Optimal. Leaf size=26 \[ x \left (2+x-\frac {x \log (x)}{4 \log \left (\frac {e^5}{3+x^4}\right )}\right ) \]
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Rubi [F] time = 3.26, 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 {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 (1+x)-\frac {x^5 \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}-\frac {x (1+2 \log (x))}{4 \left (5+\log \left (\frac {1}{3+x^4}\right )\right )}\right ) \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x (1+2 \log (x))}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\int \frac {x^5 \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ &=(1+x)^2-\frac {1}{4} \int \left (\frac {x}{5+\log \left (\frac {1}{3+x^4}\right )}+\frac {2 x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )}\right ) \, dx-\int \left (\frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}-\frac {3 x \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx+3 \int \frac {x \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx+3 \int \left (-\frac {i x \log (x)}{2 \sqrt {3} \left (-i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}+\frac {i x \log (x)}{2 \sqrt {3} \left (i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \left (i \sqrt {3}\right ) \int \frac {x \log (x)}{\left (-i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx+\frac {1}{2} \left (i \sqrt {3}\right ) \int \frac {x \log (x)}{\left (i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \left (i \sqrt {3}\right ) \int \left (-\frac {\log (x)}{2 \left (\sqrt [4]{-3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}+\frac {\log (x)}{2 \left (\sqrt [4]{-3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx+\frac {1}{2} \left (i \sqrt {3}\right ) \int \left (-\frac {\log (x)}{2 \left (-(-1)^{3/4} \sqrt [4]{3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}+\frac {\log (x)}{2 \left (-(-1)^{3/4} \sqrt [4]{3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx+\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (\sqrt [4]{-3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (-(-1)^{3/4} \sqrt [4]{3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (\sqrt [4]{-3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx+\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (-(-1)^{3/4} \sqrt [4]{3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.84, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{4} x \left (8+4 x-\frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 42, normalized size = 1.62 \begin {gather*} -\frac {x^{2} \log \relax (x) - 4 \, {\left (x^{2} + 2 \, x\right )} \log \left (\frac {e^{5}}{x^{4} + 3}\right )}{4 \, \log \left (\frac {e^{5}}{x^{4} + 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 24, normalized size = 0.92 \begin {gather*} x^{2} + \frac {x^{2} \log \relax (x)}{4 \, {\left (\log \left (x^{4} + 3\right ) - 5\right )}} + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 30, normalized size = 1.15
method | result | size |
risch | \(x^{2}+2 x -\frac {i x^{2} \ln \relax (x )}{2 \left (-2 i \ln \left (x^{4}+3\right )+10 i\right )}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 42, normalized size = 1.62 \begin {gather*} \frac {x^{2} \log \relax (x) - 20 \, x^{2} + 4 \, {\left (x^{2} + 2 \, x\right )} \log \left (x^{4} + 3\right ) - 40 \, x}{4 \, {\left (\log \left (x^{4} + 3\right ) - 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 77, normalized size = 2.96 \begin {gather*} 2\,x+\frac {3}{16\,x^2}+\frac {17\,x^2}{16}-\frac {\frac {x^2\,\ln \relax (x)}{4}+\frac {\ln \left (\frac {{\mathrm {e}}^5}{x^4+3}\right )\,\left (x^4+3\right )\,\left (2\,\ln \relax (x)+1\right )}{16\,x^2}}{\ln \left (\frac {{\mathrm {e}}^5}{x^4+3}\right )}+\frac {\ln \relax (x)\,\left (\frac {x^4}{8}+\frac {3}{8}\right )}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 24, normalized size = 0.92 \begin {gather*} - \frac {x^{2} \log {\relax (x )}}{4 \log {\left (\frac {e^{5}}{x^{4} + 3} \right )}} + x^{2} + 2 x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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