3.27.81 \(\int \frac {2 x^2 \log (12)+(2 x^2+(-4 x+x^2) \log (12)) \log (x)+(-2 x^2+(2-2 x) \log (12)) \log ^2(x)+\log (12) \log ^3(x)+(x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)) \log (\frac {5}{x \log ^2(x)})}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx\)

Optimal. Leaf size=29 \[ \frac {2 x}{x-\log (x)}-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x} \]

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Rubi [F]  time = 2.16, 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 {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x^2*Log[12] + (2*x^2 + (-4*x + x^2)*Log[12])*Log[x] + (-2*x^2 + (2 - 2*x)*Log[12])*Log[x]^2 + Log[12]*L
og[x]^3 + (x^2*Log[12]*Log[x] - 2*x*Log[12]*Log[x]^2 + Log[12]*Log[x]^3)*Log[5/(x*Log[x]^2)])/(x^4*Log[x] - 2*
x^3*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

2*ExpIntegralEi[-Log[x]]*Log[12] + ExpIntegralEi[-Log[x]]*Log[12]*Log[x] - ExpIntegralEi[-Log[x]]*Log[12]*(2 +
Log[x]) - (Log[12]*Log[5/(x*Log[x]^2)])/x - Log[12]*Defer[Int][(x - Log[x])^(-2), x] + (2 + Log[12])*Defer[In
t][(x - Log[x])^(-2), x] - 2*Defer[Int][x/(x - Log[x])^2, x] - 2*Defer[Int][(-x + Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^2 (x-\log (x))^2 \log (x)} \, dx\\ &=\int \left (\frac {-4 \log (12)+x (2+\log (12))}{x (x-\log (x))^2}+\frac {2 \log (12)}{(x-\log (x))^2 \log (x)}-\frac {2 \left (x^2-\log (12)+x \log (12)\right ) \log (x)}{x^2 (x-\log (x))^2}+\frac {\log (12) \log ^2(x)}{x^2 (x-\log (x))^2}+\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^2}\right ) \, dx\\ &=-\left (2 \int \frac {\left (x^2-\log (12)+x \log (12)\right ) \log (x)}{x^2 (x-\log (x))^2} \, dx\right )+\log (12) \int \frac {\log ^2(x)}{x^2 (x-\log (x))^2} \, dx+\log (12) \int \frac {\log \left (\frac {5}{x \log ^2(x)}\right )}{x^2} \, dx+(2 \log (12)) \int \frac {1}{(x-\log (x))^2 \log (x)} \, dx+\int \frac {-4 \log (12)+x (2+\log (12))}{x (x-\log (x))^2} \, dx\\ &=-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x}-2 \int \left (\frac {x^2-\log (12)+x \log (12)}{x (x-\log (x))^2}+\frac {-x^2+\log (12)-x \log (12)}{x^2 (x-\log (x))}\right ) \, dx+\log (12) \int \left (\frac {1}{x^2}+\frac {1}{(x-\log (x))^2}-\frac {2}{x (x-\log (x))}\right ) \, dx-\log (12) \int \frac {2+\log (x)}{x^2 \log (x)} \, dx+(2 \log (12)) \int \left (\frac {1}{x (x-\log (x))^2}+\frac {1}{x^2 (x-\log (x))}+\frac {1}{x^2 \log (x)}\right ) \, dx+\int \left (\frac {2 \left (1+\frac {\log (12)}{2}\right )}{(x-\log (x))^2}-\frac {4 \log (12)}{x (x-\log (x))^2}\right ) \, dx\\ &=-\frac {\log (12)}{x}-\text {Ei}(-\log (x)) \log (12) (2+\log (x))-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x}-2 \int \frac {x^2-\log (12)+x \log (12)}{x (x-\log (x))^2} \, dx-2 \int \frac {-x^2+\log (12)-x \log (12)}{x^2 (x-\log (x))} \, dx+\log (12) \int \frac {\text {Ei}(-\log (x))}{x} \, dx+\log (12) \int \frac {1}{(x-\log (x))^2} \, dx+(2 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx+(2 \log (12)) \int \frac {1}{x^2 (x-\log (x))} \, dx-(2 \log (12)) \int \frac {1}{x (x-\log (x))} \, dx+(2 \log (12)) \int \frac {1}{x^2 \log (x)} \, dx-(4 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx+(2+\log (12)) \int \frac {1}{(x-\log (x))^2} \, dx\\ &=-\frac {\log (12)}{x}-\text {Ei}(-\log (x)) \log (12) (2+\log (x))-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x}-2 \int \left (\frac {x}{(x-\log (x))^2}+\frac {\log (12)}{(x-\log (x))^2}-\frac {\log (12)}{x (x-\log (x))^2}\right ) \, dx-2 \int \left (\frac {\log (12)}{x^2 (x-\log (x))}-\frac {\log (12)}{x (x-\log (x))}+\frac {1}{-x+\log (x)}\right ) \, dx+\log (12) \int \frac {1}{(x-\log (x))^2} \, dx+\log (12) \operatorname {Subst}(\int \text {Ei}(-x) \, dx,x,\log (x))+(2 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx+(2 \log (12)) \int \frac {1}{x^2 (x-\log (x))} \, dx-(2 \log (12)) \int \frac {1}{x (x-\log (x))} \, dx+(2 \log (12)) \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )-(4 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx+(2+\log (12)) \int \frac {1}{(x-\log (x))^2} \, dx\\ &=2 \text {Ei}(-\log (x)) \log (12)+\text {Ei}(-\log (x)) \log (12) \log (x)-\text {Ei}(-\log (x)) \log (12) (2+\log (x))-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x}-2 \int \frac {x}{(x-\log (x))^2} \, dx-2 \int \frac {1}{-x+\log (x)} \, dx+\log (12) \int \frac {1}{(x-\log (x))^2} \, dx-(2 \log (12)) \int \frac {1}{(x-\log (x))^2} \, dx+2 \left ((2 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx\right )-(4 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx+(2+\log (12)) \int \frac {1}{(x-\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 29, normalized size = 1.00 \begin {gather*} \frac {2 x}{x-\log (x)}-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2*Log[12] + (2*x^2 + (-4*x + x^2)*Log[12])*Log[x] + (-2*x^2 + (2 - 2*x)*Log[12])*Log[x]^2 + Log
[12]*Log[x]^3 + (x^2*Log[12]*Log[x] - 2*x*Log[12]*Log[x]^2 + Log[12]*Log[x]^3)*Log[5/(x*Log[x]^2)])/(x^4*Log[x
] - 2*x^3*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

