3.47.48 \(\int \frac {-6-2 x+(18+12 x+2 x^2+(-27-18 x-3 x^2) \log (2)) \log (2 x)+3 \log (2 x) \log (\frac {25}{\log ^2(2 x)})}{(54 x+36 x^2+6 x^3+(-81 x-54 x^2-9 x^3) \log (2)) \log (2 x)+(9 x+3 x^2) \log (2 x) \log (\frac {25}{\log ^2(2 x)})} \, dx\)

Optimal. Leaf size=33 \[ \frac {1}{3} \log \left (-x+\frac {x \log \left (\frac {25}{\log ^2(2 x)}\right )}{(3+x) (-2+3 \log (2))}\right ) \]

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Rubi [F]  time = 6.26, 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 {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 - 2*x + (18 + 12*x + 2*x^2 + (-27 - 18*x - 3*x^2)*Log[2])*Log[2*x] + 3*Log[2*x]*Log[25/Log[2*x]^2])/((
54*x + 36*x^2 + 6*x^3 + (-81*x - 54*x^2 - 9*x^3)*Log[2])*Log[2*x] + (9*x + 3*x^2)*Log[2*x]*Log[25/Log[2*x]^2])
,x]

[Out]

Log[x]/3 - Log[3 + x]/3 - ((2 - Log[8])*Defer[Int][(-2*x*(1 - Log[8]/2) - 6*(1 - Log[512]/6) - Log[25/Log[2*x]
^2])^(-1), x])/3 - ((6 - Log[512])*Defer[Int][(-2*x*(1 - Log[8]/2) - 6*(1 - Log[512]/6) - Log[25/Log[2*x]^2])^
(-1), x])/9 + (2*Defer[Int][1/(Log[2*x]*(-2*x*(1 - Log[8]/2) - 6*(1 - Log[512]/6) - Log[25/Log[2*x]^2])), x])/
9 + (2*Defer[Int][1/(x*Log[2*x]*(-2*x*(1 - Log[8]/2) - 6*(1 - Log[512]/6) - Log[25/Log[2*x]^2])), x])/3 - ((6
- Log[512])*Defer[Int][(2*x*(1 - Log[8]/2) + 6*(1 - Log[512]/6) + Log[25/Log[2*x]^2])^(-1), x])/9 - ((6 - Log[
512])*Defer[Int][1/((-3 - x)*(2*x*(1 - Log[8]/2) + 6*(1 - Log[512]/6) + Log[25/Log[2*x]^2])), x])/3 - (2 - Log
[8])*Defer[Int][1/((3 + x)*(2*x*(1 - Log[8]/2) + 6*(1 - Log[512]/6) + Log[25/Log[2*x]^2])), x] + (2*Defer[Int]
[1/(Log[2*x]*(2*x*(1 - Log[8]/2) + 6*(1 - Log[512]/6) + Log[25/Log[2*x]^2])), x])/9 + (2*Defer[Int][1/((-3 - x
)*Log[2*x]*(2*x*(1 - Log[8]/2) + 6*(1 - Log[512]/6) + Log[25/Log[2*x]^2])), x])/3 + (2*Defer[Int][1/((3 + x)*L
og[2*x]*(2*x*(1 - Log[8]/2) + 6*(1 - Log[512]/6) + Log[25/Log[2*x]^2])), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 (3+x)+\log (2 x) \left ((3+x)^2 (-2+\log (8))-3 \log \left (\frac {25}{\log ^2(2 x)}\right )\right )}{3 x (3+x) \log (2 x) \left (x (-2+\log (8))-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx\\ &=\frac {1}{3} \int \frac {2 (3+x)+\log (2 x) \left ((3+x)^2 (-2+\log (8))-3 \log \left (\frac {25}{\log ^2(2 x)}\right )\right )}{x (3+x) \log (2 x) \left (x (-2+\log (8))-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx\\ &=\frac {1}{3} \int \left (\frac {3}{x (3+x)}+\frac {-6-2 x+2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)+6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{x (3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {-6-2 x+2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)+6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{x (3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\int \frac {1}{x (3+x)} \, dx\\ &=\frac {1}{3} \int \frac {1}{x} \, dx-\frac {1}{3} \int \frac {1}{3+x} \, dx+\frac {1}{3} \int \frac {2 (3+x)+x (-6+x (-2+\log (8))+\log (512)) \log (2 x)}{x (3+x) \log (2 x) \left (x (-2+\log (8))-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx\\ &=\frac {\log (x)}{3}-\frac {1}{3} \log (3+x)+\frac {1}{3} \int \left (\frac {6+2 x-2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)-6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{3 (3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {-6-2 x+2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)+6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{3 x \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}\right ) \, dx\\ &=\frac {\log (x)}{3}-\frac {1}{3} \log (3+x)+\frac {1}{9} \int \frac {6+2 x-2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)-6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{(3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\frac {1}{9} \int \frac {-6-2 x+2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)+6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{x \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx\\ &=\frac {\log (x)}{3}-\frac {1}{3} \log (3+x)+\frac {1}{9} \int \frac {2 (3+x)+x (-6+x (-2+\log (8))+\log (512)) \log (2 x)}{x \log (2 x) \left (x (-2+\log (8))-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\frac {1}{9} \int \left (\frac {x^2 (-2+\log (8))}{(3+x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {x (-6+\log (512))}{(3+x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {6}{(3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {2 x}{(3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}\right ) \, dx\\ &=\frac {\log (x)}{3}-\frac {1}{3} \log (3+x)+\frac {1}{9} \int \left (\frac {\left (1-\frac {6}{\log (512)}\right ) \log (512)}{-2 x \left (1-\frac {\log (8)}{2}\right )-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )}+\frac {2}{\log (2 x) \left (-2 x \left (1-\frac {\log (8)}{2}\right )-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {6}{x \log (2 x) \left (-2 x \left (1-\frac {\log (8)}{2}\right )-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {x (2-\log (8))}{2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )}\right ) \, dx+\frac {2}{9} \int \frac {x}{(3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\frac {2}{3} \int \frac {1}{(3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\frac {1}{9} (-2+\log (8)) \int \frac {x^2}{(3+x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\frac {1}{9} (-6+\log (512)) \int \frac {x}{(3+x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.92, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-6 - 2*x + (18 + 12*x + 2*x^2 + (-27 - 18*x - 3*x^2)*Log[2])*Log[2*x] + 3*Log[2*x]*Log[25/Log[2*x]^
2])/((54*x + 36*x^2 + 6*x^3 + (-81*x - 54*x^2 - 9*x^3)*Log[2])*Log[2*x] + (9*x + 3*x^2)*Log[2*x]*Log[25/Log[2*
x]^2]),x]

