3.64.30 \(\int \frac {-96-24 x+(64+32 x+4 x^2) \log (\frac {1}{x})+(-32-8 x) \log ^2(\frac {1}{x})+(-64-32 x-4 x^2+(64+16 x) \log (\frac {1}{x})) \log (x)+(-32-8 x) \log ^2(x)}{9 x^3+6 x^3 \log ^2(\frac {1}{x})+x^3 \log ^4(\frac {1}{x})+(-12 x^3 \log (\frac {1}{x})-4 x^3 \log ^3(\frac {1}{x})) \log (x)+(6 x^3+6 x^3 \log ^2(\frac {1}{x})) \log ^2(x)-4 x^3 \log (\frac {1}{x}) \log ^3(x)+x^3 \log ^4(x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {(4+x)^2}{x^2 \left (3+\left (-\log \left (\frac {1}{x}\right )+\log (x)\right )^2\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 2.66, 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 {-96-24 x+\left (64+32 x+4 x^2\right ) \log \left (\frac {1}{x}\right )+(-32-8 x) \log ^2\left (\frac {1}{x}\right )+\left (-64-32 x-4 x^2+(64+16 x) \log \left (\frac {1}{x}\right )\right ) \log (x)+(-32-8 x) \log ^2(x)}{9 x^3+6 x^3 \log ^2\left (\frac {1}{x}\right )+x^3 \log ^4\left (\frac {1}{x}\right )+\left (-12 x^3 \log \left (\frac {1}{x}\right )-4 x^3 \log ^3\left (\frac {1}{x}\right )\right ) \log (x)+\left (6 x^3+6 x^3 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)-4 x^3 \log \left (\frac {1}{x}\right ) \log ^3(x)+x^3 \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-96 - 24*x + (64 + 32*x + 4*x^2)*Log[x^(-1)] + (-32 - 8*x)*Log[x^(-1)]^2 + (-64 - 32*x - 4*x^2 + (64 + 16
*x)*Log[x^(-1)])*Log[x] + (-32 - 8*x)*Log[x]^2)/(9*x^3 + 6*x^3*Log[x^(-1)]^2 + x^3*Log[x^(-1)]^4 + (-12*x^3*Lo
g[x^(-1)] - 4*x^3*Log[x^(-1)]^3)*Log[x] + (6*x^3 + 6*x^3*Log[x^(-1)]^2)*Log[x]^2 - 4*x^3*Log[x^(-1)]*Log[x]^3
+ x^3*Log[x]^4),x]

[Out]

