Optimal. Leaf size=20 \[ -2+x+x^2-\frac {2 (x+\log (3+4 x))}{\log ^2(x)} \]
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Rubi [F] time = 0.39, 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 {12 x+16 x^2+\left (-14 x-8 x^2\right ) \log (x)+\left (3 x+10 x^2+8 x^3\right ) \log ^3(x)+(12+16 x) \log (3+4 x)}{\left (3 x+4 x^2\right ) \log ^3(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12 x+16 x^2+\left (-14 x-8 x^2\right ) \log (x)+\left (3 x+10 x^2+8 x^3\right ) \log ^3(x)+(12+16 x) \log (3+4 x)}{x (3+4 x) \log ^3(x)} \, dx\\ &=\int \left (1+2 x-\frac {2 (7+4 x)}{(3+4 x) \log ^2(x)}+\frac {4 (x+\log (3+4 x))}{x \log ^3(x)}\right ) \, dx\\ &=x+x^2-2 \int \frac {7+4 x}{(3+4 x) \log ^2(x)} \, dx+4 \int \frac {x+\log (3+4 x)}{x \log ^3(x)} \, dx\\ &=x+x^2-2 \int \frac {7+4 x}{(3+4 x) \log ^2(x)} \, dx+4 \int \left (\frac {1}{\log ^3(x)}+\frac {\log (3+4 x)}{x \log ^3(x)}\right ) \, dx\\ &=x+x^2-2 \int \frac {7+4 x}{(3+4 x) \log ^2(x)} \, dx+4 \int \frac {1}{\log ^3(x)} \, dx+4 \int \frac {\log (3+4 x)}{x \log ^3(x)} \, dx\\ &=x+x^2-\frac {2 x}{\log ^2(x)}+2 \int \frac {1}{\log ^2(x)} \, dx-2 \int \frac {7+4 x}{(3+4 x) \log ^2(x)} \, dx+4 \int \frac {\log (3+4 x)}{x \log ^3(x)} \, dx\\ &=x+x^2-\frac {2 x}{\log ^2(x)}-\frac {2 x}{\log (x)}-2 \int \frac {7+4 x}{(3+4 x) \log ^2(x)} \, dx+2 \int \frac {1}{\log (x)} \, dx+4 \int \frac {\log (3+4 x)}{x \log ^3(x)} \, dx\\ &=x+x^2-\frac {2 x}{\log ^2(x)}-\frac {2 x}{\log (x)}+2 \text {li}(x)-2 \int \frac {7+4 x}{(3+4 x) \log ^2(x)} \, dx+4 \int \frac {\log (3+4 x)}{x \log ^3(x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.65, size = 24, normalized size = 1.20 \begin {gather*} x+x^2-\frac {2 x}{\log ^2(x)}-\frac {2 \log (3+4 x)}{\log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 27, normalized size = 1.35 \begin {gather*} \frac {{\left (x^{2} + x\right )} \log \relax (x)^{2} - 2 \, x - 2 \, \log \left (4 \, x + 3\right )}{\log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 24, normalized size = 1.20 \begin {gather*} x^{2} + x - \frac {2 \, x}{\log \relax (x)^{2}} - \frac {2 \, \log \left (4 \, x + 3\right )}{\log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 32, normalized size = 1.60
method | result | size |
risch | \(-\frac {2 \ln \left (3+4 x \right )}{\ln \relax (x )^{2}}+\frac {x \left (x \ln \relax (x )^{2}+\ln \relax (x )^{2}-2\right )}{\ln \relax (x )^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 27, normalized size = 1.35 \begin {gather*} \frac {{\left (x^{2} + x\right )} \log \relax (x)^{2} - 2 \, x - 2 \, \log \left (4 \, x + 3\right )}{\log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.38, size = 24, normalized size = 1.20 \begin {gather*} x-\frac {2\,x}{{\ln \relax (x)}^2}-\frac {2\,\ln \left (4\,x+3\right )}{{\ln \relax (x)}^2}+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 26, normalized size = 1.30 \begin {gather*} x^{2} + x - \frac {2 x}{\log {\relax (x )}^{2}} - \frac {2 \log {\left (4 x + 3 \right )}}{\log {\relax (x )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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