(2*x)/(x - Log[x]) - (Log[12]*Log[5/(x*Log[x]^2)])/x

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fricas [A]  time = 0.55, size = 41, normalized size = 1.41 \begin {gather*} \frac {2 \, x^{2} - {\left (x \log \left (12\right ) - \log \left (12\right ) \log \relax (x)\right )} \log \left (\frac {5}{x \log \relax (x)^{2}}\right )}{x^{2} - x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(12)*log(x)^3-2*x*log(12)*log(x)^2+x^2*log(12)*log(x))*log(5/x/log(x)^2)+log(12)*log(x)^3+((-2*
x+2)*log(12)-2*x^2)*log(x)^2+((x^2-4*x)*log(12)+2*x^2)*log(x)+2*x^2*log(12))/(x^2*log(x)^3-2*x^3*log(x)^2+x^4*
log(x)),x, algorithm="fricas")

[Out]

(2*x^2 - (x*log(12) - log(12)*log(x))*log(5/(x*log(x)^2)))/(x^2 - x*log(x))

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giac [A]  time = 0.96, size = 40, normalized size = 1.38 \begin {gather*} -\frac {\log \left (12\right ) \log \relax (5)}{x} + \frac {\log \left (12\right ) \log \left (\log \relax (x)^{2}\right )}{x} + \frac {\log \left (12\right ) \log \relax (x)}{x} + \frac {2 \, x}{x - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(12)*log(x)^3-2*x*log(12)*log(x)^2+x^2*log(12)*log(x))*log(5/x/log(x)^2)+log(12)*log(x)^3+((-2*
x+2)*log(12)-2*x^2)*log(x)^2+((x^2-4*x)*log(12)+2*x^2)*log(x)+2*x^2*log(12))/(x^2*log(x)^3-2*x^3*log(x)^2+x^4*
log(x)),x, algorithm="giac")

[Out]

-log(12)*log(5)/x + log(12)*log(log(x)^2)/x + log(12)*log(x)/x + 2*x/(x - log(x))

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maple [A]  time = 0.46, size = 38, normalized size = 1.31

method	result	size
default	\(-\frac {\ln \left (12\right ) \ln \left (\frac {1}{x \ln \relax (x )^{2}}\right )}{x}-\frac {\ln \left (12\right ) \ln \relax (5)}{x}+\frac {2 x}{x -\ln \relax (x )}\)	\(38\)
risch	\(\frac {2 \left (\ln \relax (3)+2 \ln \relax (2)\right ) \ln \left (\ln \relax (x )\right )}{x}+\frac {2 x \ln \relax (3) \ln \relax (x )+2 \ln \relax (3) \ln \relax (5) \ln \relax (x )+4 x \ln \relax (2) \ln \relax (x )-4 x \ln \relax (2) \ln \relax (5)-2 \ln \relax (3) \ln \relax (x )^{2}+4 x^{2}-4 \ln \relax (2) \ln \relax (x )^{2}-2 x \ln \relax (3) \ln \relax (5)+2 i \pi \ln \relax (2) x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )-2 i \pi \ln \relax (2) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right ) \ln \relax (x )-i \pi \ln \relax (3) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right ) \ln \relax (x )+i \pi \ln \relax (3) x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )-i \pi \ln \relax (3) x \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{2}+2 i \pi \ln \relax (2) \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right ) \ln \relax (x )-4 i \pi \ln \relax (2) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2} \ln \relax (x )+i \pi \ln \relax (3) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{2} \ln \relax (x )+i \pi \ln \relax (3) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{2} \ln \relax (x )+4 \ln \relax (2) \ln \relax (5) \ln \relax (x )-i \pi \ln \relax (3) x \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3}-i \pi \ln \relax (3) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{3} \ln \relax (x )-2 i \pi \ln \relax (2) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{3} \ln \relax (x )+2 i \pi \ln \relax (2) x \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{3}-2 i \pi \ln \relax (2) x \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3}+2 i \pi \ln \relax (2) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3} \ln \relax (x )+i \pi \ln \relax (3) x \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{3}+i \pi \ln \relax (3) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3} \ln \relax (x )+i \pi \ln \relax (3) \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right ) \ln \relax (x )-2 i \pi \ln \relax (3) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2} \ln \relax (x )+2 i \pi \ln \relax (2) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{2} \ln \relax (x )-2 i \pi \ln \relax (2) x \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{2}+2 i \pi \ln \relax (3) x \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2}-2 i \pi \ln \relax (2) x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{2}-2 i \pi \ln \relax (2) x \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )-i \pi \ln \relax (3) x \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )+4 i \pi \ln \relax (2) x \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2}-i \pi \ln \relax (3) x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{2}+2 i \pi \ln \relax (2) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )^{2}}\right )^{2} \ln \relax (x )}{2 x \left (x -\ln \relax (x )\right )}\)	\(794\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((ln(12)*ln(x)^3-2*x*ln(12)*ln(x)^2+x^2*ln(12)*ln(x))*ln(5/x/ln(x)^2)+ln(12)*ln(x)^3+((-2*x+2)*ln(12)-2*x^
2)*ln(x)^2+((x^2-4*x)*ln(12)+2*x^2)*ln(x)+2*x^2*ln(12))/(x^2*ln(x)^3-2*x^3*ln(x)^2+x^4*ln(x)),x,method=_RETURN
VERBOSE)

