3.30.47 \(\int \frac {8-4 x+4 \log (3)-\log ^2(5)-4 x \log (x)}{(-8192 x+12288 x^2-6144 x^3+1024 x^4+(-12288 x+12288 x^2-3072 x^3) \log (3)+(-6144 x+3072 x^2) \log ^2(3)-1024 x \log ^3(3)+(3072 x-3072 x^2+768 x^3+(3072 x-1536 x^2) \log (3)+768 x \log ^2(3)) \log ^2(5)+(-384 x+192 x^2-192 x \log (3)) \log ^4(5)+16 x \log ^6(5)) \log ^3(x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {1}{4} \left (3+\frac {1}{8 \left (4 (-2+x-\log (3))+\log ^2(5)\right )^2 \log ^2(x)}\right ) \]

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Rubi [F]  time = 0.81, 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 {8-4 x+4 \log (3)-\log ^2(5)-4 x \log (x)}{\left (-8192 x+12288 x^2-6144 x^3+1024 x^4+\left (-12288 x+12288 x^2-3072 x^3\right ) \log (3)+\left (-6144 x+3072 x^2\right ) \log ^2(3)-1024 x \log ^3(3)+\left (3072 x-3072 x^2+768 x^3+\left (3072 x-1536 x^2\right ) \log (3)+768 x \log ^2(3)\right ) \log ^2(5)+\left (-384 x+192 x^2-192 x \log (3)\right ) \log ^4(5)+16 x \log ^6(5)\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8 - 4*x + 4*Log[3] - Log[5]^2 - 4*x*Log[x])/((-8192*x + 12288*x^2 - 6144*x^3 + 1024*x^4 + (-12288*x + 122
88*x^2 - 3072*x^3)*Log[3] + (-6144*x + 3072*x^2)*Log[3]^2 - 1024*x*Log[3]^3 + (3072*x - 3072*x^2 + 768*x^3 + (
3072*x - 1536*x^2)*Log[3] + 768*x*Log[3]^2)*Log[5]^2 + (-384*x + 192*x^2 - 192*x*Log[3])*Log[5]^4 + 16*x*Log[5
]^6)*Log[x]^3),x]

[Out]

