3.87.53 \(\int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+(-2 x^5 \log (3)+14 x^3 \log ^2(3)) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx\)

Optimal. Leaf size=22 \[ \left (x-\frac {x}{2-\frac {x^2 \log (x)}{\log (3)}}\right )^2 \]

________________________________________________________________________________________

Rubi [F]  time = 0.89, 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 {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x^3*Log[3]^2 - 4*x*Log[3]^3 + (-2*x^5*Log[3] + 14*x^3*Log[3]^2)*Log[x] - 12*x^5*Log[3]*Log[x]^2 + 2*x^7
*Log[x]^3)/(-8*Log[3]^3 + 12*x^2*Log[3]^2*Log[x] - 6*x^4*Log[3]*Log[x]^2 + x^6*Log[x]^3),x]

[Out]

x^2 - 2*Log[9]*(6*Log[3]^2 - 6*Log[3]*Log[9] + Log[9]^2)*Defer[Int][x/(Log[9] - x^2*Log[x])^3, x] + 2*Log[3]^2
*Defer[Int][x^3/(Log[9] - x^2*Log[x])^3, x] + 2*(7*Log[3]^2 - 12*Log[3]*Log[9] + 3*Log[9]^2)*Defer[Int][x/(Log
[9] - x^2*Log[x])^2, x] - 2*Log[3]*Defer[Int][x^3/(Log[9] - x^2*Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (x \log ^2(3) \left (-x^2+\log (9)\right )+x^3 \left (x^2-7 \log (3)\right ) \log (3) \log (x)+6 x^5 \log (3) \log ^2(x)-x^7 \log ^3(x)\right )}{\left (\log (9)-x^2 \log (x)\right )^3} \, dx\\ &=2 \int \frac {x \log ^2(3) \left (-x^2+\log (9)\right )+x^3 \left (x^2-7 \log (3)\right ) \log (3) \log (x)+6 x^5 \log (3) \log ^2(x)-x^7 \log ^3(x)}{\left (\log (9)-x^2 \log (x)\right )^3} \, dx\\ &=2 \int \left (x+\frac {x \left (x^2 \log ^2(3)-\log (9) \left (6 \log ^2(3)-6 \log (3) \log (9)+\log ^2(9)\right )\right )}{\left (\log (9)-x^2 \log (x)\right )^3}-\frac {x \left (x^2 \log (3)-7 \log ^2(3)+12 \log (3) \log (9)-3 \log ^2(9)\right )}{\left (\log (9)-x^2 \log (x)\right )^2}\right ) \, dx\\ &=x^2+2 \int \frac {x \left (x^2 \log ^2(3)-\log (9) \left (6 \log ^2(3)-6 \log (3) \log (9)+\log ^2(9)\right )\right )}{\left (\log (9)-x^2 \log (x)\right )^3} \, dx-2 \int \frac {x \left (x^2 \log (3)-7 \log ^2(3)+12 \log (3) \log (9)-3 \log ^2(9)\right )}{\left (\log (9)-x^2 \log (x)\right )^2} \, dx\\ &=x^2+2 \int \left (\frac {x^3 \log ^2(3)}{\left (\log (9)-x^2 \log (x)\right )^3}-\frac {x \log (9) \left (6 \log ^2(3)-6 \log (3) \log (9)+\log ^2(9)\right )}{\left (\log (9)-x^2 \log (x)\right )^3}\right ) \, dx-2 \int \left (\frac {x^3 \log (3)}{\left (\log (9)-x^2 \log (x)\right )^2}-\frac {x \left (7 \log ^2(3)-12 \log (3) \log (9)+3 \log ^2(9)\right )}{\left (\log (9)-x^2 \log (x)\right )^2}\right ) \, dx\\ &=x^2-(2 \log (3)) \int \frac {x^3}{\left (\log (9)-x^2 \log (x)\right )^2} \, dx+\left (2 \log ^2(3)\right ) \int \frac {x^3}{\left (\log (9)-x^2 \log (x)\right )^3} \, dx-\left (2 \log (9) \left (6 \log ^2(3)-6 \log (3) \log (9)+\log ^2(9)\right )\right ) \int \frac {x}{\left (\log (9)-x^2 \log (x)\right )^3} \, dx+\left (2 \left (7 \log ^2(3)-12 \log (3) \log (9)+3 \log ^2(9)\right )\right ) \int \frac {x}{\left (\log (9)-x^2 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.58, size = 182, normalized size = 8.27 \begin {gather*} \frac {x^2 \left (8 \log ^2(3) \log ^3(9)-x^6 \left (\log ^2(3)+\log (3) \log (9)-\log ^2(9)\right )+4 x^2 \log ^2(9) \left (\log ^2(3)-\log (3) \log (9)+\log ^2(9)\right )+x^4 \left (\log ^3(9)+\log ^2(3) \log (81)\right )-\left (x^8 \log (9)+32 x^2 \log ^2(3) \log ^2(9)+6 x^6 \left (2 \log ^2(3)-\log (3) \log (9)+\log ^2(9)\right )+8 x^4 \log (9) \left (4 \log ^2(3)-\log (3) \log (9)+\log ^2(9)\right )\right ) \log (x)+x^4 \left (x^2+\log (81)\right )^3 \log ^2(x)\right )}{\left (x^2+\log (81)\right )^3 \left (\log (9)-x^2 \log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^3*Log[3]^2 - 4*x*Log[3]^3 + (-2*x^5*Log[3] + 14*x^3*Log[3]^2)*Log[x] - 12*x^5*Log[3]*Log[x]^2 +
2*x^7*Log[x]^3)/(-8*Log[3]^3 + 12*x^2*Log[3]^2*Log[x] - 6*x^4*Log[3]*Log[x]^2 + x^6*Log[x]^3),x]

