3.62.72 \(\int \frac {48-26 x+4 x^2+(16-11 x+2 x^2) \log (x)+(12-4 x+(4-2 x) \log (x)) \log (\frac {1}{4} (4 x^2+4 x^2 \log (x)+x^2 \log ^2(x)))}{2 x+x \log (x)} \, dx\)

Optimal. Leaf size=25 \[ 5+x+\left (4-x+\log \left (x^2 \left (-1-\frac {\log (x)}{2}\right )^2\right )\right )^2 \]

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Rubi [F]  time = 1.16, 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 {48-26 x+4 x^2+\left (16-11 x+2 x^2\right ) \log (x)+(12-4 x+(4-2 x) \log (x)) \log \left (\frac {1}{4} \left (4 x^2+4 x^2 \log (x)+x^2 \log ^2(x)\right )\right )}{2 x+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(48 - 26*x + 4*x^2 + (16 - 11*x + 2*x^2)*Log[x] + (12 - 4*x + (4 - 2*x)*Log[x])*Log[(4*x^2 + 4*x^2*Log[x]
+ x^2*Log[x]^2)/4])/(2*x + x*Log[x]),x]

[Out]

-11*x + x^2 - (4*ExpIntegralEi[2 + Log[x]])/E^2 + 16*Log[x] + 16*Log[2 + Log[x]] - 4*Defer[Int][Log[(x^2*(2 +
Log[x])^2)/4]/(2 + Log[x]), x] + 12*Defer[Int][Log[(x^2*(2 + Log[x])^2)/4]/(x*(2 + Log[x])), x] - 2*Defer[Int]
[(Log[x]*Log[(x^2*(2 + Log[x])^2)/4])/(2 + Log[x]), x] + 4*Defer[Int][(Log[x]*Log[(x^2*(2 + Log[x])^2)/4])/(x*
(2 + Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {48-26 x+4 x^2+\left (16-11 x+2 x^2\right ) \log (x)+(12-4 x+(4-2 x) \log (x)) \log \left (\frac {1}{4} \left (4 x^2+4 x^2 \log (x)+x^2 \log ^2(x)\right )\right )}{x (2+\log (x))} \, dx\\ &=\int \left (\frac {48-26 x+4 x^2+16 \log (x)-11 x \log (x)+2 x^2 \log (x)}{x (2+\log (x))}-\frac {2 (-6+2 x-2 \log (x)+x \log (x)) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))}\right ) \, dx\\ &=-\left (2 \int \frac {(-6+2 x-2 \log (x)+x \log (x)) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx\right )+\int \frac {48-26 x+4 x^2+16 \log (x)-11 x \log (x)+2 x^2 \log (x)}{x (2+\log (x))} \, dx\\ &=-\left (2 \int \left (\frac {2 \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)}-\frac {6 \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))}+\frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)}-\frac {2 \log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))}\right ) \, dx\right )+\int \left (\frac {16-11 x+2 x^2}{x}-\frac {4 (-4+x)}{x (2+\log (x))}\right ) \, dx\\ &=-\left (2 \int \frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)} \, dx\right )-4 \int \frac {-4+x}{x (2+\log (x))} \, dx-4 \int \frac {\log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx+12 \int \frac {\log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx+\int \frac {16-11 x+2 x^2}{x} \, dx\\ &=-\left (2 \int \frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)} \, dx\right )-4 \int \left (\frac {1}{2+\log (x)}-\frac {4}{x (2+\log (x))}\right ) \, dx-4 \int \frac {\log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx+12 \int \frac {\log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx+\int \left (-11+\frac {16}{x}+2 x\right ) \, dx\\ &=-11 x+x^2+16 \log (x)-2 \int \frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)} \, dx-4 \int \frac {1}{2+\log (x)} \, dx-4 \int \frac {\log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx+12 \int \frac {\log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx+16 \int \frac {1}{x (2+\log (x))} \, dx\\ &=-11 x+x^2+16 \log (x)-2 \int \frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)} \, dx-4 \int \frac {\log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx-4 \operatorname {Subst}\left (\int \frac {e^x}{2+x} \, dx,x,\log (x)\right )+12 \int \frac {\log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx+16 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,2+\log (x)\right )\\ &=-11 x+x^2-\frac {4 \text {Ei}(2+\log (x))}{e^2}+16 \log (x)+16 \log (2+\log (x))-2 \int \frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)} \, dx-4 \int \frac {\log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{2+\log (x)} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx+12 \int \frac {\log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )}{x (2+\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.15, size = 51, normalized size = 2.04 \begin {gather*} -7 x+x^2+16 \log (x)+16 \log (2+\log (x))-2 x \log \left (\frac {1}{4} x^2 (2+\log (x))^2\right )+\log ^2\left (\frac {1}{4} x^2 (2+\log (x))^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(48 - 26*x + 4*x^2 + (16 - 11*x + 2*x^2)*Log[x] + (12 - 4*x + (4 - 2*x)*Log[x])*Log[(4*x^2 + 4*x^2*L
og[x] + x^2*Log[x]^2)/4])/(2*x + x*Log[x]),x]