64*Defer[Int][Log[x^(-1)]/(x^3*(3 + Log[x^(-1)]^2 - 2*Log[x^(-1)]*Log[x] + Log[x]^2)^2), x] + 32*Defer[Int][Lo
g[x^(-1)]/(x^2*(3 + Log[x^(-1)]^2 - 2*Log[x^(-1)]*Log[x] + Log[x]^2)^2), x] + 4*Defer[Int][Log[x^(-1)]/(x*(3 +
Log[x^(-1)]^2 - 2*Log[x^(-1)]*Log[x] + Log[x]^2)^2), x] - 64*Defer[Int][Log[x]/(x^3*(3 + Log[x^(-1)]^2 - 2*Lo
g[x^(-1)]*Log[x] + Log[x]^2)^2), x] - 32*Defer[Int][Log[x]/(x^2*(3 + Log[x^(-1)]^2 - 2*Log[x^(-1)]*Log[x] + Lo
g[x]^2)^2), x] - 4*Defer[Int][Log[x]/(x*(3 + Log[x^(-1)]^2 - 2*Log[x^(-1)]*Log[x] + Log[x]^2)^2), x] - 32*Defe
r[Int][1/(x^3*(3 + Log[x^(-1)]^2 - 2*Log[x^(-1)]*Log[x] + Log[x]^2)), x] - 8*Defer[Int][1/(x^2*(3 + Log[x^(-1)
]^2 - 2*Log[x^(-1)]*Log[x] + Log[x]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 (4+x) \left (-6-2 \log ^2\left (\frac {1}{x}\right )-(4+x) \log (x)-2 \log ^2(x)+\log \left (\frac {1}{x}\right ) (4+x+4 \log (x))\right )}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx\\ &=4 \int \frac {(4+x) \left (-6-2 \log ^2\left (\frac {1}{x}\right )-(4+x) \log (x)-2 \log ^2(x)+\log \left (\frac {1}{x}\right ) (4+x+4 \log (x))\right )}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx\\ &=4 \int \left (\frac {(4+x)^2 \left (\log \left (\frac {1}{x}\right )-\log (x)\right )}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2}-\frac {2 (4+x)}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )}\right ) \, dx\\ &=4 \int \frac {(4+x)^2 \left (\log \left (\frac {1}{x}\right )-\log (x)\right )}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx-8 \int \frac {4+x}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )} \, dx\\ &=4 \int \left (\frac {16 \left (\log \left (\frac {1}{x}\right )-\log (x)\right )}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2}+\frac {8 \left (\log \left (\frac {1}{x}\right )-\log (x)\right )}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2}+\frac {\log \left (\frac {1}{x}\right )-\log (x)}{x \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2}\right ) \, dx-8 \int \left (\frac {4}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )}+\frac {1}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )}\right ) \, dx\\ &=4 \int \frac {\log \left (\frac {1}{x}\right )-\log (x)}{x \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx-8 \int \frac {1}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )} \, dx+32 \int \frac {\log \left (\frac {1}{x}\right )-\log (x)}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx-32 \int \frac {1}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )} \, dx+64 \int \frac {\log \left (\frac {1}{x}\right )-\log (x)}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx\\ &=4 \int \left (\frac {\log \left (\frac {1}{x}\right )}{x \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2}-\frac {\log (x)}{x \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2}\right ) \, dx-8 \int \frac {1}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )} \, dx-32 \int \frac {1}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )} \, dx+32 \int \left (\frac {\log \left (\frac {1}{x}\right )}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2}-\frac {\log (x)}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2}\right ) \, dx+64 \int \left (\frac {\log \left (\frac {1}{x}\right )}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2}-\frac {\log (x)}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2}\right ) \, dx\\ &=4 \int \frac {\log \left (\frac {1}{x}\right )}{x \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx-4 \int \frac {\log (x)}{x \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx-8 \int \frac {1}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )} \, dx+32 \int \frac {\log \left (\frac {1}{x}\right )}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx-32 \int \frac {\log (x)}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx-32 \int \frac {1}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )} \, dx+64 \int \frac {\log \left (\frac {1}{x}\right )}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx-64 \int \frac {\log (x)}{x^3 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.49, size = 31, normalized size = 1.29 \begin {gather*} \frac {(4+x)^2}{x^2 \left (3+\log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right ) \log (x)+\log ^2(x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-96 - 24*x + (64 + 32*x + 4*x^2)*Log[x^(-1)] + (-32 - 8*x)*Log[x^(-1)]^2 + (-64 - 32*x - 4*x^2 + (6
4 + 16*x)*Log[x^(-1)])*Log[x] + (-32 - 8*x)*Log[x]^2)/(9*x^3 + 6*x^3*Log[x^(-1)]^2 + x^3*Log[x^(-1)]^4 + (-12*
x^3*Log[x^(-1)] - 4*x^3*Log[x^(-1)]^3)*Log[x] + (6*x^3 + 6*x^3*Log[x^(-1)]^2)*Log[x]^2 - 4*x^3*Log[x^(-1)]*Log
[x]^3 + x^3*Log[x]^4),x]

[Out]

(4 + x)^2/(x^2*(3 + Log[x^(-1)]^2 - 2*Log[x^(-1)]*Log[x] + Log[x]^2))

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 28, normalized size = 1.17 \begin {gather*} \frac {x^{2} + 8 \, x + 16}{4 \, x^{2} \log \left (\frac {1}{x}\right )^{2} + 3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-32)*log(x)^2+((16*x+64)*log(1/x)-4*x^2-32*x-64)*log(x)+(-8*x-32)*log(1/x)^2+(4*x^2+32*x+64)*l
og(1/x)-24*x-96)/(x^3*log(x)^4-4*x^3*log(1/x)*log(x)^3+(6*x^3*log(1/x)^2+6*x^3)*log(x)^2+(-4*x^3*log(1/x)^3-12
*x^3*log(1/x))*log(x)+x^3*log(1/x)^4+6*x^3*log(1/x)^2+9*x^3),x, algorithm="fricas")

[Out]

(x^2 + 8*x + 16)/(4*x^2*log(1/x)^2 + 3*x^2)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 26, normalized size = 1.08 \begin {gather*} \frac {x^{2} + 8 \, x + 16}{4 \, x^{2} \log \relax (x)^{2} + 3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-32)*log(x)^2+((16*x+64)*log(1/x)-4*x^2-32*x-64)*log(x)+(-8*x-32)*log(1/x)^2+(4*x^2+32*x+64)*l
og(1/x)-24*x-96)/(x^3*log(x)^4-4*x^3*log(1/x)*log(x)^3+(6*x^3*log(1/x)^2+6*x^3)*log(x)^2+(-4*x^3*log(1/x)^3-12
*x^3*log(1/x))*log(x)+x^3*log(1/x)^4+6*x^3*log(1/x)^2+9*x^3),x, algorithm="giac")

