3.92.11 \(\int \frac {-36 x+36 x^2+(3 x-3 x^2) \log (x)+(33 x-108 x^2+(-3 x+9 x^2) \log (x)) \log (2 x)+(-36 x+3 x \log (x)+(72 x-6 x \log (x)) \log (2 x)) \log (\frac {-12+\log (x)}{8 x})}{(-12+\log (x)) \log ^2(2 x)} \, dx\)

Optimal. Leaf size=32 \[ \frac {3 x^2 \left (-1+x-\log \left (\frac {-3+\frac {\log (x)}{4}}{2 x}\right )\right )}{\log (2 x)} \]

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Rubi [F]  time = 1.71, 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 {-36 x+36 x^2+\left (3 x-3 x^2\right ) \log (x)+\left (33 x-108 x^2+\left (-3 x+9 x^2\right ) \log (x)\right ) \log (2 x)+(-36 x+3 x \log (x)+(72 x-6 x \log (x)) \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )}{(-12+\log (x)) \log ^2(2 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-36*x + 36*x^2 + (3*x - 3*x^2)*Log[x] + (33*x - 108*x^2 + (-3*x + 9*x^2)*Log[x])*Log[2*x] + (-36*x + 3*x*
Log[x] + (72*x - 6*x*Log[x])*Log[2*x])*Log[(-12 + Log[x])/(8*x)])/((-12 + Log[x])*Log[2*x]^2),x]

[Out]