[Out]

Integrate[(-6 - 2*x + (18 + 12*x + 2*x^2 + (-27 - 18*x - 3*x^2)*Log[2])*Log[2*x] + 3*Log[2*x]*Log[25/Log[2*x]^
2])/((54*x + 36*x^2 + 6*x^3 + (-81*x - 54*x^2 - 9*x^3)*Log[2])*Log[2*x] + (9*x + 3*x^2)*Log[2*x]*Log[25/Log[2*
x]^2]), x]

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fricas [A]  time = 1.01, size = 35, normalized size = 1.06 \begin {gather*} \frac {1}{3} \, \log \left (-3 \, {\left (x + 3\right )} \log \relax (2) + 2 \, x + \log \left (\frac {25}{\log \left (2 \, x\right )^{2}}\right ) + 6\right ) - \frac {1}{3} \, \log \left (x + 3\right ) + \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(2*x)*log(25/log(2*x)^2)+((-3*x^2-18*x-27)*log(2)+2*x^2+12*x+18)*log(2*x)-2*x-6)/((3*x^2+9*x)*
log(2*x)*log(25/log(2*x)^2)+((-9*x^3-54*x^2-81*x)*log(2)+6*x^3+36*x^2+54*x)*log(2*x)),x, algorithm="fricas")

[Out]

1/3*log(-3*(x + 3)*log(2) + 2*x + log(25/log(2*x)^2) + 6) - 1/3*log(x + 3) + 1/3*log(x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{2} - 3 \, {\left (x^{2} + 6 \, x + 9\right )} \log \relax (2) + 12 \, x + 18\right )} \log \left (2 \, x\right ) + 3 \, \log \left (2 \, x\right ) \log \left (\frac {25}{\log \left (2 \, x\right )^{2}}\right ) - 2 \, x - 6}{3 \, {\left ({\left (x^{2} + 3 \, x\right )} \log \left (2 \, x\right ) \log \left (\frac {25}{\log \left (2 \, x\right )^{2}}\right ) + {\left (2 \, x^{3} + 12 \, x^{2} - 3 \, {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} \log \relax (2) + 18 \, x\right )} \log \left (2 \, x\right )\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(2*x)*log(25/log(2*x)^2)+((-3*x^2-18*x-27)*log(2)+2*x^2+12*x+18)*log(2*x)-2*x-6)/((3*x^2+9*x)*
log(2*x)*log(25/log(2*x)^2)+((-9*x^3-54*x^2-81*x)*log(2)+6*x^3+36*x^2+54*x)*log(2*x)),x, algorithm="giac")

[Out]

integrate(1/3*((2*x^2 - 3*(x^2 + 6*x + 9)*log(2) + 12*x + 18)*log(2*x) + 3*log(2*x)*log(25/log(2*x)^2) - 2*x -
6)/((x^2 + 3*x)*log(2*x)*log(25/log(2*x)^2) + (2*x^3 + 12*x^2 - 3*(x^3 + 6*x^2 + 9*x)*log(2) + 18*x)*log(2*x)
), x)