[Out]

-ln(12)*ln(1/x/ln(x)^2)/x-ln(12)*ln(5)/x+2/(x-ln(x))*x

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maxima [B]  time = 0.62, size = 96, normalized size = 3.31 \begin {gather*} -\frac {{\left (\log \relax (3) + 2 \, \log \relax (2)\right )} \log \relax (x)^{2} + {\left (\log \relax (5) \log \relax (3) + 2 \, \log \relax (5) \log \relax (2)\right )} x - 2 \, x^{2} - {\left (x {\left (\log \relax (3) + 2 \, \log \relax (2)\right )} + \log \relax (5) \log \relax (3) + 2 \, \log \relax (5) \log \relax (2)\right )} \log \relax (x) - 2 \, {\left (x {\left (\log \relax (3) + 2 \, \log \relax (2)\right )} - {\left (\log \relax (3) + 2 \, \log \relax (2)\right )} \log \relax (x)\right )} \log \left (\log \relax (x)\right )}{x^{2} - x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(12)*log(x)^3-2*x*log(12)*log(x)^2+x^2*log(12)*log(x))*log(5/x/log(x)^2)+log(12)*log(x)^3+((-2*
x+2)*log(12)-2*x^2)*log(x)^2+((x^2-4*x)*log(12)+2*x^2)*log(x)+2*x^2*log(12))/(x^2*log(x)^3-2*x^3*log(x)^2+x^4*
log(x)),x, algorithm="maxima")

[Out]

-((log(3) + 2*log(2))*log(x)^2 + (log(5)*log(3) + 2*log(5)*log(2))*x - 2*x^2 - (x*(log(3) + 2*log(2)) + log(5)
*log(3) + 2*log(5)*log(2))*log(x) - 2*(x*(log(3) + 2*log(2)) - (log(3) + 2*log(2))*log(x))*log(log(x)))/(x^2 -
x*log(x))

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mupad [B]  time = 1.95, size = 54, normalized size = 1.86 \begin {gather*} \frac {2}{x-1}-\frac {\frac {2\,x}{x-1}-\frac {2\,x\,\ln \relax (x)}{x-1}}{x-\ln \relax (x)}-\frac {\ln \left (12\right )\,\ln \left (\frac {5}{x\,{\ln \relax (x)}^2}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(12)*log(x)^3 - log(x)*(log(12)*(4*x - x^2) - 2*x^2) - log(x)^2*(log(12)*(2*x - 2) + 2*x^2) + 2*x^2*lo
g(12) + log(5/(x*log(x)^2))*(log(12)*log(x)^3 - 2*x*log(12)*log(x)^2 + x^2*log(12)*log(x)))/(x^4*log(x) + x^2*
log(x)^3 - 2*x^3*log(x)^2),x)

[Out]

2/(x - 1) - ((2*x)/(x - 1) - (2*x*log(x))/(x - 1))/(x - log(x)) - (log(12)*log(5/(x*log(x)^2)))/x

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sympy [A]  time = 0.40, size = 24, normalized size = 0.83 \begin {gather*} - \frac {2 x}{- x + \log {\relax (x )}} - \frac {\log {\left (12 \right )} \log {\left (\frac {5}{x \log {\relax (x )}^{2}} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((ln(12)*ln(x)**3-2*x*ln(12)*ln(x)**2+x**2*ln(12)*ln(x))*ln(5/x/ln(x)**2)+ln(12)*ln(x)**3+((-2*x+2)*
ln(12)-2*x**2)*ln(x)**2+((x**2-4*x)*ln(12)+2*x**2)*ln(x)+2*x**2*ln(12))/(x**2*ln(x)**3-2*x**3*ln(x)**2+x**4*ln
(x)),x)

[Out]

-2*x/(-x + log(x)) - log(12)*log(5/(x*log(x)**2))/x

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