3.91.34 \(\int \frac {-18 x-2 x^3+16 x^4-6 x^3 \log (5)+(18 x-6 x^2+48 x^3-18 x^2 \log (5)) \log (x)+(-6 x+48 x^2-18 x \log (5)) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx\)

Optimal. Leaf size=28 \[ x^2-\left (\frac {1+x}{3}+\log (5)\right )^2+\frac {x^2}{(x+\log (x))^2} \]

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Rubi [F]  time = 0.42, 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 {-18 x-2 x^3+16 x^4-6 x^3 \log (5)+\left (18 x-6 x^2+48 x^3-18 x^2 \log (5)\right ) \log (x)+\left (-6 x+48 x^2-18 x \log (5)\right ) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-18*x - 2*x^3 + 16*x^4 - 6*x^3*Log[5] + (18*x - 6*x^2 + 48*x^3 - 18*x^2*Log[5])*Log[x] + (-6*x + 48*x^2 -
18*x*Log[5])*Log[x]^2 + (-2 + 16*x - 6*Log[5])*Log[x]^3)/(9*x^3 + 27*x^2*Log[x] + 27*x*Log[x]^2 + 9*Log[x]^3)
,x]

[Out]

(8*x^2)/9 - (2*x*(1 + Log[125]))/9 - 2*Defer[Int][x/(x + Log[x])^3, x] - 2*Defer[Int][x^2/(x + Log[x])^3, x] +
2*Defer[Int][x/(x + Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-18 x+16 x^4+x^3 (-2-6 \log (5))+\left (18 x-6 x^2+48 x^3-18 x^2 \log (5)\right ) \log (x)+\left (-6 x+48 x^2-18 x \log (5)\right ) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx\\ &=\int \frac {2 \left (x \left (-9+8 x^3-x^2 (1+\log (125))\right )+3 x \left (3+8 x^2-x (1+\log (125))\right ) \log (x)+3 x (-1+8 x-3 \log (5)) \log ^2(x)+(-1+8 x-3 \log (5)) \log ^3(x)\right )}{9 (x+\log (x))^3} \, dx\\ &=\frac {2}{9} \int \frac {x \left (-9+8 x^3-x^2 (1+\log (125))\right )+3 x \left (3+8 x^2-x (1+\log (125))\right ) \log (x)+3 x (-1+8 x-3 \log (5)) \log ^2(x)+(-1+8 x-3 \log (5)) \log ^3(x)}{(x+\log (x))^3} \, dx\\ &=\frac {2}{9} \int \left (-1+8 x-\log (125)-\frac {9 x (1+x)}{(x+\log (x))^3}+\frac {9 x}{(x+\log (x))^2}\right ) \, dx\\ &=\frac {8 x^2}{9}-\frac {2}{9} x (1+\log (125))-2 \int \frac {x (1+x)}{(x+\log (x))^3} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx\\ &=\frac {8 x^2}{9}-\frac {2}{9} x (1+\log (125))+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \left (\frac {x}{(x+\log (x))^3}+\frac {x^2}{(x+\log (x))^3}\right ) \, dx\\ &=\frac {8 x^2}{9}-\frac {2}{9} x (1+\log (125))-2 \int \frac {x}{(x+\log (x))^3} \, dx-2 \int \frac {x^2}{(x+\log (x))^3} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{9} x \left (-2 (1+\log (125))+x \left (8+\frac {9}{(x+\log (x))^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18*x - 2*x^3 + 16*x^4 - 6*x^3*Log[5] + (18*x - 6*x^2 + 48*x^3 - 18*x^2*Log[5])*Log[x] + (-6*x + 48
*x^2 - 18*x*Log[5])*Log[x]^2 + (-2 + 16*x - 6*Log[5])*Log[x]^3)/(9*x^3 + 27*x^2*Log[x] + 27*x*Log[x]^2 + 9*Log
[x]^3),x]

[Out]

