3.18.61 \(\int \frac {4+x-16 x^5-8 x^6-x^7+(1-8 x^5-2 x^6) \log (3)-x^5 \log ^2(3)+(-16-5 x-4 \log (3)) \log (x)}{-16 x^6-8 x^7-x^8+(-8 x^6-2 x^7) \log (3)-x^6 \log ^2(3)+(4 x+x^2+x \log (3)) \log (x)} \, dx\)

Optimal. Leaf size=19 \[ 2+\log \left (x-\frac {\log (x)}{x^4 (4+x+\log (3))}\right ) \]

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Rubi [F]  time = 3.44, 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 {4+x-16 x^5-8 x^6-x^7+\left (1-8 x^5-2 x^6\right ) \log (3)-x^5 \log ^2(3)+(-16-5 x-4 \log (3)) \log (x)}{-16 x^6-8 x^7-x^8+\left (-8 x^6-2 x^7\right ) \log (3)-x^6 \log ^2(3)+\left (4 x+x^2+x \log (3)\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + x - 16*x^5 - 8*x^6 - x^7 + (1 - 8*x^5 - 2*x^6)*Log[3] - x^5*Log[3]^2 + (-16 - 5*x - 4*Log[3])*Log[x])
/(-16*x^6 - 8*x^7 - x^8 + (-8*x^6 - 2*x^7)*Log[3] - x^6*Log[3]^2 + (4*x + x^2 + x*Log[3])*Log[x]),x]

[Out]