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maple [C]  time = 0.28, size = 108, normalized size = 3.27

method	result	size
risch	\(\frac {\ln \relax (x )}{3}-\frac {\ln \left (3+x \right )}{3}+\frac {\ln \left (\ln \left (\ln \left (2 x \right )\right )+\frac {i \left (-\pi \mathrm {csgn}\left (i \ln \left (2 x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (2 x \right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \ln \left (2 x \right )\right ) \mathrm {csgn}\left (i \ln \left (2 x \right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \ln \left (2 x \right )^{2}\right )^{3}-6 i x \ln \relax (2)-18 i \ln \relax (2)+4 i \ln \relax (5)+4 i x +12 i\right )}{4}\right )}{3}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*ln(2*x)*ln(25/ln(2*x)^2)+((-3*x^2-18*x-27)*ln(2)+2*x^2+12*x+18)*ln(2*x)-2*x-6)/((3*x^2+9*x)*ln(2*x)*ln(
25/ln(2*x)^2)+((-9*x^3-54*x^2-81*x)*ln(2)+6*x^3+36*x^2+54*x)*ln(2*x)),x,method=_RETURNVERBOSE)

[Out]

1/3*ln(x)-1/3*ln(3+x)+1/3*ln(ln(ln(2*x))+1/4*I*(-Pi*csgn(I*ln(2*x))^2*csgn(I*ln(2*x)^2)+2*Pi*csgn(I*ln(2*x))*c
sgn(I*ln(2*x)^2)^2-Pi*csgn(I*ln(2*x)^2)^3-6*I*x*ln(2)-18*I*ln(2)+4*I*ln(5)+4*I*x+12*I))

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maxima [A]  time = 0.50, size = 39, normalized size = 1.18 \begin {gather*} \frac {1}{3} \, \log \left (\frac {1}{2} \, x {\left (3 \, \log \relax (2) - 2\right )} - \log \relax (5) + \frac {9}{2} \, \log \relax (2) + \log \left (\log \relax (2) + \log \relax (x)\right ) - 3\right ) - \frac {1}{3} \, \log \left (x + 3\right ) + \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(2*x)*log(25/log(2*x)^2)+((-3*x^2-18*x-27)*log(2)+2*x^2+12*x+18)*log(2*x)-2*x-6)/((3*x^2+9*x)*
log(2*x)*log(25/log(2*x)^2)+((-9*x^3-54*x^2-81*x)*log(2)+6*x^3+36*x^2+54*x)*log(2*x)),x, algorithm="maxima")

[Out]

1/3*log(1/2*x*(3*log(2) - 2) - log(5) + 9/2*log(2) + log(log(2) + log(x)) - 3) - 1/3*log(x + 3) + 1/3*log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {2\,x-3\,\ln \left (2\,x\right )\,\ln \left (\frac {25}{{\ln \left (2\,x\right )}^2}\right )-\ln \left (2\,x\right )\,\left (12\,x-\ln \relax (2)\,\left (3\,x^2+18\,x+27\right )+2\,x^2+18\right )+6}{\ln \left (2\,x\right )\,\left (54\,x-\ln \relax (2)\,\left (9\,x^3+54\,x^2+81\,x\right )+36\,x^2+6\,x^3\right )+\ln \left (2\,x\right )\,\ln \left (\frac {25}{{\ln \left (2\,x\right )}^2}\right )\,\left (3\,x^2+9\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 3*log(2*x)*log(25/log(2*x)^2) - log(2*x)*(12*x - log(2)*(18*x + 3*x^2 + 27) + 2*x^2 + 18) + 6)/(lo
g(2*x)*(54*x - log(2)*(81*x + 54*x^2 + 9*x^3) + 36*x^2 + 6*x^3) + log(2*x)*log(25/log(2*x)^2)*(9*x + 3*x^2)),x
)

[Out]

-int((2*x - 3*log(2*x)*log(25/log(2*x)^2) - log(2*x)*(12*x - log(2)*(18*x + 3*x^2 + 27) + 2*x^2 + 18) + 6)/(lo
g(2*x)*(54*x - log(2)*(81*x + 54*x^2 + 9*x^3) + 36*x^2 + 6*x^3) + log(2*x)*log(25/log(2*x)^2)*(9*x + 3*x^2)),
x)

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sympy [A]  time = 0.49, size = 41, normalized size = 1.24 \begin {gather*} \frac {\log {\relax (x )}}{3} - \frac {\log {\left (x + 3 \right )}}{3} + \frac {\log {\left (- 3 x \log {\relax (2 )} + 2 x + \log {\left (\frac {25}{\log {\left (2 x \right )}^{2}} \right )} - 9 \log {\relax (2 )} + 6 \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*ln(2*x)*ln(25/ln(2*x)**2)+((-3*x**2-18*x-27)*ln(2)+2*x**2+12*x+18)*ln(2*x)-2*x-6)/((3*x**2+9*x)*l
n(2*x)*ln(25/ln(2*x)**2)+((-9*x**3-54*x**2-81*x)*ln(2)+6*x**3+36*x**2+54*x)*ln(2*x)),x)

[Out]

log(x)/3 - log(x + 3)/3 + log(-3*x*log(2) + 2*x + log(25/log(2*x)**2) - 9*log(2) + 6)/3

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