[Out]

-7*x + x^2 + 16*Log[x] + 16*Log[2 + Log[x]] - 2*x*Log[(x^2*(2 + Log[x])^2)/4] + Log[(x^2*(2 + Log[x])^2)/4]^2

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fricas [B]  time = 0.58, size = 54, normalized size = 2.16 \begin {gather*} x^{2} - 2 \, {\left (x - 4\right )} \log \left (\frac {1}{4} \, x^{2} \log \relax (x)^{2} + x^{2} \log \relax (x) + x^{2}\right ) + \log \left (\frac {1}{4} \, x^{2} \log \relax (x)^{2} + x^{2} \log \relax (x) + x^{2}\right )^{2} - 7 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4-2*x)*log(x)-4*x+12)*log(1/4*x^2*log(x)^2+x^2*log(x)+x^2)+(2*x^2-11*x+16)*log(x)+4*x^2-26*x+48)/
(x*log(x)+2*x),x, algorithm="fricas")

[Out]

x^2 - 2*(x - 4)*log(1/4*x^2*log(x)^2 + x^2*log(x) + x^2) + log(1/4*x^2*log(x)^2 + x^2*log(x) + x^2)^2 - 7*x

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giac [B]  time = 0.48, size = 74, normalized size = 2.96 \begin {gather*} x^{2} + x {\left (4 \, \log \relax (2) - 7\right )} - 2 \, {\left (x - 2 \, \log \relax (x)\right )} \log \left (\log \relax (x)^{2} + 4 \, \log \relax (x) + 4\right ) + \log \left (\log \relax (x)^{2} + 4 \, \log \relax (x) + 4\right )^{2} - 4 \, x \log \relax (x) - 8 \, {\left (\log \relax (2) - 2\right )} \log \relax (x) + 4 \, \log \relax (x)^{2} - 8 \, {\left (\log \relax (2) - 2\right )} \log \left (\log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4-2*x)*log(x)-4*x+12)*log(1/4*x^2*log(x)^2+x^2*log(x)+x^2)+(2*x^2-11*x+16)*log(x)+4*x^2-26*x+48)/
(x*log(x)+2*x),x, algorithm="giac")

[Out]

x^2 + x*(4*log(2) - 7) - 2*(x - 2*log(x))*log(log(x)^2 + 4*log(x) + 4) + log(log(x)^2 + 4*log(x) + 4)^2 - 4*x*
log(x) - 8*(log(2) - 2)*log(x) + 4*log(x)^2 - 8*(log(2) - 2)*log(log(x) + 2)

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maple [B]  time = 0.44, size = 75, normalized size = 3.00

method	result	size
norman	\(x^{2}+\ln \left (\frac {x^{2} \ln \relax (x )^{2}}{4}+x^{2} \ln \relax (x )+x^{2}\right )^{2}+8 \ln \left (\frac {x^{2} \ln \relax (x )^{2}}{4}+x^{2} \ln \relax (x )+x^{2}\right )-7 x -2 \ln \left (\frac {x^{2} \ln \relax (x )^{2}}{4}+x^{2} \ln \relax (x )+x^{2}\right ) x\)	\(75\)
risch	\(-7 x +16 \ln \relax (x )+4 \ln \relax (x )^{2}+x^{2}-8 \ln \relax (2) \ln \relax (x )-4 x \ln \relax (x )+4 x \ln \relax (2)+i x \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+i \pi x \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right )^{3}+i \pi x \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )^{3}-2 i \pi \ln \left (\ln \relax (x )+2\right ) \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi \ln \left (\ln \relax (x )+2\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right )^{3}-2 i \pi \ln \left (\ln \relax (x )+2\right ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )^{3}-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right )^{3}-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )^{3}+16 \ln \left (\ln \relax (x )+2\right )-2 i \pi \ln \left (\ln \relax (x )+2\right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )+i \pi x \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )+2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )^{2}-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )\right )^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right )+4 i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right )^{2}+2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )^{2}-i \pi x \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )^{2}-2 i \pi \ln \left (\ln \relax (x )+2\right ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+i \pi x \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )\right )^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right )-2 i \pi x \,\mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right )^{2}-i \pi x \,\mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )^{2}+4 \ln \left (\ln \relax (x )+2\right )^{2}+\left (-4 x +8 \ln \relax (x )\right ) \ln \left (\ln \relax (x )+2\right )-8 \ln \relax (2) \ln \left (\ln \relax (x )+2\right )+4 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+4 i \pi \ln \left (\ln \relax (x )+2\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+2 i \pi \ln \left (\ln \relax (x )+2\right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )^{2}-2 i \pi \ln \left (\ln \relax (x )+2\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )\right )^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right )+4 i \pi \ln \left (\ln \relax (x )+2\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right )^{2}+2 i \pi \ln \left (\ln \relax (x )+2\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+2\right )^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\ln \relax (x )+2\right )^{2}\right )^{2}-2 i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i x \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )\)	\(816\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4-2*x)*ln(x)-4*x+12)*ln(1/4*x^2*ln(x)^2+x^2*ln(x)+x^2)+(2*x^2-11*x+16)*ln(x)+4*x^2-26*x+48)/(x*ln(x)+2*
x),x,method=_RETURNVERBOSE)

