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

Optimal. Leaf size=22 \[ 2-\frac {4}{\left (x \left (1+x^2\right )-\log (3)\right ) \log (x)} \]

________________________________________________________________________________________

Rubi [F]  time = 0.50, 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+4 x^3-4 \log (3)+\left (4 x+12 x^3\right ) \log (x)}{\left (x^3+2 x^5+x^7+\left (-2 x^2-2 x^4\right ) \log (3)+x \log ^2(3)\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 19, normalized size = 0.86 \begin {gather*} \frac {4}{\left (-x-x^3+\log (3)\right ) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x + 4*x^3 - 4*Log[3] + (4*x + 12*x^3)*Log[x])/((x^3 + 2*x^5 + x^7 + (-2*x^2 - 2*x^4)*Log[3] + x*L
og[3]^2)*Log[x]^2),x]

[Out]

4/((-x - x^3 + Log[3])*Log[x])

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 17, normalized size = 0.77 \begin {gather*} -\frac {4}{{\left (x^{3} + x - \log \relax (3)\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/((x^3 + x - log(3))*log(x))

________________________________________________________________________________________

giac [A]  time = 0.15, size = 21, normalized size = 0.95 \begin {gather*} -\frac {4}{x^{3} \log \relax (x) + x \log \relax (x) - \log \relax (3) \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.15, size = 20, normalized size = 0.91

method	result	size
norman	\(\frac {4}{\left (-x^{3}+\ln \relax (3)-x \right ) \ln \relax (x )}\)	\(20\)
risch	\(\frac {4}{\left (-x^{3}+\ln \relax (3)-x \right ) \ln \relax (x )}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^3+4*x)*ln(x)-4*ln(3)+4*x^3+4*x)/(x*ln(3)^2+(-2*x^4-2*x^2)*ln(3)+x^7+2*x^5+x^3)/ln(x)^2,x,method=_RE
TURNVERBOSE)

[Out]

4/(-x^3+ln(3)-x)/ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.48, size = 17, normalized size = 0.77 \begin {gather*} -\frac {4}{{\left (x^{3} + x - \log \relax (3)\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/((x^3 + x - log(3))*log(x))

________________________________________________________________________________________

mupad [B]  time = 3.49, size = 30, normalized size = 1.36 \begin {gather*} -\frac {4\,x^3+4\,x-\ln \left (81\right )}{\ln \relax (x)\,{\left (x^3+x-\ln \relax (3)\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*x - log(81) + 4*x^3)/(log(x)*(x - log(3) + x^3)^2)

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 14, normalized size = 0.64 \begin {gather*} - \frac {4}{\left (x^{3} + x - \log {\relax (3 )}\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/((x**3 + x - log(3))*log(x))

________________________________________________________________________________________