[Out]

(x^2 + 8*x + 16)/(4*x^2*log(x)^2 + 3*x^2)

________________________________________________________________________________________

maple [A]  time = 0.25, size = 23, normalized size = 0.96

method	result	size
risch	\(\frac {x^{2}+8 x +16}{x^{2} \left (4 \ln \relax (x )^{2}+3\right )}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x-32)*ln(x)^2+((16*x+64)*ln(1/x)-4*x^2-32*x-64)*ln(x)+(-8*x-32)*ln(1/x)^2+(4*x^2+32*x+64)*ln(1/x)-24*
x-96)/(x^3*ln(x)^4-4*x^3*ln(1/x)*ln(x)^3+(6*x^3*ln(1/x)^2+6*x^3)*ln(x)^2+(-4*x^3*ln(1/x)^3-12*x^3*ln(1/x))*ln(
x)+x^3*ln(1/x)^4+6*x^3*ln(1/x)^2+9*x^3),x,method=_RETURNVERBOSE)

[Out]

(x^2+8*x+16)/x^2/(4*ln(x)^2+3)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 26, normalized size = 1.08 \begin {gather*} \frac {x^{2} + 8 \, x + 16}{4 \, x^{2} \log \relax (x)^{2} + 3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-32)*log(x)^2+((16*x+64)*log(1/x)-4*x^2-32*x-64)*log(x)+(-8*x-32)*log(1/x)^2+(4*x^2+32*x+64)*l
og(1/x)-24*x-96)/(x^3*log(x)^4-4*x^3*log(1/x)*log(x)^3+(6*x^3*log(1/x)^2+6*x^3)*log(x)^2+(-4*x^3*log(1/x)^3-12
*x^3*log(1/x))*log(x)+x^3*log(1/x)^4+6*x^3*log(1/x)^2+9*x^3),x, algorithm="maxima")

[Out]

(x^2 + 8*x + 16)/(4*x^2*log(x)^2 + 3*x^2)

________________________________________________________________________________________

mupad [B]  time = 4.60, size = 46, normalized size = 1.92 \begin {gather*} \frac {x^2+8\,x+16}{x^2\,{\ln \left (\frac {1}{x}\right )}^2-2\,x^2\,\ln \left (\frac {1}{x}\right )\,\ln \relax (x)+x^2\,{\ln \relax (x)}^2+3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(24*x - log(1/x)*(32*x + 4*x^2 + 64) + log(1/x)^2*(8*x + 32) + log(x)*(32*x + 4*x^2 - log(1/x)*(16*x + 64
) + 64) + log(x)^2*(8*x + 32) + 96)/(log(x)^2*(6*x^3 + 6*x^3*log(1/x)^2) + x^3*log(x)^4 + 9*x^3 + 6*x^3*log(1/
x)^2 + x^3*log(1/x)^4 - log(x)*(12*x^3*log(1/x) + 4*x^3*log(1/x)^3) - 4*x^3*log(1/x)*log(x)^3),x)

[Out]

(8*x + x^2 + 16)/(x^2*log(x)^2 + 3*x^2 + x^2*log(1/x)^2 - 2*x^2*log(1/x)*log(x))

________________________________________________________________________________________

sympy [A]  time = 0.31, size = 22, normalized size = 0.92 \begin {gather*} \frac {x^{2} + 8 x + 16}{4 x^{2} \log {\relax (x )}^{2} + 3 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-32)*ln(x)**2+((16*x+64)*ln(1/x)-4*x**2-32*x-64)*ln(x)+(-8*x-32)*ln(1/x)**2+(4*x**2+32*x+64)*l
n(1/x)-24*x-96)/(x**3*ln(x)**4-4*x**3*ln(1/x)*ln(x)**3+(6*x**3*ln(1/x)**2+6*x**3)*ln(x)**2+(-4*x**3*ln(1/x)**3
-12*x**3*ln(1/x))*ln(x)+x**3*ln(1/x)**4+6*x**3*ln(1/x)**2+9*x**3),x)

[Out]

(x**2 + 8*x + 16)/(4*x**2*log(x)**2 + 3*x**2)

________________________________________________________________________________________