Optimal. Leaf size=29 \[ (4+x) \left (3+25 x^4+\frac {x}{\log \left (\frac {\left (x+4 x^2\right )^2}{x^4}\right )}\right ) \]
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Rubi [F] time = 0.49, 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 {8+2 x+\left (4+18 x+8 x^2\right ) \log \left (\frac {1+8 x+16 x^2}{x^2}\right )+\left (3+12 x+400 x^3+1725 x^4+500 x^5\right ) \log ^2\left (\frac {1+8 x+16 x^2}{x^2}\right )}{(1+4 x) \log ^2\left (\frac {1+8 x+16 x^2}{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 {8+2 x+\left (4+18 x+8 x^2\right ) \log \left (\frac {1+8 x+16 x^2}{x^2}\right )+\left (3+12 x+400 x^3+1725 x^4+500 x^5\right ) \log ^2\left (\frac {1+8 x+16 x^2}{x^2}\right )}{(1+4 x) \log ^2\left (16+\frac {1}{x^2}+\frac {8}{x}\right )} \, dx\\ &=\int \left (3+400 x^3+125 x^4+\frac {2 (4+x)}{(1+4 x) \log ^2\left (\frac {(1+4 x)^2}{x^2}\right )}+\frac {2 (2+x)}{\log \left (\frac {(1+4 x)^2}{x^2}\right )}\right ) \, dx\\ &=3 x+100 x^4+25 x^5+2 \int \frac {4+x}{(1+4 x) \log ^2\left (\frac {(1+4 x)^2}{x^2}\right )} \, dx+2 \int \frac {2+x}{\log \left (\frac {(1+4 x)^2}{x^2}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 33, normalized size = 1.14 \begin {gather*} 3 x+100 x^4+25 x^5+\frac {x (4+x)}{\log \left (\frac {(1+4 x)^2}{x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 55, normalized size = 1.90 \begin {gather*} \frac {x^{2} + {\left (25 \, x^{5} + 100 \, x^{4} + 3 \, x\right )} \log \left (\frac {16 \, x^{2} + 8 \, x + 1}{x^{2}}\right ) + 4 \, x}{\log \left (\frac {16 \, x^{2} + 8 \, x + 1}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 39, normalized size = 1.34 \begin {gather*} 25 \, x^{5} + 100 \, x^{4} + 3 \, x + \frac {x^{2} + 4 \, x}{\log \left (\frac {16 \, x^{2} + 8 \, x + 1}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 37, normalized size = 1.28
method | result | size |
risch | \(25 x^{5}+100 x^{4}+3 x +\frac {\left (4+x \right ) x}{\ln \left (\frac {16 x^{2}+8 x +1}{x^{2}}\right )}\) | \(37\) |
norman | \(\frac {x^{2}+4 x +100 x^{4} \ln \left (\frac {16 x^{2}+8 x +1}{x^{2}}\right )+25 x^{5} \ln \left (\frac {16 x^{2}+8 x +1}{x^{2}}\right )+3 \ln \left (\frac {16 x^{2}+8 x +1}{x^{2}}\right ) x}{\ln \left (\frac {16 x^{2}+8 x +1}{x^{2}}\right )}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 62, normalized size = 2.14 \begin {gather*} \frac {x^{2} + 2 \, {\left (25 \, x^{5} + 100 \, x^{4} + 3 \, x\right )} \log \left (4 \, x + 1\right ) - 2 \, {\left (25 \, x^{5} + 100 \, x^{4} + 3 \, x\right )} \log \relax (x) + 4 \, x}{2 \, {\left (\log \left (4 \, x + 1\right ) - \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 37, normalized size = 1.28 \begin {gather*} x\,\left (25\,x^4+100\,x^3+3\right )+\frac {x\,\left (x+4\right )}{\ln \left (\frac {16\,x^2+8\,x+1}{x^2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 34, normalized size = 1.17 \begin {gather*} 25 x^{5} + 100 x^{4} + 3 x + \frac {x^{2} + 4 x}{\log {\left (\frac {16 x^{2} + 8 x + 1}{x^{2}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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