(3*ExpIntegralEi[2*Log[2*x]])/2 - (9*ExpIntegralEi[3*Log[2*x]])/8 - (3*x^2)/Log[2*x] + (3*x^3)/Log[2*x] + 33*D
efer[Int][x/((-12 + Log[x])*Log[2*x]), x] - 108*Defer[Int][x^2/((-12 + Log[x])*Log[2*x]), x] - 3*Defer[Int][(x
*Log[x])/((-12 + Log[x])*Log[2*x]), x] + 9*Defer[Int][(x^2*Log[x])/((-12 + Log[x])*Log[2*x]), x] + 3*Defer[Int
][(x*Log[(-12 + Log[x])/(8*x)])/Log[2*x]^2, x] - 6*Defer[Int][(x*Log[(-12 + Log[x])/(8*x)])/Log[2*x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x \left (12-12 x+(-1+x) \log (x)-(11-36 x+(-1+3 x) \log (x)) \log (2 x)+(-12+\log (x)) (-1+2 \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )\right )}{(12-\log (x)) \log ^2(2 x)} \, dx\\ &=3 \int \frac {x \left (12-12 x+(-1+x) \log (x)-(11-36 x+(-1+3 x) \log (x)) \log (2 x)+(-12+\log (x)) (-1+2 \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )\right )}{(12-\log (x)) \log ^2(2 x)} \, dx\\ &=3 \int \left (\frac {x (-12+12 x+\log (x)-x \log (x)+11 \log (2 x)-36 x \log (2 x)-\log (x) \log (2 x)+3 x \log (x) \log (2 x))}{(-12+\log (x)) \log ^2(2 x)}-\frac {x (-1+2 \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log ^2(2 x)}\right ) \, dx\\ &=3 \int \frac {x (-12+12 x+\log (x)-x \log (x)+11 \log (2 x)-36 x \log (2 x)-\log (x) \log (2 x)+3 x \log (x) \log (2 x))}{(-12+\log (x)) \log ^2(2 x)} \, dx-3 \int \frac {x (-1+2 \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log ^2(2 x)} \, dx\\ &=3 \int \left (-\frac {(-1+x) x}{\log ^2(2 x)}+\frac {x (11-36 x-\log (x)+3 x \log (x))}{(-12+\log (x)) \log (2 x)}\right ) \, dx-3 \int \left (-\frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log ^2(2 x)}+\frac {2 x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log (2 x)}\right ) \, dx\\ &=-\left (3 \int \frac {(-1+x) x}{\log ^2(2 x)} \, dx\right )+3 \int \frac {x (11-36 x-\log (x)+3 x \log (x))}{(-12+\log (x)) \log (2 x)} \, dx+3 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log ^2(2 x)} \, dx-6 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log (2 x)} \, dx\\ &=-\left (3 \int \left (-\frac {x}{\log ^2(2 x)}+\frac {x^2}{\log ^2(2 x)}\right ) \, dx\right )+3 \int \left (\frac {11 x}{(-12+\log (x)) \log (2 x)}-\frac {36 x^2}{(-12+\log (x)) \log (2 x)}-\frac {x \log (x)}{(-12+\log (x)) \log (2 x)}+\frac {3 x^2 \log (x)}{(-12+\log (x)) \log (2 x)}\right ) \, dx+3 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log ^2(2 x)} \, dx-6 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log (2 x)} \, dx\\ &=3 \int \frac {x}{\log ^2(2 x)} \, dx-3 \int \frac {x^2}{\log ^2(2 x)} \, dx-3 \int \frac {x \log (x)}{(-12+\log (x)) \log (2 x)} \, dx+3 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log ^2(2 x)} \, dx-6 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log (2 x)} \, dx+9 \int \frac {x^2 \log (x)}{(-12+\log (x)) \log (2 x)} \, dx+33 \int \frac {x}{(-12+\log (x)) \log (2 x)} \, dx-108 \int \frac {x^2}{(-12+\log (x)) \log (2 x)} \, dx\\ &=-\frac {3 x^2}{\log (2 x)}+\frac {3 x^3}{\log (2 x)}-3 \int \frac {x \log (x)}{(-12+\log (x)) \log (2 x)} \, dx+3 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log ^2(2 x)} \, dx+6 \int \frac {x}{\log (2 x)} \, dx-6 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log (2 x)} \, dx-9 \int \frac {x^2}{\log (2 x)} \, dx+9 \int \frac {x^2 \log (x)}{(-12+\log (x)) \log (2 x)} \, dx+33 \int \frac {x}{(-12+\log (x)) \log (2 x)} \, dx-108 \int \frac {x^2}{(-12+\log (x)) \log (2 x)} \, dx\\ &=-\frac {3 x^2}{\log (2 x)}+\frac {3 x^3}{\log (2 x)}-\frac {9}{8} \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (2 x)\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (2 x)\right )-3 \int \frac {x \log (x)}{(-12+\log (x)) \log (2 x)} \, dx+3 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log ^2(2 x)} \, dx-6 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log (2 x)} \, dx+9 \int \frac {x^2 \log (x)}{(-12+\log (x)) \log (2 x)} \, dx+33 \int \frac {x}{(-12+\log (x)) \log (2 x)} \, dx-108 \int \frac {x^2}{(-12+\log (x)) \log (2 x)} \, dx\\ &=\frac {3}{2} \text {Ei}(2 \log (2 x))-\frac {9}{8} \text {Ei}(3 \log (2 x))-\frac {3 x^2}{\log (2 x)}+\frac {3 x^3}{\log (2 x)}-3 \int \frac {x \log (x)}{(-12+\log (x)) \log (2 x)} \, dx+3 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log ^2(2 x)} \, dx-6 \int \frac {x \log \left (\frac {-12+\log (x)}{8 x}\right )}{\log (2 x)} \, dx+9 \int \frac {x^2 \log (x)}{(-12+\log (x)) \log (2 x)} \, dx+33 \int \frac {x}{(-12+\log (x)) \log (2 x)} \, dx-108 \int \frac {x^2}{(-12+\log (x)) \log (2 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 28, normalized size = 0.88 \begin {gather*} \frac {3 x^2 \left (-1+x-\log \left (\frac {-12+\log (x)}{8 x}\right )\right )}{\log (2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36*x + 36*x^2 + (3*x - 3*x^2)*Log[x] + (33*x - 108*x^2 + (-3*x + 9*x^2)*Log[x])*Log[2*x] + (-36*x
+ 3*x*Log[x] + (72*x - 6*x*Log[x])*Log[2*x])*Log[(-12 + Log[x])/(8*x)])/((-12 + Log[x])*Log[2*x]^2),x]

