Optimal. Leaf size=25 \[ 5 \left (3-x^2+\frac {x}{\log \left (\frac {x}{2}+x^2\right )}\right )^2 \]
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Rubi [F]
time = 0.53, 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 {-10 x-40 x^2+\left (-30-110 x+30 x^2+40 x^3\right ) \log \left (\frac {1}{2} \left (x+2 x^2\right )\right )+\left (30+60 x-30 x^2-60 x^3\right ) \log ^2\left (\frac {1}{2} \left (x+2 x^2\right )\right )+\left (-60 x-120 x^2+20 x^3+40 x^4\right ) \log ^3\left (\frac {1}{2} \left (x+2 x^2\right )\right )}{(1+2 x) \log ^3\left (\frac {1}{2} \left (x+2 x^2\right )\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 {-10 x-40 x^2+\left (-30-110 x+30 x^2+40 x^3\right ) \log \left (\frac {1}{2} \left (x+2 x^2\right )\right )+\left (30+60 x-30 x^2-60 x^3\right ) \log ^2\left (\frac {1}{2} \left (x+2 x^2\right )\right )+\left (-60 x-120 x^2+20 x^3+40 x^4\right ) \log ^3\left (\frac {1}{2} \left (x+2 x^2\right )\right )}{(1+2 x) \log ^3\left (\frac {1}{2} x (1+2 x)\right )} \, dx\\ &=\int \left (20 x \left (-3+x^2\right )+\frac {10 (-1-4 x) x}{(1+2 x) \log ^3\left (x \left (\frac {1}{2}+x\right )\right )}+\frac {10 \left (-3-11 x+3 x^2+4 x^3\right )}{(1+2 x) \log ^2\left (x \left (\frac {1}{2}+x\right )\right )}+\frac {30 \left (1-x^2\right )}{\log \left (x \left (\frac {1}{2}+x\right )\right )}\right ) \, dx\\ &=10 \int \frac {(-1-4 x) x}{(1+2 x) \log ^3\left (x \left (\frac {1}{2}+x\right )\right )} \, dx+10 \int \frac {-3-11 x+3 x^2+4 x^3}{(1+2 x) \log ^2\left (x \left (\frac {1}{2}+x\right )\right )} \, dx+20 \int x \left (-3+x^2\right ) \, dx+30 \int \frac {1-x^2}{\log \left (x \left (\frac {1}{2}+x\right )\right )} \, dx\\ &=10 \int \frac {(-1-4 x) x}{(1+2 x) \log ^3\left (x \left (\frac {1}{2}+x\right )\right )} \, dx+10 \int \frac {-3-11 x+3 x^2+4 x^3}{(1+2 x) \log ^2\left (x \left (\frac {1}{2}+x\right )\right )} \, dx+20 \int \left (-3 x+x^3\right ) \, dx+30 \int \frac {1-x^2}{\log \left (x \left (\frac {1}{2}+x\right )\right )} \, dx\\ &=-30 x^2+5 x^4+10 \int \frac {(-1-4 x) x}{(1+2 x) \log ^3\left (x \left (\frac {1}{2}+x\right )\right )} \, dx+10 \int \frac {-3-11 x+3 x^2+4 x^3}{(1+2 x) \log ^2\left (x \left (\frac {1}{2}+x\right )\right )} \, dx+30 \int \frac {1-x^2}{\log \left (x \left (\frac {1}{2}+x\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 43, normalized size = 1.72 \begin {gather*} 5 x \left (-6 x+x^3+\frac {x}{\log ^2\left (\frac {x}{2}+x^2\right )}-\frac {2 \left (-3+x^2\right )}{\log \left (\frac {x}{2}+x^2\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs.
\(2(23)=46\).
time = 1.26, size = 53, normalized size = 2.12
method | result | size |
risch | \(5 x^{4}-30 x^{2}+45-\frac {5 \left (2 \ln \left (x^{2}+\frac {1}{2} x \right ) x^{2}-x -6 \ln \left (x^{2}+\frac {1}{2} x \right )\right ) x}{\ln \left (x^{2}+\frac {1}{2} x \right )^{2}}\) | \(53\) |
norman | \(\frac {5 x^{2}+30 \ln \left (x^{2}+\frac {1}{2} x \right ) x -10 \ln \left (x^{2}+\frac {1}{2} x \right ) x^{3}-30 \ln \left (x^{2}+\frac {1}{2} x \right )^{2} x^{2}+5 \ln \left (x^{2}+\frac {1}{2} x \right )^{2} x^{4}}{\ln \left (x^{2}+\frac {1}{2} x \right )^{2}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs.
\(2 (22) = 44\).
time = 0.91, size = 173, normalized size = 6.92 \begin {gather*} \frac {5 \, {\left (x^{4} \log \left (2\right )^{2} + 2 \, x^{3} \log \left (2\right ) - {\left (6 \, \log \left (2\right )^{2} - 1\right )} x^{2} + {\left (x^{4} - 6 \, x^{2}\right )} \log \left (2 \, x + 1\right )^{2} + {\left (x^{4} - 6 \, x^{2}\right )} \log \left (x\right )^{2} - 6 \, x \log \left (2\right ) - 2 \, {\left (x^{4} \log \left (2\right ) + x^{3} - 6 \, x^{2} \log \left (2\right ) - {\left (x^{4} - 6 \, x^{2}\right )} \log \left (x\right ) - 3 \, x\right )} \log \left (2 \, x + 1\right ) - 2 \, {\left (x^{4} \log \left (2\right ) + x^{3} - 6 \, x^{2} \log \left (2\right ) - 3 \, x\right )} \log \left (x\right )\right )}}{\log \left (2\right )^{2} - 2 \, {\left (\log \left (2\right ) - \log \left (x\right )\right )} \log \left (2 \, x + 1\right ) + \log \left (2 \, x + 1\right )^{2} - 2 \, \log \left (2\right ) \log \left (x\right ) + \log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (22) = 44\).
time = 0.33, size = 53, normalized size = 2.12 \begin {gather*} \frac {5 \, {\left ({\left (x^{4} - 6 \, x^{2}\right )} \log \left (x^{2} + \frac {1}{2} \, x\right )^{2} + x^{2} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \left (x^{2} + \frac {1}{2} \, x\right )\right )}}{\log \left (x^{2} + \frac {1}{2} \, x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (17) = 34\).
time = 0.06, size = 41, normalized size = 1.64 \begin {gather*} 5 x^{4} - 30 x^{2} + \frac {5 x^{2} + \left (- 10 x^{3} + 30 x\right ) \log {\left (x^{2} + \frac {x}{2} \right )}}{\log {\left (x^{2} + \frac {x}{2} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (22) = 44\).
time = 0.57, size = 53, normalized size = 2.12 \begin {gather*} 5 \, x^{4} - 30 \, x^{2} - \frac {5 \, {\left (2 \, x^{3} \log \left (x^{2} + \frac {1}{2} \, x\right ) - x^{2} - 6 \, x \log \left (x^{2} + \frac {1}{2} \, x\right )\right )}}{\log \left (x^{2} + \frac {1}{2} \, x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.67, size = 48, normalized size = 1.92 \begin {gather*} \frac {5\,x^2-5\,x\,\ln \left (x^2+\frac {x}{2}\right )\,\left (2\,x^2-6\right )}{{\ln \left (x^2+\frac {x}{2}\right )}^2}-5\,x\,\left (6\,x-x^3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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