(x*(-2*(1 + Log[125]) + x*(8 + 9/(x + Log[x])^2)))/9

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fricas [B]  time = 0.43, size = 82, normalized size = 2.93 \begin {gather*} \frac {8 \, x^{4} - 6 \, x^{3} \log \relax (5) - 2 \, x^{3} + 2 \, {\left (4 \, x^{2} - 3 \, x \log \relax (5) - x\right )} \log \relax (x)^{2} + 9 \, x^{2} + 4 \, {\left (4 \, x^{3} - 3 \, x^{2} \log \relax (5) - x^{2}\right )} \log \relax (x)}{9 \, {\left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*log(5)+16*x-2)*log(x)^3+(-18*x*log(5)+48*x^2-6*x)*log(x)^2+(-18*x^2*log(5)+48*x^3-6*x^2+18*x)*l
og(x)-6*x^3*log(5)+16*x^4-2*x^3-18*x)/(9*log(x)^3+27*x*log(x)^2+27*x^2*log(x)+9*x^3),x, algorithm="fricas")

[Out]

1/9*(8*x^4 - 6*x^3*log(5) - 2*x^3 + 2*(4*x^2 - 3*x*log(5) - x)*log(x)^2 + 9*x^2 + 4*(4*x^3 - 3*x^2*log(5) - x^
2)*log(x))/(x^2 + 2*x*log(x) + log(x)^2)

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giac [B]  time = 0.15, size = 54, normalized size = 1.93 \begin {gather*} \frac {8}{9} \, x^{2} - \frac {2}{9} \, x {\left (3 \, \log \relax (5) + 1\right )} + \frac {x^{3} + x^{2}}{x^{3} + 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*log(5)+16*x-2)*log(x)^3+(-18*x*log(5)+48*x^2-6*x)*log(x)^2+(-18*x^2*log(5)+48*x^3-6*x^2+18*x)*l
og(x)-6*x^3*log(5)+16*x^4-2*x^3-18*x)/(9*log(x)^3+27*x*log(x)^2+27*x^2*log(x)+9*x^3),x, algorithm="giac")

[Out]

8/9*x^2 - 2/9*x*(3*log(5) + 1) + (x^3 + x^2)/(x^3 + 2*x^2*log(x) + x*log(x)^2 + x^2 + 2*x*log(x) + log(x)^2)

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maple [A]  time = 0.13, size = 25, normalized size = 0.89

method	result	size
risch	\(-\frac {2 x \ln \relax (5)}{3}+\frac {8 x^{2}}{9}-\frac {2 x}{9}+\frac {x^{2}}{\left (x +\ln \relax (x )\right )^{2}}\)	\(25\)
default	\(\frac {-9 \ln \relax (x )^{2}-18 x \ln \relax (x )-2 x^{3}+8 x^{4}-2 x \ln \relax (x )^{2}+16 x^{3} \ln \relax (x )-4 x^{2} \ln \relax (x )+8 x^{2} \ln \relax (x )^{2}}{9 \left (x +\ln \relax (x )\right )^{2}}-\frac {2 x \ln \relax (5)}{3}\)	\(67\)
norman	\(\frac {-\ln \relax (x )^{2}+\left (-\frac {2}{9}-\frac {2 \ln \relax (5)}{3}\right ) x^{3}-2 x \ln \relax (x )+\left (-\frac {4}{9}-\frac {4 \ln \relax (5)}{3}\right ) x^{2} \ln \relax (x )+\left (-\frac {2}{9}-\frac {2 \ln \relax (5)}{3}\right ) x \ln \relax (x )^{2}+\frac {8 x^{4}}{9}+\frac {8 x^{2} \ln \relax (x )^{2}}{9}+\frac {16 x^{3} \ln \relax (x )}{9}}{\left (x +\ln \relax (x )\right )^{2}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*ln(5)+16*x-2)*ln(x)^3+(-18*x*ln(5)+48*x^2-6*x)*ln(x)^2+(-18*x^2*ln(5)+48*x^3-6*x^2+18*x)*ln(x)-6*x^3*
ln(5)+16*x^4-2*x^3-18*x)/(9*ln(x)^3+27*x*ln(x)^2+27*x^2*ln(x)+9*x^3),x,method=_RETURNVERBOSE)

