3.47.38 \(\int \frac {-8 x+(-24-8 x-8 x^2 \log (3)) \log (3+x)+(-24 x-8 x^2) \log (3) \log ^2(3+x)+(3 x^3+x^4) \log ^3(3+x)}{(12 x^3+4 x^4) \log ^3(3+x)} \, dx\)

Optimal. Leaf size=21 \[ \frac {x}{4}+\left (\log (3)+\frac {1}{x \log (3+x)}\right )^2 \]

________________________________________________________________________________________

Rubi [F]  time = 0.74, 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 {-8 x+\left (-24-8 x-8 x^2 \log (3)\right ) \log (3+x)+\left (-24 x-8 x^2\right ) \log (3) \log ^2(3+x)+\left (3 x^3+x^4\right ) \log ^3(3+x)}{\left (12 x^3+4 x^4\right ) \log ^3(3+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*x + (-24 - 8*x - 8*x^2*Log[3])*Log[3 + x] + (-24*x - 8*x^2)*Log[3]*Log[3 + x]^2 + (3*x^3 + x^4)*Log[3
+ x]^3)/((12*x^3 + 4*x^4)*Log[3 + x]^3),x]

[Out]

x/4 + 1/(9*Log[3 + x]^2) - (2*Log[3])/(3*Log[3 + x]) - (2*Defer[Int][1/(x^2*Log[3 + x]^3), x])/3 + (2*Defer[In
t][1/(x*Log[3 + x]^3), x])/9 - 2*Defer[Int][1/(x^3*Log[3 + x]^2), x] - (2*Log[3]*Defer[Int][1/(x*Log[3 + x]^2)
, x])/3 - Log[9]*Defer[Int][1/(x^2*Log[3 + x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8 x+\left (-24-8 x-8 x^2 \log (3)\right ) \log (3+x)+\left (-24 x-8 x^2\right ) \log (3) \log ^2(3+x)+\left (3 x^3+x^4\right ) \log ^3(3+x)}{x^3 (12+4 x) \log ^3(3+x)} \, dx\\ &=\int \left (\frac {1}{4}-\frac {2}{x^2 (3+x) \log ^3(3+x)}-\frac {2 \left (3+x+x^2 \log (3)\right )}{x^3 (3+x) \log ^2(3+x)}-\frac {\log (9)}{x^2 \log (3+x)}\right ) \, dx\\ &=\frac {x}{4}-2 \int \frac {1}{x^2 (3+x) \log ^3(3+x)} \, dx-2 \int \frac {3+x+x^2 \log (3)}{x^3 (3+x) \log ^2(3+x)} \, dx-\log (9) \int \frac {1}{x^2 \log (3+x)} \, dx\\ &=\frac {x}{4}-2 \int \left (\frac {1}{3 x^2 \log ^3(3+x)}-\frac {1}{9 x \log ^3(3+x)}+\frac {1}{9 (3+x) \log ^3(3+x)}\right ) \, dx-2 \int \left (\frac {1}{x^3 \log ^2(3+x)}+\frac {\log (3)}{3 x \log ^2(3+x)}-\frac {\log (3)}{3 (3+x) \log ^2(3+x)}\right ) \, dx-\log (9) \int \frac {1}{x^2 \log (3+x)} \, dx\\ &=\frac {x}{4}+\frac {2}{9} \int \frac {1}{x \log ^3(3+x)} \, dx-\frac {2}{9} \int \frac {1}{(3+x) \log ^3(3+x)} \, dx-\frac {2}{3} \int \frac {1}{x^2 \log ^3(3+x)} \, dx-2 \int \frac {1}{x^3 \log ^2(3+x)} \, dx-\frac {1}{3} (2 \log (3)) \int \frac {1}{x \log ^2(3+x)} \, dx+\frac {1}{3} (2 \log (3)) \int \frac {1}{(3+x) \log ^2(3+x)} \, dx-\log (9) \int \frac {1}{x^2 \log (3+x)} \, dx\\ &=\frac {x}{4}+\frac {2}{9} \int \frac {1}{x \log ^3(3+x)} \, dx-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{x \log ^3(x)} \, dx,x,3+x\right )-\frac {2}{3} \int \frac {1}{x^2 \log ^3(3+x)} \, dx-2 \int \frac {1}{x^3 \log ^2(3+x)} \, dx-\frac {1}{3} (2 \log (3)) \int \frac {1}{x \log ^2(3+x)} \, dx+\frac {1}{3} (2 \log (3)) \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,3+x\right )-\log (9) \int \frac {1}{x^2 \log (3+x)} \, dx\\ &=\frac {x}{4}+\frac {2}{9} \int \frac {1}{x \log ^3(3+x)} \, dx-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (3+x)\right )-\frac {2}{3} \int \frac {1}{x^2 \log ^3(3+x)} \, dx-2 \int \frac {1}{x^3 \log ^2(3+x)} \, dx-\frac {1}{3} (2 \log (3)) \int \frac {1}{x \log ^2(3+x)} \, dx+\frac {1}{3} (2 \log (3)) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (3+x)\right )-\log (9) \int \frac {1}{x^2 \log (3+x)} \, dx\\ &=\frac {x}{4}+\frac {1}{9 \log ^2(3+x)}-\frac {2 \log (3)}{3 \log (3+x)}+\frac {2}{9} \int \frac {1}{x \log ^3(3+x)} \, dx-\frac {2}{3} \int \frac {1}{x^2 \log ^3(3+x)} \, dx-2 \int \frac {1}{x^3 \log ^2(3+x)} \, dx-\frac {1}{3} (2 \log (3)) \int \frac {1}{x \log ^2(3+x)} \, dx-\log (9) \int \frac {1}{x^2 \log (3+x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.89, size = 29, normalized size = 1.38 \begin {gather*} \frac {x}{4}+\frac {1}{x^2 \log ^2(3+x)}+\frac {2 \log (3)}{x \log (3+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x + (-24 - 8*x - 8*x^2*Log[3])*Log[3 + x] + (-24*x - 8*x^2)*Log[3]*Log[3 + x]^2 + (3*x^3 + x^4)*
Log[3 + x]^3)/((12*x^3 + 4*x^4)*Log[3 + x]^3),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.66, size = 32, normalized size = 1.52 \begin {gather*} \frac {x^{3} \log \left (x + 3\right )^{2} + 8 \, x \log \relax (3) \log \left (x + 3\right ) + 4}{4 \, x^{2} \log \left (x + 3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+3*x^3)*log(3+x)^3+(-8*x^2-24*x)*log(3)*log(3+x)^2+(-8*x^2*log(3)-8*x-24)*log(3+x)-8*x)/(4*x^4+
12*x^3)/log(3+x)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.19, size = 25, normalized size = 1.19 \begin {gather*} \frac {1}{4} \, x + \frac {2 \, x \log \relax (3) \log \left (x + 3\right ) + 1}{x^{2} \log \left (x + 3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+3*x^3)*log(3+x)^3+(-8*x^2-24*x)*log(3)*log(3+x)^2+(-8*x^2*log(3)-8*x-24)*log(3+x)-8*x)/(4*x^4+
12*x^3)/log(3+x)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.14, size = 26, normalized size = 1.24

