Optimal. Leaf size=19 \[ 2+x+\frac {x \left (-2+x^4\right )}{25 \log \left (x^2\right )} \]
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Rubi [A] time = 0.19, antiderivative size = 26, normalized size of antiderivative = 1.37, number of steps used = 17, number of rules used = 8, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {12, 6742, 2330, 2297, 2300, 2178, 2306, 2310} \begin {gather*} -\frac {2 x}{25 \log \left (x^2\right )}+\frac {x^5}{25 \log \left (x^2\right )}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2297
Rule 2300
Rule 2306
Rule 2310
Rule 2330
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int \frac {4-2 x^4+\left (-2+5 x^4\right ) \log \left (x^2\right )+25 \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )} \, dx\\ &=\frac {1}{25} \int \left (25-\frac {2 \left (-2+x^4\right )}{\log ^2\left (x^2\right )}+\frac {-2+5 x^4}{\log \left (x^2\right )}\right ) \, dx\\ &=x+\frac {1}{25} \int \frac {-2+5 x^4}{\log \left (x^2\right )} \, dx-\frac {2}{25} \int \frac {-2+x^4}{\log ^2\left (x^2\right )} \, dx\\ &=x+\frac {1}{25} \int \left (-\frac {2}{\log \left (x^2\right )}+\frac {5 x^4}{\log \left (x^2\right )}\right ) \, dx-\frac {2}{25} \int \left (-\frac {2}{\log ^2\left (x^2\right )}+\frac {x^4}{\log ^2\left (x^2\right )}\right ) \, dx\\ &=x-\frac {2}{25} \int \frac {x^4}{\log ^2\left (x^2\right )} \, dx-\frac {2}{25} \int \frac {1}{\log \left (x^2\right )} \, dx+\frac {4}{25} \int \frac {1}{\log ^2\left (x^2\right )} \, dx+\frac {1}{5} \int \frac {x^4}{\log \left (x^2\right )} \, dx\\ &=x-\frac {2 x}{25 \log \left (x^2\right )}+\frac {x^5}{25 \log \left (x^2\right )}+\frac {2}{25} \int \frac {1}{\log \left (x^2\right )} \, dx-\frac {1}{5} \int \frac {x^4}{\log \left (x^2\right )} \, dx+\frac {x^5 \operatorname {Subst}\left (\int \frac {e^{5 x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{10 \left (x^2\right )^{5/2}}-\frac {x \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{25 \sqrt {x^2}}\\ &=x-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{25 \sqrt {x^2}}+\frac {x^5 \text {Ei}\left (\frac {5 \log \left (x^2\right )}{2}\right )}{10 \left (x^2\right )^{5/2}}-\frac {2 x}{25 \log \left (x^2\right )}+\frac {x^5}{25 \log \left (x^2\right )}-\frac {x^5 \operatorname {Subst}\left (\int \frac {e^{5 x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{10 \left (x^2\right )^{5/2}}+\frac {x \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{25 \sqrt {x^2}}\\ &=x-\frac {2 x}{25 \log \left (x^2\right )}+\frac {x^5}{25 \log \left (x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 18, normalized size = 0.95 \begin {gather*} x+\frac {x \left (-2+x^4\right )}{25 \log \left (x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 22, normalized size = 1.16 \begin {gather*} \frac {x^{5} + 25 \, x \log \left (x^{2}\right ) - 2 \, x}{25 \, \log \left (x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 17, normalized size = 0.89 \begin {gather*} x + \frac {x^{5} - 2 \, x}{25 \, \log \left (x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 17, normalized size = 0.89
method | result | size |
risch | \(x +\frac {x \left (x^{4}-2\right )}{25 \ln \left (x^{2}\right )}\) | \(17\) |
norman | \(\frac {x \ln \left (x^{2}\right )-\frac {2 x}{25}+\frac {x^{5}}{25}}{\ln \left (x^{2}\right )}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 15, normalized size = 0.79 \begin {gather*} x + \frac {x^{5} - 2 \, x}{50 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.75, size = 16, normalized size = 0.84 \begin {gather*} x+\frac {x\,\left (x^4-2\right )}{25\,\ln \left (x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 14, normalized size = 0.74 \begin {gather*} x + \frac {x^{5} - 2 x}{25 \log {\left (x^{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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