[Out]

-2/3*x*ln(5)+8/9*x^2-2/9*x+x^2/(x+ln(x))^2

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maxima [B]  time = 0.47, size = 81, normalized size = 2.89 \begin {gather*} \frac {8 \, x^{4} - 2 \, x^{3} {\left (3 \, \log \relax (5) + 1\right )} + 2 \, {\left (4 \, x^{2} - x {\left (3 \, \log \relax (5) + 1\right )}\right )} \log \relax (x)^{2} + 9 \, x^{2} + 4 \, {\left (4 \, x^{3} - x^{2} {\left (3 \, \log \relax (5) + 1\right )}\right )} \log \relax (x)}{9 \, {\left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*log(5)+16*x-2)*log(x)^3+(-18*x*log(5)+48*x^2-6*x)*log(x)^2+(-18*x^2*log(5)+48*x^3-6*x^2+18*x)*l
og(x)-6*x^3*log(5)+16*x^4-2*x^3-18*x)/(9*log(x)^3+27*x*log(x)^2+27*x^2*log(x)+9*x^3),x, algorithm="maxima")

[Out]

1/9*(8*x^4 - 2*x^3*(3*log(5) + 1) + 2*(4*x^2 - x*(3*log(5) + 1))*log(x)^2 + 9*x^2 + 4*(4*x^3 - x^2*(3*log(5) +
1))*log(x))/(x^2 + 2*x*log(x) + log(x)^2)

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mupad [B]  time = 7.67, size = 89, normalized size = 3.18 \begin {gather*} \frac {8\,x^5+16\,x^4\,\ln \relax (x)+\left (-6\,\ln \relax (5)-2\right )\,x^4+8\,x^3\,{\ln \relax (x)}^2+\left (-12\,\ln \relax (5)-4\right )\,x^3\,\ln \relax (x)+9\,x^3+\left (-6\,\ln \relax (5)-2\right )\,x^2\,{\ln \relax (x)}^2}{9\,x^3+18\,x^2\,\ln \relax (x)+9\,x\,{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(18*x + log(x)^2*(6*x + 18*x*log(5) - 48*x^2) - log(x)*(18*x - 18*x^2*log(5) - 6*x^2 + 48*x^3) + 6*x^3*lo
g(5) + log(x)^3*(6*log(5) - 16*x + 2) + 2*x^3 - 16*x^4)/(27*x*log(x)^2 + 27*x^2*log(x) + 9*log(x)^3 + 9*x^3),x
)

[Out]

(16*x^4*log(x) - x^4*(6*log(5) + 2) + 8*x^3*log(x)^2 + 9*x^3 + 8*x^5 - x^3*log(x)*(12*log(5) + 4) - x^2*log(x)
^2*(6*log(5) + 2))/(9*x*log(x)^2 + 18*x^2*log(x) + 9*x^3)

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sympy [A]  time = 0.17, size = 37, normalized size = 1.32 \begin {gather*} \frac {8 x^{2}}{9} + \frac {x^{2}}{x^{2} + 2 x \log {\relax (x )} + \log {\relax (x )}^{2}} + x \left (- \frac {2 \log {\relax (5 )}}{3} - \frac {2}{9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*ln(5)+16*x-2)*ln(x)**3+(-18*x*ln(5)+48*x**2-6*x)*ln(x)**2+(-18*x**2*ln(5)+48*x**3-6*x**2+18*x)*
ln(x)-6*x**3*ln(5)+16*x**4-2*x**3-18*x)/(9*ln(x)**3+27*x*ln(x)**2+27*x**2*ln(x)+9*x**3),x)

[Out]

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

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