Optimal. Leaf size=23 \[ \frac {1}{\left (-x+x \left (1+\log (1+x)+\log \left (\frac {x^2}{\log ^2(x)}\right )\right )\right )^2} \]
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Rubi [A]
time = 0.38, antiderivative size = 20, normalized size of antiderivative = 0.87, number of steps
used = 2, number of rules used = 2, integrand size = 141, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6820, 6819}
\begin {gather*} \frac {1}{x^2 \left (\log \left (\frac {x^2}{\log ^2(x)}\right )+\log (x+1)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 6819
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 (1+x)-2 \log (x) \left (2+3 x+(1+x) \log (1+x)+(1+x) \log \left (\frac {x^2}{\log ^2(x)}\right )\right )}{x^3 (1+x) \log (x) \left (\log (1+x)+\log \left (\frac {x^2}{\log ^2(x)}\right )\right )^3} \, dx\\ &=\frac {1}{x^2 \left (\log (1+x)+\log \left (\frac {x^2}{\log ^2(x)}\right )\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.12, size = 20, normalized size = 0.87 \begin {gather*} \frac {1}{x^2 \left (\log (1+x)+\log \left (\frac {x^2}{\log ^2(x)}\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 11.51, size = 212, normalized size = 9.22
method | result | size |
risch | \(-\frac {4}{x^{2} \left (-4 i \ln \left (x \right )+\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right )^{2}+\pi \mathrm {csgn}\left (i \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right )+\pi \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-\pi \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right )^{3}+4 i \ln \left (\ln \left (x \right )\right )-2 i \ln \left (x +1\right )\right )^{2}}\) | \(212\) |
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. 63 vs.
\(2 (23) = 46\).
time = 0.37, size = 63, normalized size = 2.74 \begin {gather*} \frac {1}{x^{2} \log \left (x + 1\right )^{2} + 4 \, x^{2} \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 4 \, x^{2} \log \left (\log \left (x\right )\right )^{2} + 4 \, {\left (x^{2} \log \left (x\right ) - x^{2} \log \left (\log \left (x\right )\right )\right )} \log \left (x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 46, normalized size = 2.00 \begin {gather*} \frac {1}{x^{2} \log \left (x + 1\right )^{2} + 2 \, x^{2} \log \left (x + 1\right ) \log \left (\frac {x^{2}}{\log \left (x\right )^{2}}\right ) + x^{2} \log \left (\frac {x^{2}}{\log \left (x\right )^{2}}\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. 46 vs.
\(2 (22) = 44\).
time = 0.15, size = 46, normalized size = 2.00 \begin {gather*} \frac {1}{x^{2} \log {\left (\frac {x^{2}}{\log {\left (x \right )}^{2}} \right )}^{2} + 2 x^{2} \log {\left (\frac {x^{2}}{\log {\left (x \right )}^{2}} \right )} \log {\left (x + 1 \right )} + x^{2} \log {\left (x + 1 \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. 315 vs.
\(2 (23) = 46\).
time = 1.31, size = 315, normalized size = 13.70 \begin {gather*} -\frac {3 \, x \log \left (x\right ) - 2 \, x + 2 \, \log \left (x\right ) - 2}{12 \, x^{3} \log \left (x\right )^{2} \log \left (\log \left (x\right )^{2}\right ) - 3 \, x^{3} \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right )^{2} + 6 \, x^{3} \log \left (\log \left (x\right )^{2}\right ) \log \left (x + 1\right ) \log \left (x\right ) - 3 \, x^{3} \log \left (x + 1\right )^{2} \log \left (x\right ) - 12 \, x^{3} \log \left (x + 1\right ) \log \left (x\right )^{2} - 12 \, x^{3} \log \left (x\right )^{3} - 8 \, x^{3} \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right ) + 8 \, x^{2} \log \left (x\right )^{2} \log \left (\log \left (x\right )^{2}\right ) + 2 \, x^{3} \log \left (\log \left (x\right )^{2}\right )^{2} - 2 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right )^{2} - 4 \, x^{3} \log \left (\log \left (x\right )^{2}\right ) \log \left (x + 1\right ) + 2 \, x^{3} \log \left (x + 1\right )^{2} + 8 \, x^{3} \log \left (x + 1\right ) \log \left (x\right ) + 4 \, x^{2} \log \left (\log \left (x\right )^{2}\right ) \log \left (x + 1\right ) \log \left (x\right ) - 2 \, x^{2} \log \left (x + 1\right )^{2} \log \left (x\right ) + 8 \, x^{3} \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x + 1\right ) \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x\right )^{3} - 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right ) + 2 \, x^{2} \log \left (\log \left (x\right )^{2}\right )^{2} - 4 \, x^{2} \log \left (\log \left (x\right )^{2}\right ) \log \left (x + 1\right ) + 2 \, x^{2} \log \left (x + 1\right )^{2} + 8 \, x^{2} \log \left (x + 1\right ) \log \left (x\right ) + 8 \, x^{2} \log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \left (x\right )\,\left (6\,x+4\right )-4\,x+\ln \left (x\right )\,\ln \left (\frac {x^2}{{\ln \left (x\right )}^2}\right )\,\left (2\,x+2\right )+\ln \left (x+1\right )\,\ln \left (x\right )\,\left (2\,x+2\right )-4}{\ln \left (x\right )\,\left (x^4+x^3\right )\,{\ln \left (x+1\right )}^3+\ln \left (x\right )\,\left (3\,x^4+3\,x^3\right )\,{\ln \left (x+1\right )}^2\,\ln \left (\frac {x^2}{{\ln \left (x\right )}^2}\right )+\ln \left (x\right )\,\left (3\,x^4+3\,x^3\right )\,\ln \left (x+1\right )\,{\ln \left (\frac {x^2}{{\ln \left (x\right )}^2}\right )}^2+\ln \left (x\right )\,\left (x^4+x^3\right )\,{\ln \left (\frac {x^2}{{\ln \left (x\right )}^2}\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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