[Out]

(3*x^2*(-1 + x - Log[(-12 + Log[x])/(8*x)]))/Log[2*x]

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fricas [A]  time = 0.84, size = 33, normalized size = 1.03 \begin {gather*} \frac {3 \, {\left (x^{3} - x^{2} \log \left (\frac {\log \relax (x) - 12}{8 \, x}\right ) - x^{2}\right )}}{\log \relax (2) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x*log(x)+72*x)*log(2*x)+3*x*log(x)-36*x)*log(1/8*(log(x)-12)/x)+((9*x^2-3*x)*log(x)-108*x^2+33
*x)*log(2*x)+(-3*x^2+3*x)*log(x)+36*x^2-36*x)/(log(x)-12)/log(2*x)^2,x, algorithm="fricas")

[Out]

3*(x^3 - x^2*log(1/8*(log(x) - 12)/x) - x^2)/(log(2) + log(x))

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giac [B]  time = 0.23, size = 70, normalized size = 2.19 \begin {gather*} \frac {3 \, x^{3}}{\log \relax (2) + \log \relax (x)} + \frac {9 \, x^{2} \log \relax (2)}{\log \relax (2) + \log \relax (x)} + \frac {3 \, x^{2} \log \relax (x)}{\log \relax (2) + \log \relax (x)} - \frac {3 \, x^{2} \log \left (\log \relax (x) - 12\right )}{\log \relax (2) + \log \relax (x)} - \frac {3 \, x^{2}}{\log \relax (2) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x*log(x)+72*x)*log(2*x)+3*x*log(x)-36*x)*log(1/8*(log(x)-12)/x)+((9*x^2-3*x)*log(x)-108*x^2+33
*x)*log(2*x)+(-3*x^2+3*x)*log(x)+36*x^2-36*x)/(log(x)-12)/log(2*x)^2,x, algorithm="giac")

[Out]

3*x^3/(log(2) + log(x)) + 9*x^2*log(2)/(log(2) + log(x)) + 3*x^2*log(x)/(log(2) + log(x)) - 3*x^2*log(log(x) -
12)/(log(2) + log(x)) - 3*x^2/(log(2) + log(x))

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maple [C]  time = 0.29, size = 172, normalized size = 5.38

method	result	size
risch	\(-\frac {6 i x^{2} \ln \left (\ln \relax (x )-12\right )}{2 i \ln \relax (2)+2 i \ln \relax (x )}+\frac {18 x^{2} \ln \relax (2)+6 x^{2} \ln \relax (x )+3 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )-12\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-12\right )}{x}\right )-3 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-12\right )}{x}\right )^{2}-3 i \pi \,x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-12\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-12\right )}{x}\right )^{2}+3 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-12\right )}{x}\right )^{3}+6 x^{3}-6 x^{2}}{2 \ln \relax (2)+2 \ln \relax (x )}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-6*x*ln(x)+72*x)*ln(2*x)+3*x*ln(x)-36*x)*ln(1/8*(ln(x)-12)/x)+((9*x^2-3*x)*ln(x)-108*x^2+33*x)*ln(2*x)+
(-3*x^2+3*x)*ln(x)+36*x^2-36*x)/(ln(x)-12)/ln(2*x)^2,x,method=_RETURNVERBOSE)

[Out]