-1/16*Defer[Int][1/(x*(-8 + 4*x + Log[5]^2 - Log[81])^2*Log[x]^3), x] - Defer[Int][1/((-8 + 4*x - 4*Log[3] + L
og[5]^2)^3*Log[x]^2), x]/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8-4 x+4 \log (3)-\log ^2(5)-4 x \log (x)}{\left (12288 x^2-6144 x^3+1024 x^4+\left (-12288 x+12288 x^2-3072 x^3\right ) \log (3)+\left (-6144 x+3072 x^2\right ) \log ^2(3)+x \left (-8192-1024 \log ^3(3)\right )+\left (3072 x-3072 x^2+768 x^3+\left (3072 x-1536 x^2\right ) \log (3)+768 x \log ^2(3)\right ) \log ^2(5)+\left (-384 x+192 x^2-192 x \log (3)\right ) \log ^4(5)+16 x \log ^6(5)\right ) \log ^3(x)} \, dx\\ &=\int \frac {8-4 x+4 \log (3)-\log ^2(5)-4 x \log (x)}{\left (12288 x^2-6144 x^3+1024 x^4+\left (-12288 x+12288 x^2-3072 x^3\right ) \log (3)+\left (-6144 x+3072 x^2\right ) \log ^2(3)+\left (3072 x-3072 x^2+768 x^3+\left (3072 x-1536 x^2\right ) \log (3)+768 x \log ^2(3)\right ) \log ^2(5)+\left (-384 x+192 x^2-192 x \log (3)\right ) \log ^4(5)+x \left (-8192-1024 \log ^3(3)+16 \log ^6(5)\right )\right ) \log ^3(x)} \, dx\\ &=\int \frac {4 x-8 \left (1+\frac {1}{8} \left (-\log ^2(5)+\log (81)\right )\right )+4 x \log (x)}{16 x \left (8-4 x+4 \log (3)-\log ^2(5)\right )^3 \log ^3(x)} \, dx\\ &=\frac {1}{16} \int \frac {4 x-8 \left (1+\frac {1}{8} \left (-\log ^2(5)+\log (81)\right )\right )+4 x \log (x)}{x \left (8-4 x+4 \log (3)-\log ^2(5)\right )^3 \log ^3(x)} \, dx\\ &=\frac {1}{16} \int \left (-\frac {1}{x \left (-8+4 x+\log ^2(5)-\log (81)\right )^2 \log ^3(x)}-\frac {4}{\left (-8+4 x-4 \log (3)+\log ^2(5)\right )^3 \log ^2(x)}\right ) \, dx\\ &=-\left (\frac {1}{16} \int \frac {1}{x \left (-8+4 x+\log ^2(5)-\log (81)\right )^2 \log ^3(x)} \, dx\right )-\frac {1}{4} \int \frac {1}{\left (-8+4 x-4 \log (3)+\log ^2(5)\right )^3 \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 38, normalized size = 1.27 \begin {gather*} \frac {\left (8-4 x-\log ^2(5)+\log (81)\right )^2}{32 \left (-8+4 x-4 \log (3)+\log ^2(5)\right )^4 \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 - 4*x + 4*Log[3] - Log[5]^2 - 4*x*Log[x])/((-8192*x + 12288*x^2 - 6144*x^3 + 1024*x^4 + (-12288*x
+ 12288*x^2 - 3072*x^3)*Log[3] + (-6144*x + 3072*x^2)*Log[3]^2 - 1024*x*Log[3]^3 + (3072*x - 3072*x^2 + 768*x
^3 + (3072*x - 1536*x^2)*Log[3] + 768*x*Log[3]^2)*Log[5]^2 + (-384*x + 192*x^2 - 192*x*Log[3])*Log[5]^4 + 16*x
*Log[5]^6)*Log[x]^3),x]

[Out]

(8 - 4*x - Log[5]^2 + Log[81])^2/(32*(-8 + 4*x - 4*Log[3] + Log[5]^2)^4*Log[x]^2)

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fricas [B]  time = 0.50, size = 48, normalized size = 1.60 \begin {gather*} \frac {1}{32 \, {\left (\log \relax (5)^{4} + 8 \, {\left (x - \log \relax (3) - 2\right )} \log \relax (5)^{2} + 16 \, x^{2} - 32 \, {\left (x - 2\right )} \log \relax (3) + 16 \, \log \relax (3)^{2} - 64 \, x + 64\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)-log(5)^2+4*log(3)-4*x+8)/(16*x*log(5)^6+(-192*x*log(3)+192*x^2-384*x)*log(5)^4+(768*x*l
og(3)^2+(-1536*x^2+3072*x)*log(3)+768*x^3-3072*x^2+3072*x)*log(5)^2-1024*x*log(3)^3+(3072*x^2-6144*x)*log(3)^2
+(-3072*x^3+12288*x^2-12288*x)*log(3)+1024*x^4-6144*x^3+12288*x^2-8192*x)/log(x)^3,x, algorithm="fricas")

[Out]

1/32/((log(5)^4 + 8*(x - log(3) - 2)*log(5)^2 + 16*x^2 - 32*(x - 2)*log(3) + 16*log(3)^2 - 64*x + 64)*log(x)^2
)