[Out]

(x^2*(8*Log[3]^2*Log[9]^3 - x^6*(Log[3]^2 + Log[3]*Log[9] - Log[9]^2) + 4*x^2*Log[9]^2*(Log[3]^2 - Log[3]*Log[
9] + Log[9]^2) + x^4*(Log[9]^3 + Log[3]^2*Log[81]) - (x^8*Log[9] + 32*x^2*Log[3]^2*Log[9]^2 + 6*x^6*(2*Log[3]^
2 - Log[3]*Log[9] + Log[9]^2) + 8*x^4*Log[9]*(4*Log[3]^2 - Log[3]*Log[9] + Log[9]^2))*Log[x] + x^4*(x^2 + Log[
81])^3*Log[x]^2))/((x^2 + Log[81])^3*(Log[9] - x^2*Log[x])^2)

________________________________________________________________________________________

fricas [B]  time = 0.58, size = 53, normalized size = 2.41 \begin {gather*} \frac {x^{6} \log \relax (x)^{2} - 2 \, x^{4} \log \relax (3) \log \relax (x) + x^{2} \log \relax (3)^{2}}{x^{4} \log \relax (x)^{2} - 4 \, x^{2} \log \relax (3) \log \relax (x) + 4 \, \log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7*log(x)^3-12*x^5*log(3)*log(x)^2+(14*x^3*log(3)^2-2*x^5*log(3))*log(x)-4*x*log(3)^3+2*x^3*log(
3)^2)/(x^6*log(x)^3-6*x^4*log(3)*log(x)^2+12*x^2*log(3)^2*log(x)-8*log(3)^3),x, algorithm="fricas")

[Out]

(x^6*log(x)^2 - 2*x^4*log(3)*log(x) + x^2*log(3)^2)/(x^4*log(x)^2 - 4*x^2*log(3)*log(x) + 4*log(3)^2)

________________________________________________________________________________________

giac [B]  time = 0.15, size = 50, normalized size = 2.27 \begin {gather*} x^{2} + \frac {2 \, x^{4} \log \relax (3) \log \relax (x) - 3 \, x^{2} \log \relax (3)^{2}}{x^{4} \log \relax (x)^{2} - 4 \, x^{2} \log \relax (3) \log \relax (x) + 4 \, \log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7*log(x)^3-12*x^5*log(3)*log(x)^2+(14*x^3*log(3)^2-2*x^5*log(3))*log(x)-4*x*log(3)^3+2*x^3*log(
3)^2)/(x^6*log(x)^3-6*x^4*log(3)*log(x)^2+12*x^2*log(3)^2*log(x)-8*log(3)^3),x, algorithm="giac")

[Out]

x^2 + (2*x^4*log(3)*log(x) - 3*x^2*log(3)^2)/(x^4*log(x)^2 - 4*x^2*log(3)*log(x) + 4*log(3)^2)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 38, normalized size = 1.73

method	result	size
risch	\(x^{2}-\frac {\left (-2 x^{2} \ln \relax (x )+3 \ln \relax (3)\right ) \ln \relax (3) x^{2}}{\left (-x^{2} \ln \relax (x )+2 \ln \relax (3)\right )^{2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^7*ln(x)^3-12*x^5*ln(3)*ln(x)^2+(14*x^3*ln(3)^2-2*x^5*ln(3))*ln(x)-4*x*ln(3)^3+2*x^3*ln(3)^2)/(x^6*ln(
x)^3-6*x^4*ln(3)*ln(x)^2+12*x^2*ln(3)^2*ln(x)-8*ln(3)^3),x,method=_RETURNVERBOSE)