[Out]

x^2+ln(1/4*x^2*ln(x)^2+x^2*ln(x)+x^2)^2+8*ln(1/4*x^2*ln(x)^2+x^2*ln(x)+x^2)-7*x-2*ln(1/4*x^2*ln(x)^2+x^2*ln(x)
+x^2)*x

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maxima [B]  time = 0.47, size = 63, normalized size = 2.52 \begin {gather*} x^{2} + x {\left (4 \, \log \relax (2) - 7\right )} - 4 \, {\left (x + 2 \, \log \relax (2) - 4\right )} \log \relax (x) + 4 \, \log \relax (x)^{2} - 4 \, {\left (x + 2 \, \log \relax (2) - 2 \, \log \relax (x) + 8\right )} \log \left (\log \relax (x) + 2\right ) + 4 \, \log \left (\log \relax (x) + 2\right )^{2} + 48 \, \log \left (\log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4-2*x)*log(x)-4*x+12)*log(1/4*x^2*log(x)^2+x^2*log(x)+x^2)+(2*x^2-11*x+16)*log(x)+4*x^2-26*x+48)/
(x*log(x)+2*x),x, algorithm="maxima")

[Out]

x^2 + x*(4*log(2) - 7) - 4*(x + 2*log(2) - 4)*log(x) + 4*log(x)^2 - 4*(x + 2*log(2) - 2*log(x) + 8)*log(log(x)
+ 2) + 4*log(log(x) + 2)^2 + 48*log(log(x) + 2)

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mupad [B]  time = 4.25, size = 63, normalized size = 2.52 \begin {gather*} 16\,\ln \left (\ln \relax (x)+2\right )-7\,x+16\,\ln \relax (x)-2\,x\,\ln \left (\frac {x^2\,{\ln \relax (x)}^2}{4}+x^2\,\ln \relax (x)+x^2\right )+{\ln \left (\frac {x^2\,{\ln \relax (x)}^2}{4}+x^2\,\ln \relax (x)+x^2\right )}^2+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(2*x^2 - 11*x + 16) - log(x^2*log(x) + (x^2*log(x)^2)/4 + x^2)*(4*x + log(x)*(2*x - 4) - 12) - 26*
x + 4*x^2 + 48)/(2*x + x*log(x)),x)

[Out]

16*log(log(x) + 2) - 7*x + 16*log(x) - 2*x*log(x^2*log(x) + (x^2*log(x)^2)/4 + x^2) + log(x^2*log(x) + (x^2*lo
g(x)^2)/4 + x^2)^2 + x^2

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sympy [B]  time = 0.40, size = 68, normalized size = 2.72 \begin {gather*} x^{2} - 2 x \log {\left (\frac {x^{2} \log {\relax (x )}^{2}}{4} + x^{2} \log {\relax (x )} + x^{2} \right )} - 7 x + 16 \log {\relax (x )} + 16 \log {\left (\log {\relax (x )} + 2 \right )} + \log {\left (\frac {x^{2} \log {\relax (x )}^{2}}{4} + x^{2} \log {\relax (x )} + x^{2} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4-2*x)*ln(x)-4*x+12)*ln(1/4*x**2*ln(x)**2+x**2*ln(x)+x**2)+(2*x**2-11*x+16)*ln(x)+4*x**2-26*x+48)
/(x*ln(x)+2*x),x)

[Out]

x**2 - 2*x*log(x**2*log(x)**2/4 + x**2*log(x) + x**2) - 7*x + 16*log(x) + 16*log(log(x) + 2) + log(x**2*log(x)
**2/4 + x**2*log(x) + x**2)**2

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