Optimal. Leaf size=22 \[ \frac {(-x-x \log (2)+\log (3))^2}{4+x+x^2} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(22)=44\).
time = 0.17, antiderivative size = 54, normalized size of antiderivative = 2.45, number of steps
used = 4, number of rules used = 4, integrand size = 82, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {1694, 12, 1828,
8} \begin {gather*} -\frac {2 \left (2 \left (x+\frac {1}{2}\right ) (1+\log (2)) (1+\log (18))+7-2 \log ^2(3)+7 \log ^2(2)+\log \left (\frac {16384}{9}\right )-\log (3) \log (4)\right )}{4 \left (x+\frac {1}{2}\right )^2+15} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 1694
Rule 1828
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\text {Subst}\left (\int \frac {4 \left (-15 (1+\log (2)) (1+\log (18))+4 x^2 (1+\log (2)) (1+\log (18))+4 x \left (7+7 \log ^2(2)-2 \log ^2(3)-\log (3) \log (4)+\log \left (\frac {16384}{9}\right )\right )\right )}{\left (15+4 x^2\right )^2} \, dx,x,\frac {1}{2}+x\right )\\ &=4 \text {Subst}\left (\int \frac {-15 (1+\log (2)) (1+\log (18))+4 x^2 (1+\log (2)) (1+\log (18))+4 x \left (7+7 \log ^2(2)-2 \log ^2(3)-\log (3) \log (4)+\log \left (\frac {16384}{9}\right )\right )}{\left (15+4 x^2\right )^2} \, dx,x,\frac {1}{2}+x\right )\\ &=-\frac {2 \left (7+7 \log ^2(2)-2 \log ^2(3)-\log (3) \log (4)+(1+2 x) (1+\log (2)) (1+\log (18))+\log \left (\frac {16384}{9}\right )\right )}{15+(1+2 x)^2}-\frac {2}{15} \text {Subst}\left (\int 0 \, dx,x,\frac {1}{2}+x\right )\\ &=-\frac {2 \left (7+7 \log ^2(2)-2 \log ^2(3)-\log (3) \log (4)+(1+2 x) (1+\log (2)) (1+\log (18))+\log \left (\frac {16384}{9}\right )\right )}{15+(1+2 x)^2}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(22)=44\).
time = 0.05, size = 85, normalized size = 3.86 \begin {gather*} -\frac {60+60 \log ^2(2)+8 \log (3)-16 \log ^2(3)-4 \log (36)-\log (4) \log (81)+16 \log (256)+\log (3) \log (768)+x \left (15+15 \log ^2(2)-2 \log ^2(3)+\log (3) (16+7 \log (4))+7 \log (36)+\log (9) \log (768)+\log (65536)\right )}{15 \left (4+x+x^2\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. \(50\) vs.
\(2(22)=44\).
time = 0.17, size = 51, normalized size = 2.32
method | result | size |
norman | \(\frac {\left (-2 \ln \left (2\right ) \ln \left (3\right )-\ln \left (2\right )^{2}-2 \ln \left (3\right )-2 \ln \left (2\right )-1\right ) x -4-8 \ln \left (2\right )-4 \ln \left (2\right )^{2}+\ln \left (3\right )^{2}}{x^{2}+x +4}\) | \(50\) |
risch | \(\frac {\left (-2 \ln \left (2\right ) \ln \left (3\right )-\ln \left (2\right )^{2}-2 \ln \left (3\right )-2 \ln \left (2\right )-1\right ) x -4-8 \ln \left (2\right )-4 \ln \left (2\right )^{2}+\ln \left (3\right )^{2}}{x^{2}+x +4}\) | \(50\) |
default | \(-\frac {4+\left (2 \ln \left (2\right ) \ln \left (3\right )+\ln \left (2\right )^{2}+2 \ln \left (3\right )+2 \ln \left (2\right )+1\right ) x -\ln \left (3\right )^{2}+4 \ln \left (2\right )^{2}+8 \ln \left (2\right )}{x^{2}+x +4}\) | \(51\) |
gosper | \(\frac {-2 x \ln \left (2\right ) \ln \left (3\right )-x \ln \left (2\right )^{2}+\ln \left (3\right )^{2}-2 x \ln \left (3\right )-4 \ln \left (2\right )^{2}-2 x \ln \left (2\right )-8 \ln \left (2\right )-x -4}{x^{2}+x +4}\) | \(53\) |
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. 48 vs.
\(2 (21) = 42\).
time = 0.28, size = 48, normalized size = 2.18 \begin {gather*} -\frac {{\left (2 \, {\left (\log \left (2\right ) + 1\right )} \log \left (3\right ) + \log \left (2\right )^{2} + 2 \, \log \left (2\right ) + 1\right )} x - \log \left (3\right )^{2} + 4 \, \log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 4}{x^{2} + x + 4} \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. 44 vs.
\(2 (21) = 42\).
time = 0.41, size = 44, normalized size = 2.00 \begin {gather*} -\frac {{\left (x + 4\right )} \log \left (2\right )^{2} + 2 \, {\left (x \log \left (2\right ) + x\right )} \log \left (3\right ) - \log \left (3\right )^{2} + 2 \, {\left (x + 4\right )} \log \left (2\right ) + x + 4}{x^{2} + x + 4} \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. 53 vs.
\(2 (17) = 34\).
time = 1.86, size = 53, normalized size = 2.41 \begin {gather*} \frac {x \left (- 2 \log {\left (3 \right )} - 2 \log {\left (2 \right )} \log {\left (3 \right )} - 2 \log {\left (2 \right )} - 1 - \log {\left (2 \right )}^{2}\right ) - 8 \log {\left (2 \right )} - 4 - 4 \log {\left (2 \right )}^{2} + \log {\left (3 \right )}^{2}}{x^{2} + x + 4} \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. 52 vs.
\(2 (21) = 42\).
time = 0.43, size = 52, normalized size = 2.36 \begin {gather*} -\frac {2 \, x \log \left (3\right ) \log \left (2\right ) + x \log \left (2\right )^{2} + 2 \, x \log \left (3\right ) - \log \left (3\right )^{2} + 2 \, x \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + x + 8 \, \log \left (2\right ) + 4}{x^{2} + x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 68, normalized size = 3.09 \begin {gather*} -\frac {\ln \left (256\right )+\ln \left (3\right )\,\ln \left (2^{8/15}\right )+4\,{\ln \left (2\right )}^2-{\ln \left (3\right )}^2+\ln \left (\frac {{177147}^{1/15}}{3}\right )\,\ln \left (4\right )+x\,\left (\ln \left (36\right )+\ln \left (4\right )\,\ln \left (3^{7/15}\right )+{\ln \left (2\right )}^2+\ln \left (2\,2^{1/15}\right )\,\ln \left (3\right )+1\right )+4}{x^2+x+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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