[Out]

x^2-(-2*x^2*ln(x)+3*ln(3))*ln(3)*x^2/(-x^2*ln(x)+2*ln(3))^2

________________________________________________________________________________________

maxima [B]  time = 0.50, size = 53, normalized size = 2.41 \begin {gather*} \frac {x^{6} \log \relax (x)^{2} - 2 \, x^{4} \log \relax (3) \log \relax (x) + x^{2} \log \relax (3)^{2}}{x^{4} \log \relax (x)^{2} - 4 \, x^{2} \log \relax (3) \log \relax (x) + 4 \, \log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7*log(x)^3-12*x^5*log(3)*log(x)^2+(14*x^3*log(3)^2-2*x^5*log(3))*log(x)-4*x*log(3)^3+2*x^3*log(
3)^2)/(x^6*log(x)^3-6*x^4*log(3)*log(x)^2+12*x^2*log(3)^2*log(x)-8*log(3)^3),x, algorithm="maxima")

[Out]

(x^6*log(x)^2 - 2*x^4*log(3)*log(x) + x^2*log(3)^2)/(x^4*log(x)^2 - 4*x^2*log(3)*log(x) + 4*log(3)^2)

________________________________________________________________________________________

mupad [B]  time = 5.79, size = 229, normalized size = 10.41 \begin {gather*} \frac {\frac {52\,x^2\,{\ln \relax (3)}^3+11\,x^4\,{\ln \relax (3)}^2-8\,{\ln \relax (3)}^3\,\ln \relax (9)+x^6\,\ln \relax (3)+64\,{\ln \relax (3)}^4-4\,x^2\,{\ln \relax (3)}^2\,\ln \relax (9)}{{\left (x^2+4\,\ln \relax (3)\right )}^3}-\frac {2\,x^4\,{\ln \relax (3)}^2\,\ln \relax (x)}{{\left (x^2+4\,\ln \relax (3)\right )}^3}}{\ln \relax (x)-\frac {2\,\ln \relax (3)}{x^2}}+x^2-\frac {\frac {{\ln \relax (3)}^2\,x^2-{\ln \relax (3)}^2\,\ln \relax (9)+8\,{\ln \relax (3)}^3}{x^2\,\left (x^2+4\,\ln \relax (3)\right )}-\frac {\ln \relax (x)\,\left (\ln \relax (3)\,x^2+5\,{\ln \relax (3)}^2\right )}{x^2+4\,\ln \relax (3)}}{\frac {4\,{\ln \relax (3)}^2}{x^4}+{\ln \relax (x)}^2-\frac {4\,\ln \relax (3)\,\ln \relax (x)}{x^2}}+\frac {2\,x^4\,{\ln \relax (3)}^2}{x^6+12\,\ln \relax (3)\,x^4+48\,{\ln \relax (3)}^2\,x^2+64\,{\ln \relax (3)}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3*log(3)^2 + log(x)*(14*x^3*log(3)^2 - 2*x^5*log(3)) + 2*x^7*log(x)^3 - 4*x*log(3)^3 - 12*x^5*log(3)*
log(x)^2)/(x^6*log(x)^3 - 8*log(3)^3 + 12*x^2*log(3)^2*log(x) - 6*x^4*log(3)*log(x)^2),x)

[Out]

((52*x^2*log(3)^3 + 11*x^4*log(3)^2 - 8*log(3)^3*log(9) + x^6*log(3) + 64*log(3)^4 - 4*x^2*log(3)^2*log(9))/(4
*log(3) + x^2)^3 - (2*x^4*log(3)^2*log(x))/(4*log(3) + x^2)^3)/(log(x) - (2*log(3))/x^2) + x^2 - ((x^2*log(3)^
2 - log(3)^2*log(9) + 8*log(3)^3)/(x^2*(4*log(3) + x^2)) - (log(x)*(x^2*log(3) + 5*log(3)^2))/(4*log(3) + x^2)
)/((4*log(3)^2)/x^4 + log(x)^2 - (4*log(3)*log(x))/x^2) + (2*x^4*log(3)^2)/(48*x^2*log(3)^2 + 12*x^4*log(3) +
64*log(3)^3 + x^6)

________________________________________________________________________________________

sympy [B]  time = 0.17, size = 51, normalized size = 2.32 \begin {gather*} x^{2} + \frac {2 x^{4} \log {\relax (3 )} \log {\relax (x )} - 3 x^{2} \log {\relax (3 )}^{2}}{x^{4} \log {\relax (x )}^{2} - 4 x^{2} \log {\relax (3 )} \log {\relax (x )} + 4 \log {\relax (3 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**7*ln(x)**3-12*x**5*ln(3)*ln(x)**2+(14*x**3*ln(3)**2-2*x**5*ln(3))*ln(x)-4*x*ln(3)**3+2*x**3*ln
(3)**2)/(x**6*ln(x)**3-6*x**4*ln(3)*ln(x)**2+12*x**2*ln(3)**2*ln(x)-8*ln(3)**3),x)

[Out]

x**2 + (2*x**4*log(3)*log(x) - 3*x**2*log(3)**2)/(x**4*log(x)**2 - 4*x**2*log(3)*log(x) + 4*log(3)**2)

________________________________________________________________________________________