Optimal. Leaf size=20 \[ (1-3 x) \log \left (\frac {3}{5 x^3 \log (4+x)}\right ) \]
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Rubi [A] time = 0.42, antiderivative size = 29, normalized size of antiderivative = 1.45, number of steps used = 17, number of rules used = 8, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1593, 6688, 2411, 2353, 2298, 2302, 29, 2549} \begin {gather*} -3 x \log \left (\frac {3}{5 x^3 \log (x+4)}\right )-3 \log (x)-\log (\log (x+4)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 1593
Rule 2298
Rule 2302
Rule 2353
Rule 2411
Rule 2549
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+3 x^2+\left (-12+33 x+9 x^2\right ) \log (4+x)+\left (-12 x-3 x^2\right ) \log (4+x) \log \left (\frac {3}{5 x^3 \log (4+x)}\right )}{x (4+x) \log (4+x)} \, dx\\ &=\int \left (9-\frac {3}{x}+\frac {-1+3 x}{(4+x) \log (4+x)}-3 \log \left (\frac {3}{5 x^3 \log (4+x)}\right )\right ) \, dx\\ &=9 x-3 \log (x)-3 \int \log \left (\frac {3}{5 x^3 \log (4+x)}\right ) \, dx+\int \frac {-1+3 x}{(4+x) \log (4+x)} \, dx\\ &=9 x-3 \log (x)-3 x \log \left (\frac {3}{5 x^3 \log (4+x)}\right )+3 \int \left (-3-\frac {x}{(4+x) \log (4+x)}\right ) \, dx+\operatorname {Subst}\left (\int \frac {-13+3 x}{x \log (x)} \, dx,x,4+x\right )\\ &=-3 \log (x)-3 x \log \left (\frac {3}{5 x^3 \log (4+x)}\right )-3 \int \frac {x}{(4+x) \log (4+x)} \, dx+\operatorname {Subst}\left (\int \left (\frac {3}{\log (x)}-\frac {13}{x \log (x)}\right ) \, dx,x,4+x\right )\\ &=-3 \log (x)-3 x \log \left (\frac {3}{5 x^3 \log (4+x)}\right )+3 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,4+x\right )-3 \operatorname {Subst}\left (\int \frac {-4+x}{x \log (x)} \, dx,x,4+x\right )-13 \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,4+x\right )\\ &=-3 \log (x)-3 x \log \left (\frac {3}{5 x^3 \log (4+x)}\right )+3 \text {li}(4+x)-3 \operatorname {Subst}\left (\int \left (\frac {1}{\log (x)}-\frac {4}{x \log (x)}\right ) \, dx,x,4+x\right )-13 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (4+x)\right )\\ &=-3 \log (x)-3 x \log \left (\frac {3}{5 x^3 \log (4+x)}\right )-13 \log (\log (4+x))+3 \text {li}(4+x)-3 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,4+x\right )+12 \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,4+x\right )\\ &=-3 \log (x)-3 x \log \left (\frac {3}{5 x^3 \log (4+x)}\right )-13 \log (\log (4+x))+12 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (4+x)\right )\\ &=-3 \log (x)-3 x \log \left (\frac {3}{5 x^3 \log (4+x)}\right )-\log (\log (4+x))\\ \end {aligned} \end {gather*}
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Mathematica [C] time = 0.06, size = 42, normalized size = 2.10 \begin {gather*} -3 \text {Ei}(\log (4+x))-3 \log (x)-3 x \log \left (\frac {3}{5 x^3 \log (4+x)}\right )-\log (\log (4+x))+3 \text {li}(4+x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 19, normalized size = 0.95 \begin {gather*} -{\left (3 \, x - 1\right )} \log \left (\frac {3}{5 \, x^{3} \log \left (x + 4\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 30, normalized size = 1.50 \begin {gather*} -3 \, x \log \relax (3) + 3 \, x \log \left (5 \, x^{3} \log \left (x + 4\right )\right ) - 3 \, \log \relax (x) - \log \left (\log \left (x + 4\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 28, normalized size = 1.40
method | result | size |
norman | \(-3 x \ln \left (\frac {3}{5 x^{3} \ln \left (4+x \right )}\right )-3 \ln \relax (x )-\ln \left (\ln \left (4+x \right )\right )\) | \(28\) |
default | \(-\ln \left (\ln \left (4+x \right )\right )-3 \ln \relax (x )-3 x \ln \relax (3)+3 x \ln \relax (5)-3 \ln \left (\frac {1}{x^{3} \ln \left (4+x \right )}\right ) x -36\) | \(38\) |
risch | \(3 x \ln \left (\ln \left (4+x \right )\right )+9 x \ln \relax (x )-\frac {3 i \pi x \,\mathrm {csgn}\left (\frac {i}{x^{3}}\right ) \mathrm {csgn}\left (\frac {i}{x^{3} \ln \left (4+x \right )}\right )^{2}}{2}-\frac {3 i \pi x \,\mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i}{x^{3} \ln \left (4+x \right )}\right )^{2}}{2}-\frac {3 i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{2}-\frac {3 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )}{2}+\frac {3 i \pi x \,\mathrm {csgn}\left (\frac {i}{x^{3}}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i}{x^{3} \ln \left (4+x \right )}\right )}{2}+\frac {3 i \pi x \mathrm {csgn}\left (\frac {i}{x^{3} \ln \left (4+x \right )}\right )^{3}}{2}-\frac {3 i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}+3 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\frac {3 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}}{2}-\frac {3 i \pi x \mathrm {csgn}\left (i x^{3}\right )^{3}}{2}+\frac {3 i \pi x \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}}{2}+3 x \ln \relax (5)-3 x \ln \relax (3)-3 \ln \relax (x )-\ln \left (\ln \left (4+x \right )\right )\) | \(278\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 35, normalized size = 1.75 \begin {gather*} 3 \, x {\left (\log \relax (5) - \log \relax (3)\right )} + 3 \, {\left (3 \, x - 1\right )} \log \relax (x) + 3 \, x \log \left (\log \left (x + 4\right )\right ) - \log \left (\log \left (x + 4\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 27, normalized size = 1.35 \begin {gather*} -\ln \left (\ln \left (x+4\right )\right )-3\,\ln \relax (x)-3\,x\,\ln \left (\frac {3}{5\,x^3\,\ln \left (x+4\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 32, normalized size = 1.60 \begin {gather*} \left (- 3 x - 2\right ) \log {\left (\frac {3}{5 x^{3} \log {\left (x + 4 \right )}} \right )} - 9 \log {\relax (x )} - 3 \log {\left (\log {\left (x + 4 \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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