-(((16 + Log[81])*Log[x])/(4 + Log[3])) - Log[4 + x + Log[3]] + 5*(4 + Log[3])*Defer[Int][x^4/(x^5*(4 + x + Lo
g[3]) - Log[x]), x] + 6*Defer[Int][x^5/(x^5*(4 + x + Log[3]) - Log[x]), x] + Log[3]^4*Log[243]*(16 + Log[3] +
(32*Log[6561])/Log[3]^3 - (1 + 480*Log[3]^4 + 80*Log[3]^5 + 5*Log[3]^6 + 32*Log[3]^3*(40 - Log[14348907]))/(Lo
g[3]^4*Log[243]))*Defer[Int][(x^6*(4 + x + Log[3]) - x*Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4+x-8 x^6-x^7+\left (1-8 x^5-2 x^6\right ) \log (3)+x^5 \left (-16-\log ^2(3)\right )+(-16-5 x-4 \log (3)) \log (x)}{-16 x^6-8 x^7-x^8+\left (-8 x^6-2 x^7\right ) \log (3)-x^6 \log ^2(3)+\left (4 x+x^2+x \log (3)\right ) \log (x)} \, dx\\ &=\int \frac {4+x-8 x^6-x^7+\left (1-8 x^5-2 x^6\right ) \log (3)+x^5 \left (-16-\log ^2(3)\right )+(-16-5 x-4 \log (3)) \log (x)}{-8 x^7-x^8+\left (-8 x^6-2 x^7\right ) \log (3)+x^6 \left (-16-\log ^2(3)\right )+\left (4 x+x^2+x \log (3)\right ) \log (x)} \, dx\\ &=\int \frac {-x+x^7-4 \left (1+\frac {\log (3)}{4}\right )+x^5 (4+\log (3))^2+x^6 (8+\log (9))+(16+5 x+\log (81)) \log (x)}{x (4+x+\log (3)) \left (x^5 (4+x+\log (3))-\log (x)\right )} \, dx\\ &=\int \left (\frac {-16-5 x-\log (81)}{x (4+x+\log (3))}+\frac {-x+6 x^7-4 \left (1+\frac {\log (3)}{4}\right )+44 x^6 \left (1+\frac {\log (3)}{4}\right )+80 x^5 \left (1+\frac {1}{80} \log (3) (40+\log (243))\right )}{x (4+x+\log (3)) \left (x^6+4 x^5 \left (1+\frac {\log (3)}{4}\right )-\log (x)\right )}\right ) \, dx\\ &=\int \frac {-16-5 x-\log (81)}{x (4+x+\log (3))} \, dx+\int \frac {-x+6 x^7-4 \left (1+\frac {\log (3)}{4}\right )+44 x^6 \left (1+\frac {\log (3)}{4}\right )+80 x^5 \left (1+\frac {1}{80} \log (3) (40+\log (243))\right )}{x (4+x+\log (3)) \left (x^6+4 x^5 \left (1+\frac {\log (3)}{4}\right )-\log (x)\right )} \, dx\\ &=\int \left (\frac {1}{-4-x-\log (3)}+\frac {-16-\log (81)}{x (4+\log (3))}\right ) \, dx+\int \frac {-1+6 x^6-1280 \log ^2(3)-1280 \log ^3(3)-480 \log ^4(3)-80 \log ^5(3)-5 \log ^6(3)+x^5 (20+5 \log (3))+256 \log (3) \log (243)+256 \log ^2(3) \log (243)+96 \log ^3(3) \log (243)+16 \log ^4(3) \log (243)+\log ^5(3) \log (243)+x^4 \left (-5 \log ^2(3)+\log (3) \log (243)\right )+x^3 \left (20 \log ^2(3)+5 \log ^3(3)-4 \log (3) \log (243)-\log ^2(3) \log (243)\right )+x^2 \left (-80 \log ^2(3)-40 \log ^3(3)-5 \log ^4(3)+16 \log (3) \log (243)+8 \log ^2(3) \log (243)+\log ^3(3) \log (243)\right )+x \left (320 \log ^2(3)+240 \log ^3(3)+60 \log ^4(3)+5 \log ^5(3)-64 \log (3) \log (243)-48 \log ^2(3) \log (243)-12 \log ^3(3) \log (243)-\log ^4(3) \log (243)\right )}{x \left (x^6+4 x^5 \left (1+\frac {\log (3)}{4}\right )-\log (x)\right )} \, dx\\ &=-\frac {(16+\log (81)) \log (x)}{4+\log (3)}-\log (4+x+\log (3))+\int \frac {-1+6 x^6-5 \log ^6(3)+5 x^5 (4+\log (3))+\log (3) \left (\log ^4(3) (-80+\log (243))+16 \log ^3(3) (-30+\log (243))+32 \log (243) \log (6561)+32 \log ^2(3) (-40+\log (14348907))\right )}{x^6 (4+x+\log (3))-x \log (x)} \, dx\\ &=-\frac {(16+\log (81)) \log (x)}{4+\log (3)}-\log (4+x+\log (3))+\int \left (\frac {6 x^5}{x^6+4 x^5 \left (1+\frac {\log (3)}{4}\right )-\log (x)}+\frac {5 x^4 (4+\log (3))}{x^6+4 x^5 \left (1+\frac {\log (3)}{4}\right )-\log (x)}+\frac {16 \log ^4(3) \log (243) \left (1+\frac {1}{16} \log (3) \left (1+\frac {32 \log (6561)}{\log ^4(3)}-\frac {1+480 \log ^4(3)+80 \log ^5(3)+5 \log ^6(3)-32 \log ^3(3) (-40+\log (14348907))}{\log ^5(3) \log (243)}\right )\right )}{x \left (x^6+4 x^5 \left (1+\frac {\log (3)}{4}\right )-\log (x)\right )}\right ) \, dx\\ &=-\frac {(16+\log (81)) \log (x)}{4+\log (3)}-\log (4+x+\log (3))+6 \int \frac {x^5}{x^6+4 x^5 \left (1+\frac {\log (3)}{4}\right )-\log (x)} \, dx+(5 (4+\log (3))) \int \frac {x^4}{x^6+4 x^5 \left (1+\frac {\log (3)}{4}\right )-\log (x)} \, dx+\left (16 \log ^4(3) \log (243) \left (1+\frac {1}{16} \log (3) \left (1+\frac {32 \log (6561)}{\log ^4(3)}-\frac {1+480 \log ^4(3)+80 \log ^5(3)+5 \log ^6(3)-32 \log ^3(3) (-40+\log (14348907))}{\log ^5(3) \log (243)}\right )\right )\right ) \int \frac {1}{x \left (x^6+4 x^5 \left (1+\frac {\log (3)}{4}\right )-\log (x)\right )} \, dx\\ &=-\frac {(16+\log (81)) \log (x)}{4+\log (3)}-\log (4+x+\log (3))+6 \int \frac {x^5}{x^5 (4+x+\log (3))-\log (x)} \, dx+(5 (4+\log (3))) \int \frac {x^4}{x^5 (4+x+\log (3))-\log (x)} \, dx+\left (16 \log ^4(3) \log (243) \left (1+\frac {1}{16} \log (3) \left (1+\frac {32 \log (6561)}{\log ^4(3)}-\frac {1+480 \log ^4(3)+80 \log ^5(3)+5 \log ^6(3)-32 \log ^3(3) (-40+\log (14348907))}{\log ^5(3) \log (243)}\right )\right )\right ) \int \frac {1}{x^6 (4+x+\log (3))-x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4+x-16 x^5-8 x^6-x^7+\left (1-8 x^5-2 x^6\right ) \log (3)-x^5 \log ^2(3)+(-16-5 x-4 \log (3)) \log (x)}{-16 x^6-8 x^7-x^8+\left (-8 x^6-2 x^7\right ) \log (3)-x^6 \log ^2(3)+\left (4 x+x^2+x \log (3)\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(4 + x - 16*x^5 - 8*x^6 - x^7 + (1 - 8*x^5 - 2*x^6)*Log[3] - x^5*Log[3]^2 + (-16 - 5*x - 4*Log[3])*L
og[x])/(-16*x^6 - 8*x^7 - x^8 + (-8*x^6 - 2*x^7)*Log[3] - x^6*Log[3]^2 + (4*x + x^2 + x*Log[3])*Log[x]),x]

