frac {1}{29} \left (690 e^5\right ) \int \frac {1}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx-\frac {e^5 \int \frac {1}{\left (\sqrt {5}-x\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx}{\sqrt {5}}-\frac {e^5 \int \frac {1}{\left (\sqrt {5}+x\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx}{\sqrt {5}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 35, normalized size = 1.25 \begin {gather*} -\frac {e^5 \log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 35, normalized size = 1.25 \begin {gather*} -\frac {e^{5} \log \left (x + \log \left (-\frac {4 \, x^{3} - 5 \, x^{2} - 20 \, x + 29}{x^{2} - 5}\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.78, size = 35, normalized size = 1.25 \begin {gather*} -\frac {e^{5} \log \left (x + \log \left (-\frac {4 \, x^{3} - 5 \, x^{2} - 20 \, x + 29}{x^{2} - 5}\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (4 x^{5}-5 x^{4}-40 x^{3}+54 x^{2}+100 x -145\right ) {\mathrm e}^{5} \ln \left (\frac {-4 x^{3}+5 x^{2}+20 x -29}{x^{2}-5}\right )+\left (4 x^{6}-5 x^{5}-40 x^{4}+54 x^{3}+100 x^{2}-145 x \right ) {\mathrm e}^{5}\right ) \ln \left (\ln \left (\frac {-4 x^{3}+5 x^{2}+20 x -29}{x^{2}-5}\right )+x \right )+\left (-4 x^{6}+x^{5}+40 x^{4}-14 x^{3}-92 x^{2}+45 x \right ) {\mathrm e}^{5}}{\left (4 x^{7}-5 x^{6}-40 x^{5}+54 x^{4}+100 x^{3}-145 x^{2}\right ) \ln \left (\frac {-4 x^{3}+5 x^{2}+20 x -29}{x^{2}-5}\right )+4 x^{8}-5 x^{7}-40 x^{6}+54 x^{5}+100 x^{4}-145 x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 34, normalized size = 1.21 \begin {gather*} -\frac {e^{5} \log \left (x + \log \left (-4 \, x^{3} + 5 \, x^{2} + 20 \, x - 29\right ) - \log \left (x^{2} - 5\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.86, size = 34, normalized size = 1.21 \begin {gather*} -\frac {\ln \left (x+\ln \left (\frac {-4\,x^3+5\,x^2+20\,x-29}{x^2-5}\right )\right )\,{\mathrm {e}}^5}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.23, size = 31, normalized size = 1.11 \begin {gather*} - \frac {e^{5} \log {\left (x + \log {\left (\frac {- 4 x^{3} + 5 x^{2} + 20 x - 29}{x^{2} - 5} \right )} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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