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giac [B]  time = 0.18, size = 96, normalized size = 3.20 \begin {gather*} \frac {1}{32 \, {\left (\log \relax (5)^{4} \log \relax (x)^{2} + 8 \, x \log \relax (5)^{2} \log \relax (x)^{2} - 8 \, \log \relax (5)^{2} \log \relax (3) \log \relax (x)^{2} + 16 \, x^{2} \log \relax (x)^{2} - 16 \, \log \relax (5)^{2} \log \relax (x)^{2} - 32 \, x \log \relax (3) \log \relax (x)^{2} + 16 \, \log \relax (3)^{2} \log \relax (x)^{2} - 64 \, x \log \relax (x)^{2} + 64 \, \log \relax (3) \log \relax (x)^{2} + 64 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)-log(5)^2+4*log(3)-4*x+8)/(16*x*log(5)^6+(-192*x*log(3)+192*x^2-384*x)*log(5)^4+(768*x*l
og(3)^2+(-1536*x^2+3072*x)*log(3)+768*x^3-3072*x^2+3072*x)*log(5)^2-1024*x*log(3)^3+(3072*x^2-6144*x)*log(3)^2
+(-3072*x^3+12288*x^2-12288*x)*log(3)+1024*x^4-6144*x^3+12288*x^2-8192*x)/log(x)^3,x, algorithm="giac")

[Out]

1/32/(log(5)^4*log(x)^2 + 8*x*log(5)^2*log(x)^2 - 8*log(5)^2*log(3)*log(x)^2 + 16*x^2*log(x)^2 - 16*log(5)^2*l
og(x)^2 - 32*x*log(3)*log(x)^2 + 16*log(3)^2*log(x)^2 - 64*x*log(x)^2 + 64*log(3)*log(x)^2 + 64*log(x)^2)

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maple [A]  time = 0.63, size = 22, normalized size = 0.73

method	result	size
norman	\(\frac {1}{32 \left (4 x -8-4 \ln \relax (3)+\ln \relax (5)^{2}\right )^{2} \ln \relax (x )^{2}}\)	\(22\)
risch	\(\frac {1}{32 \left (\ln \relax (5)^{4}-8 \ln \relax (3) \ln \relax (5)^{2}+8 x \ln \relax (5)^{2}-16 \ln \relax (5)^{2}+16 \ln \relax (3)^{2}-32 x \ln \relax (3)+16 x^{2}+64 \ln \relax (3)-64 x +64\right ) \ln \relax (x )^{2}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x*ln(x)-ln(5)^2+4*ln(3)-4*x+8)/(16*x*ln(5)^6+(-192*x*ln(3)+192*x^2-384*x)*ln(5)^4+(768*x*ln(3)^2+(-153
6*x^2+3072*x)*ln(3)+768*x^3-3072*x^2+3072*x)*ln(5)^2-1024*x*ln(3)^3+(3072*x^2-6144*x)*ln(3)^2+(-3072*x^3+12288
*x^2-12288*x)*ln(3)+1024*x^4-6144*x^3+12288*x^2-8192*x)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

1/32/(4*x-8-4*ln(3)+ln(5)^2)^2/ln(x)^2

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maxima [B]  time = 0.90, size = 54, normalized size = 1.80 \begin {gather*} \frac {1}{32 \, {\left (\log \relax (5)^{4} + 8 \, {\left (\log \relax (5)^{2} - 4 \, \log \relax (3) - 8\right )} x + 16 \, x^{2} - 16 \, \log \relax (5)^{2} - 8 \, {\left (\log \relax (5)^{2} - 8\right )} \log \relax (3) + 16 \, \log \relax (3)^{2} + 64\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)-log(5)^2+4*log(3)-4*x+8)/(16*x*log(5)^6+(-192*x*log(3)+192*x^2-384*x)*log(5)^4+(768*x*l
og(3)^2+(-1536*x^2+3072*x)*log(3)+768*x^3-3072*x^2+3072*x)*log(5)^2-1024*x*log(3)^3+(3072*x^2-6144*x)*log(3)^2
+(-3072*x^3+12288*x^2-12288*x)*log(3)+1024*x^4-6144*x^3+12288*x^2-8192*x)/log(x)^3,x, algorithm="maxima")