[Out]

Integrate[(4 + x - 16*x^5 - 8*x^6 - x^7 + (1 - 8*x^5 - 2*x^6)*Log[3] - x^5*Log[3]^2 + (-16 - 5*x - 4*Log[3])*L
og[x])/(-16*x^6 - 8*x^7 - x^8 + (-8*x^6 - 2*x^7)*Log[3] - x^6*Log[3]^2 + (4*x + x^2 + x*Log[3])*Log[x]), x]

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fricas [A]  time = 0.85, size = 34, normalized size = 1.79 \begin {gather*} \log \left (-x^{6} - x^{5} \log \relax (3) - 4 \, x^{5} + \log \relax (x)\right ) - \log \left (x + \log \relax (3) + 4\right ) - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(3)-5*x-16)*log(x)-x^5*log(3)^2+(-2*x^6-8*x^5+1)*log(3)-x^7-8*x^6-16*x^5+x+4)/((x*log(3)+x^2
+4*x)*log(x)-x^6*log(3)^2+(-2*x^7-8*x^6)*log(3)-x^8-8*x^7-16*x^6),x, algorithm="fricas")

[Out]

log(-x^6 - x^5*log(3) - 4*x^5 + log(x)) - log(x + log(3) + 4) - 4*log(x)

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giac [A]  time = 0.31, size = 34, normalized size = 1.79 \begin {gather*} \log \left (-x^{6} - x^{5} \log \relax (3) - 4 \, x^{5} + \log \relax (x)\right ) - \log \left (x + \log \relax (3) + 4\right ) - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(3)-5*x-16)*log(x)-x^5*log(3)^2+(-2*x^6-8*x^5+1)*log(3)-x^7-8*x^6-16*x^5+x+4)/((x*log(3)+x^2
+4*x)*log(x)-x^6*log(3)^2+(-2*x^7-8*x^6)*log(3)-x^8-8*x^7-16*x^6),x, algorithm="giac")

[Out]

log(-x^6 - x^5*log(3) - 4*x^5 + log(x)) - log(x + log(3) + 4) - 4*log(x)

