Optimal. Leaf size=22 \[ -5+\frac {16 (2+2 x)^2}{5 x \log ^2\left (x^2\right )} \]
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Rubi [A] time = 0.48, antiderivative size = 35, normalized size of antiderivative = 1.59, number of steps used = 25, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {12, 6741, 6742, 2353, 2297, 2300, 2178, 2306, 2310, 2302, 30} \begin {gather*} \frac {64 x}{5 \log ^2\left (x^2\right )}+\frac {128}{5 \log ^2\left (x^2\right )}+\frac {64}{5 x \log ^2\left (x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2178
Rule 2297
Rule 2300
Rule 2302
Rule 2306
Rule 2310
Rule 2353
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {-256-512 x-256 x^2+\left (-64+64 x^2\right ) \log \left (x^2\right )}{x^2 \log ^3\left (x^2\right )} \, dx\\ &=\frac {1}{5} \int \frac {64 (1+x) \left (-4-4 x-\log \left (x^2\right )+x \log \left (x^2\right )\right )}{x^2 \log ^3\left (x^2\right )} \, dx\\ &=\frac {64}{5} \int \frac {(1+x) \left (-4-4 x-\log \left (x^2\right )+x \log \left (x^2\right )\right )}{x^2 \log ^3\left (x^2\right )} \, dx\\ &=\frac {64}{5} \int \left (-\frac {4 (1+x)^2}{x^2 \log ^3\left (x^2\right )}+\frac {-1+x^2}{x^2 \log ^2\left (x^2\right )}\right ) \, dx\\ &=\frac {64}{5} \int \frac {-1+x^2}{x^2 \log ^2\left (x^2\right )} \, dx-\frac {256}{5} \int \frac {(1+x)^2}{x^2 \log ^3\left (x^2\right )} \, dx\\ &=\frac {64}{5} \int \left (\frac {1}{\log ^2\left (x^2\right )}-\frac {1}{x^2 \log ^2\left (x^2\right )}\right ) \, dx-\frac {256}{5} \int \left (\frac {1}{\log ^3\left (x^2\right )}+\frac {1}{x^2 \log ^3\left (x^2\right )}+\frac {2}{x \log ^3\left (x^2\right )}\right ) \, dx\\ &=\frac {64}{5} \int \frac {1}{\log ^2\left (x^2\right )} \, dx-\frac {64}{5} \int \frac {1}{x^2 \log ^2\left (x^2\right )} \, dx-\frac {256}{5} \int \frac {1}{\log ^3\left (x^2\right )} \, dx-\frac {256}{5} \int \frac {1}{x^2 \log ^3\left (x^2\right )} \, dx-\frac {512}{5} \int \frac {1}{x \log ^3\left (x^2\right )} \, dx\\ &=\frac {64}{5 x \log ^2\left (x^2\right )}+\frac {64 x}{5 \log ^2\left (x^2\right )}+\frac {32}{5 x \log \left (x^2\right )}-\frac {32 x}{5 \log \left (x^2\right )}+\frac {32}{5} \int \frac {1}{\log \left (x^2\right )} \, dx+\frac {32}{5} \int \frac {1}{x^2 \log \left (x^2\right )} \, dx-\frac {64}{5} \int \frac {1}{\log ^2\left (x^2\right )} \, dx+\frac {64}{5} \int \frac {1}{x^2 \log ^2\left (x^2\right )} \, dx-\frac {256}{5} \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log \left (x^2\right )\right )\\ &=\frac {128}{5 \log ^2\left (x^2\right )}+\frac {64}{5 x \log ^2\left (x^2\right )}+\frac {64 x}{5 \log ^2\left (x^2\right )}-\frac {32}{5} \int \frac {1}{\log \left (x^2\right )} \, dx-\frac {32}{5} \int \frac {1}{x^2 \log \left (x^2\right )} \, dx+\frac {(16 x) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{5 \sqrt {x^2}}+\frac {\left (16 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{5 x}\\ &=\frac {16 \sqrt {x^2} \text {Ei}\left (-\frac {1}{2} \log \left (x^2\right )\right )}{5 x}+\frac {16 x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{5 \sqrt {x^2}}+\frac {128}{5 \log ^2\left (x^2\right )}+\frac {64}{5 x \log ^2\left (x^2\right )}+\frac {64 x}{5 \log ^2\left (x^2\right )}-\frac {(16 x) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{5 \sqrt {x^2}}-\frac {\left (16 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{5 x}\\ &=\frac {128}{5 \log ^2\left (x^2\right )}+\frac {64}{5 x \log ^2\left (x^2\right )}+\frac {64 x}{5 \log ^2\left (x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 18, normalized size = 0.82 \begin {gather*} \frac {64 (1+x)^2}{5 x \log ^2\left (x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 19, normalized size = 0.86 \begin {gather*} \frac {64 \, {\left (x^{2} + 2 \, x + 1\right )}}{5 \, x \log \left (x^{2}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 19, normalized size = 0.86 \begin {gather*} \frac {64 \, {\left (x^{2} + 2 \, x + 1\right )}}{5 \, x \log \left (x^{2}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 20, normalized size = 0.91
method | result | size |
risch | \(\frac {\frac {64}{5}+\frac {128}{5} x +\frac {64}{5} x^{2}}{x \ln \left (x^{2}\right )^{2}}\) | \(20\) |
norman | \(\frac {\frac {64}{5}+\frac {128}{5} x +\frac {64}{5} x^{2}}{x \ln \left (x^{2}\right )^{2}}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 17, normalized size = 0.77 \begin {gather*} \frac {16 \, {\left (x^{2} + 2 \, x + 1\right )}}{5 \, x \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 21, normalized size = 0.95 \begin {gather*} \frac {64\,x^2+128\,x+64}{5\,x\,{\ln \left (x^2\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 19, normalized size = 0.86 \begin {gather*} \frac {64 x^{2} + 128 x + 64}{5 x \log {\left (x^{2} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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