method	result	size
risch	\(\frac {x}{4}+\frac {2 x \ln \relax (3) \ln \left (3+x \right )+1}{x^{2} \ln \left (3+x \right )^{2}}\)	\(26\)
norman	\(\frac {1+\frac {x^{3} \ln \left (3+x \right )^{2}}{4}+2 x \ln \relax (3) \ln \left (3+x \right )}{\ln \left (3+x \right )^{2} x^{2}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4+3*x^3)*ln(3+x)^3+(-8*x^2-24*x)*ln(3)*ln(3+x)^2+(-8*x^2*ln(3)-8*x-24)*ln(3+x)-8*x)/(4*x^4+12*x^3)/ln(
3+x)^3,x,method=_RETURNVERBOSE)

[Out]

1/4*x+1/x^2*(2*x*ln(3)*ln(3+x)+1)/ln(3+x)^2

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 32, normalized size = 1.52 \begin {gather*} \frac {x^{3} \log \left (x + 3\right )^{2} + 8 \, x \log \relax (3) \log \left (x + 3\right ) + 4}{4 \, x^{2} \log \left (x + 3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+3*x^3)*log(3+x)^3+(-8*x^2-24*x)*log(3)*log(3+x)^2+(-8*x^2*log(3)-8*x-24)*log(3+x)-8*x)/(4*x^4+
12*x^3)/log(3+x)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.37, size = 25, normalized size = 1.19 \begin {gather*} \frac {x}{4}+\frac {2\,x\,\ln \left (x+3\right )\,\ln \relax (3)+1}{x^2\,{\ln \left (x+3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x - log(x + 3)^3*(3*x^3 + x^4) + log(x + 3)*(8*x + 8*x^2*log(3) + 24) + log(x + 3)^2*log(3)*(24*x + 8*
x^2))/(log(x + 3)^3*(12*x^3 + 4*x^4)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 26, normalized size = 1.24 \begin {gather*} \frac {x}{4} + \frac {2 x \log {\relax (3 )} \log {\left (x + 3 \right )} + 1}{x^{2} \log {\left (x + 3 \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**4+3*x**3)*ln(3+x)**3+(-8*x**2-24*x)*ln(3)*ln(3+x)**2+(-8*x**2*ln(3)-8*x-24)*ln(3+x)-8*x)/(4*x**
4+12*x**3)/ln(3+x)**3,x)

[Out]

x/4 + (2*x*log(3)*log(x + 3) + 1)/(x**2*log(x + 3)**2)

________________________________________________________________________________________