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maple [A]  time = 0.21, size = 34, normalized size = 1.79

method	result	size
norman	\(-4 \ln \relax (x )-\ln \left (\ln \relax (3)+4+x \right )+\ln \left (x^{5} \ln \relax (3)+x^{6}+4 x^{5}-\ln \relax (x )\right )\)	\(34\)
risch	\(-4 \ln \relax (x )-\ln \left (\ln \relax (3)+4+x \right )+\ln \left (-x^{5} \ln \relax (3)-x^{6}-4 x^{5}+\ln \relax (x )\right )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*ln(3)-5*x-16)*ln(x)-x^5*ln(3)^2+(-2*x^6-8*x^5+1)*ln(3)-x^7-8*x^6-16*x^5+x+4)/((x*ln(3)+x^2+4*x)*ln(x)
-x^6*ln(3)^2+(-2*x^7-8*x^6)*ln(3)-x^8-8*x^7-16*x^6),x,method=_RETURNVERBOSE)

[Out]

-4*ln(x)-ln(ln(3)+4+x)+ln(x^5*ln(3)+x^6+4*x^5-ln(x))

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maxima [A]  time = 0.54, size = 31, normalized size = 1.63 \begin {gather*} \log \left (-x^{6} - x^{5} {\left (\log \relax (3) + 4\right )} + \log \relax (x)\right ) - \log \left (x + \log \relax (3) + 4\right ) - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(3)-5*x-16)*log(x)-x^5*log(3)^2+(-2*x^6-8*x^5+1)*log(3)-x^7-8*x^6-16*x^5+x+4)/((x*log(3)+x^2
+4*x)*log(x)-x^6*log(3)^2+(-2*x^7-8*x^6)*log(3)-x^8-8*x^7-16*x^6),x, algorithm="maxima")

[Out]

log(-x^6 - x^5*(log(3) + 4) + log(x)) - log(x + log(3) + 4) - 4*log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x^5\,{\ln \relax (3)}^2-x+\ln \relax (x)\,\left (5\,x+4\,\ln \relax (3)+16\right )+\ln \relax (3)\,\left (2\,x^6+8\,x^5-1\right )+16\,x^5+8\,x^6+x^7-4}{x^6\,{\ln \relax (3)}^2-\ln \relax (x)\,\left (4\,x+x\,\ln \relax (3)+x^2\right )+\ln \relax (3)\,\left (2\,x^7+8\,x^6\right )+16\,x^6+8\,x^7+x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*log(3)^2 - x + log(x)*(5*x + 4*log(3) + 16) + log(3)*(8*x^5 + 2*x^6 - 1) + 16*x^5 + 8*x^6 + x^7 - 4)/
(x^6*log(3)^2 - log(x)*(4*x + x*log(3) + x^2) + log(3)*(8*x^6 + 2*x^7) + 16*x^6 + 8*x^7 + x^8),x)

[Out]

int((x^5*log(3)^2 - x + log(x)*(5*x + 4*log(3) + 16) + log(3)*(8*x^5 + 2*x^6 - 1) + 16*x^5 + 8*x^6 + x^7 - 4)/
(x^6*log(3)^2 - log(x)*(4*x + x*log(3) + x^2) + log(3)*(8*x^6 + 2*x^7) + 16*x^6 + 8*x^7 + x^8), x)

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sympy [A]  time = 0.65, size = 32, normalized size = 1.68 \begin {gather*} - 4 \log {\relax (x )} - \log {\left (x + \log {\relax (3 )} + 4 \right )} + \log {\left (- x^{6} - 4 x^{5} - x^{5} \log {\relax (3 )} + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*ln(3)-5*x-16)*ln(x)-x**5*ln(3)**2+(-2*x**6-8*x**5+1)*ln(3)-x**7-8*x**6-16*x**5+x+4)/((x*ln(3)+x
**2+4*x)*ln(x)-x**6*ln(3)**2+(-2*x**7-8*x**6)*ln(3)-x**8-8*x**7-16*x**6),x)

[Out]

-4*log(x) - log(x + log(3) + 4) + log(-x**6 - 4*x**5 - x**5*log(3) + log(x))

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