[Out]

1/32/((log(5)^4 + 8*(log(5)^2 - 4*log(3) - 8)*x + 16*x^2 - 16*log(5)^2 - 8*(log(5)^2 - 8)*log(3) + 16*log(3)^2
+ 64)*log(x)^2)

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mupad [B]  time = 2.27, size = 123, normalized size = 4.10 \begin {gather*} \frac {x^2}{2\,\left (1024\,x^2\,{\ln \relax (x)}^2-1024\,x^3\,{\ln \relax (x)}^2+256\,x^4\,{\ln \relax (x)}^2+256\,x^2\,{\ln \relax (3)}^2\,{\ln \relax (x)}^2-256\,x^2\,{\ln \relax (5)}^2\,{\ln \relax (x)}^2+128\,x^3\,{\ln \relax (5)}^2\,{\ln \relax (x)}^2+16\,x^2\,{\ln \relax (5)}^4\,{\ln \relax (x)}^2+1024\,x^2\,\ln \relax (3)\,{\ln \relax (x)}^2-512\,x^3\,\ln \relax (3)\,{\ln \relax (x)}^2-128\,x^2\,\ln \relax (3)\,{\ln \relax (5)}^2\,{\ln \relax (x)}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - 4*log(3) + 4*x*log(x) + log(5)^2 - 8)/(log(x)^3*(8192*x + log(3)*(12288*x - 12288*x^2 + 3072*x^3) +
log(3)^2*(6144*x - 3072*x^2) + 1024*x*log(3)^3 - 16*x*log(5)^6 - log(5)^2*(3072*x + log(3)*(3072*x - 1536*x^2
) + 768*x*log(3)^2 - 3072*x^2 + 768*x^3) - 12288*x^2 + 6144*x^3 - 1024*x^4 + log(5)^4*(384*x + 192*x*log(3) -
192*x^2))),x)

[Out]

x^2/(2*(1024*x^2*log(x)^2 - 1024*x^3*log(x)^2 + 256*x^4*log(x)^2 + 256*x^2*log(3)^2*log(x)^2 - 256*x^2*log(5)^
2*log(x)^2 + 128*x^3*log(5)^2*log(x)^2 + 16*x^2*log(5)^4*log(x)^2 + 1024*x^2*log(3)*log(x)^2 - 512*x^3*log(3)*
log(x)^2 - 128*x^2*log(3)*log(5)^2*log(x)^2))

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sympy [B]  time = 0.24, size = 66, normalized size = 2.20 \begin {gather*} \frac {1}{\left (512 x^{2} - 2048 x - 1024 x \log {\relax (3 )} + 256 x \log {\relax (5 )}^{2} - 512 \log {\relax (5 )}^{2} - 256 \log {\relax (3 )} \log {\relax (5 )}^{2} + 32 \log {\relax (5 )}^{4} + 512 \log {\relax (3 )}^{2} + 2048 + 2048 \log {\relax (3 )}\right ) \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*ln(x)-ln(5)**2+4*ln(3)-4*x+8)/(16*x*ln(5)**6+(-192*x*ln(3)+192*x**2-384*x)*ln(5)**4+(768*x*ln(
3)**2+(-1536*x**2+3072*x)*ln(3)+768*x**3-3072*x**2+3072*x)*ln(5)**2-1024*x*ln(3)**3+(3072*x**2-6144*x)*ln(3)**
2+(-3072*x**3+12288*x**2-12288*x)*ln(3)+1024*x**4-6144*x**3+12288*x**2-8192*x)/ln(x)**3,x)

[Out]

1/((512*x**2 - 2048*x - 1024*x*log(3) + 256*x*log(5)**2 - 512*log(5)**2 - 256*log(3)*log(5)**2 + 32*log(5)**4
+ 512*log(3)**2 + 2048 + 2048*log(3))*log(x)**2)

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