-6*I*x^2/(2*I*ln(2)+2*I*ln(x))*ln(ln(x)-12)+3*(6*x^2*ln(2)+2*x^2*ln(x)+I*Pi*x^2*csgn(I/x)*csgn(I*(ln(x)-12))*c
sgn(I/x*(ln(x)-12))-I*Pi*x^2*csgn(I/x)*csgn(I/x*(ln(x)-12))^2-I*Pi*x^2*csgn(I*(ln(x)-12))*csgn(I/x*(ln(x)-12))
^2+I*Pi*x^2*csgn(I/x*(ln(x)-12))^3+2*x^3-2*x^2)/(2*ln(2)+2*ln(x))

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maxima [A]  time = 0.49, size = 39, normalized size = 1.22 \begin {gather*} \frac {3 \, {\left (x^{3} + x^{2} {\left (3 \, \log \relax (2) - 1\right )} + x^{2} \log \relax (x) - x^{2} \log \left (\log \relax (x) - 12\right )\right )}}{\log \relax (2) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x*log(x)+72*x)*log(2*x)+3*x*log(x)-36*x)*log(1/8*(log(x)-12)/x)+((9*x^2-3*x)*log(x)-108*x^2+33
*x)*log(2*x)+(-3*x^2+3*x)*log(x)+36*x^2-36*x)/(log(x)-12)/log(2*x)^2,x, algorithm="maxima")

[Out]

3*(x^3 + x^2*(3*log(2) - 1) + x^2*log(x) - x^2*log(log(x) - 12))/(log(2) + log(x))

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mupad [B]  time = 7.89, size = 115, normalized size = 3.59 \begin {gather*} 9\,x^3-6\,x^2-\frac {3\,x^2\,\left (2\,\ln \relax (x)-2\,\ln \left (2\,x\right )-x+3\,x\,\left (\ln \left (2\,x\right )-\ln \relax (x)\right )+1\right )+3\,x^2\,\ln \relax (x)\,\left (3\,x-2\right )}{\ln \left (2\,x\right )}-\frac {\ln \left (\frac {\frac {\ln \relax (x)}{8}-\frac {3}{2}}{x}\right )\,\left (6\,x^2\,\left (\ln \left (2\,x\right )-\ln \relax (x)\right )+3\,x^2\,\left (2\,\ln \relax (x)-2\,\ln \left (2\,x\right )+1\right )\right )}{\ln \left (2\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((log(x)/8 - 3/2)/x)*(log(2*x)*(72*x - 6*x*log(x)) - 36*x + 3*x*log(x)) - 36*x + log(x)*(3*x - 3*x^2)
- log(2*x)*(log(x)*(3*x - 9*x^2) - 33*x + 108*x^2) + 36*x^2)/(log(2*x)^2*(log(x) - 12)),x)

[Out]

9*x^3 - 6*x^2 - (3*x^2*(2*log(x) - 2*log(2*x) - x + 3*x*(log(2*x) - log(x)) + 1) + 3*x^2*log(x)*(3*x - 2))/log
(2*x) - (log((log(x)/8 - 3/2)/x)*(6*x^2*(log(2*x) - log(x)) + 3*x^2*(2*log(x) - 2*log(2*x) + 1)))/log(2*x)

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sympy [A]  time = 0.49, size = 39, normalized size = 1.22 \begin {gather*} - \frac {3 x^{2} \log {\left (\frac {\frac {\log {\relax (x )}}{8} - \frac {3}{2}}{x} \right )}}{\log {\relax (x )} + \log {\relax (2 )}} + \frac {3 x^{3} - 3 x^{2}}{\log {\relax (x )} + \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x*ln(x)+72*x)*ln(2*x)+3*x*ln(x)-36*x)*ln(1/8*(ln(x)-12)/x)+((9*x**2-3*x)*ln(x)-108*x**2+33*x)*
ln(2*x)+(-3*x**2+3*x)*ln(x)+36*x**2-36*x)/(ln(x)-12)/ln(2*x)**2,x)

[Out]

-3*x**2*log((log(x)/8 - 3/2)/x)/(log(x) + log(2)) + (3*x**3 - 3*x**2)/(log(x) + log(2))

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