Optimal. Leaf size=27 \[ -x+\log \left (x \left (x-x \left (-1+625 x^4-\frac {4}{-4+\log (x)}\right )\right )\right ) \]
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Rubi [F]
time = 2.16, 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 {-28+16 x+60000 x^4-10000 x^5+\left (24-12 x-30000 x^4+5000 x^5\right ) \log (x)+\left (-4+2 x+3750 x^4-625 x^5\right ) \log ^2(x)}{-16 x+10000 x^5+\left (12 x-5000 x^5\right ) \log (x)+\left (-2 x+625 x^5\right ) \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 (\frac {-4+2 x+3750 x^4-625 x^5}{x \left (-2+625 x^4\right )}-\frac {1}{x (-4+\log (x))}+\frac {4+7500 x^4+390625 x^8}{x \left (-2+625 x^4\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}\right ) \, dx\\ &=\int \frac {-4+2 x+3750 x^4-625 x^5}{x \left (-2+625 x^4\right )} \, dx-\int \frac {1}{x (-4+\log (x))} \, dx+\int \frac {4+7500 x^4+390625 x^8}{x \left (-2+625 x^4\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx\\ &=\int \left (\frac {2-625 x^4}{-2+625 x^4}+\frac {-4+3750 x^4}{x \left (-2+625 x^4\right )}\right ) \, dx+\int \left (-\frac {2}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}+\frac {625 x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)}+\frac {10000 x^3}{\left (-2+625 x^4\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,-4+\log (x)\right )\\ &=-\log (4-\log (x))-2 \int \frac {1}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+625 \int \frac {x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)} \, dx+10000 \int \frac {x^3}{\left (-2+625 x^4\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+\int \frac {2-625 x^4}{-2+625 x^4} \, dx+\int \frac {-4+3750 x^4}{x \left (-2+625 x^4\right )} \, dx\\ &=-\log (4-\log (x))+\frac {1}{4} \text {Subst}\left (\int \frac {-4+3750 x}{x (-2+625 x)} \, dx,x,x^4\right )-2 \int \frac {1}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+625 \int \frac {x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)} \, dx+10000 \int \left (\frac {x}{2 \left (-25 \sqrt {2}+625 x^2\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}+\frac {x}{2 \left (25 \sqrt {2}+625 x^2\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}\right ) \, dx-\int 1 \, dx\\ &=-x-\log (4-\log (x))+\frac {1}{4} \text {Subst}\left (\int \left (\frac {2}{x}+\frac {2500}{-2+625 x}\right ) \, dx,x,x^4\right )-2 \int \frac {1}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+625 \int \frac {x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)} \, dx+5000 \int \frac {x}{\left (-25 \sqrt {2}+625 x^2\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+5000 \int \frac {x}{\left (25 \sqrt {2}+625 x^2\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx\\ &=-x+2 \log (x)+\log \left (2-625 x^4\right )-\log (4-\log (x))-2 \int \frac {1}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+625 \int \frac {x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)} \, dx+5000 \int \left (-\frac {1}{250 \left (i \sqrt [4]{2}-5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}+\frac {1}{250 \left (i \sqrt [4]{2}+5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}\right ) \, dx+5000 \int \left (-\frac {1}{250 \left (\sqrt [4]{2}-5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}+\frac {1}{250 \left (\sqrt [4]{2}+5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}\right ) \, dx\\ &=-x+2 \log (x)+\log \left (2-625 x^4\right )-\log (4-\log (x))-2 \int \frac {1}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx-20 \int \frac {1}{\left (i \sqrt [4]{2}-5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx-20 \int \frac {1}{\left (\sqrt [4]{2}-5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+20 \int \frac {1}{\left (i \sqrt [4]{2}+5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+20 \int \frac {1}{\left (\sqrt [4]{2}+5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+625 \int \frac {x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 36, normalized size = 1.33 \begin {gather*} -x+2 \log (x)-\log (4-\log (x))+\log \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.40, size = 35, normalized size = 1.30
method | result | size |
default | \(2 \ln \left (x \right )-x +\ln \left (625 x^{4} \ln \left (x \right )-2500 x^{4}-2 \ln \left (x \right )+4\right )-\ln \left (\ln \left (x \right )-4\right )\) | \(35\) |
norman | \(2 \ln \left (x \right )-x +\ln \left (625 x^{4} \ln \left (x \right )-2500 x^{4}-2 \ln \left (x \right )+4\right )-\ln \left (\ln \left (x \right )-4\right )\) | \(35\) |
risch | \(-x +2 \ln \left (x \right )+\ln \left (625 x^{4}-2\right )+\ln \left (\ln \left (x \right )-\frac {4 \left (625 x^{4}-1\right )}{625 x^{4}-2}\right )-\ln \left (\ln \left (x \right )-4\right )\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 53, normalized size = 1.96 \begin {gather*} -x + \log \left (625 \, x^{4} - 2\right ) + 2 \, \log \left (x\right ) + \log \left (-\frac {2500 \, x^{4} - {\left (625 \, x^{4} - 2\right )} \log \left (x\right ) - 4}{625 \, x^{4} - 2}\right ) - \log \left (\log \left (x\right ) - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 53, normalized size = 1.96 \begin {gather*} -x + \log \left (625 \, x^{4} - 2\right ) + 2 \, \log \left (x\right ) + \log \left (-\frac {2500 \, x^{4} - {\left (625 \, x^{4} - 2\right )} \log \left (x\right ) - 4}{625 \, x^{4} - 2}\right ) - \log \left (\log \left (x\right ) - 4\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 = 0.44, size = 34, normalized size = 1.26 \begin {gather*} -x + \log \left (625 \, x^{4} \log \left (x\right ) - 2500 \, x^{4} - 2 \, \log \left (x\right ) + 4\right ) + 2 \, \log \left (x\right ) - \log \left (\log \left (x\right ) - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.58, size = 92, normalized size = 3.41 \begin {gather*} \ln \left (390625\,x^8+7500\,x^4+4\right )-x+\ln \left (\frac {4\,\ln \left (x\right )-1250\,x^4\,\ln \left (x\right )+5000\,x^4-8}{2\,x-625\,x^5}\right )-\ln \left (x^4-\frac {2}{625}\right )-\ln \left (\frac {\left (\ln \left (x\right )-4\right )\,\left (390625\,x^8+7500\,x^4+4\right )}{x\,{\left (625\,x^4-2\right )}^2}\right )+2\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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