3.98.7 \(\int \frac {2 x+(-3-x+24 x^2+8 x^3) \log (3+x)+(-6 x^2-2 x^3) \log (3+x) \log (\frac {4 x}{\log ^2(3+x)})}{(-12 x-4 x^2) \log (3+x)+(3 x+x^2) \log (3+x) \log (\frac {4 x}{\log ^2(3+x)})} \, dx\)

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

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Rubi [A]  time = 0.99, antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6688, 6742, 6684} \begin {gather*} -x^2-\log \left (4-\log \left (\frac {4 x}{\log ^2(x+3)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x + (-3 - x + 24*x^2 + 8*x^3)*Log[3 + x] + (-6*x^2 - 2*x^3)*Log[3 + x]*Log[(4*x)/Log[3 + x]^2])/((-12*x
- 4*x^2)*Log[3 + x] + (3*x + x^2)*Log[3 + x]*Log[(4*x)/Log[3 + x]^2]),x]

[Out]

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

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 23, normalized size = 1.10 \begin {gather*} -x^2-\log \left (4-\log \left (\frac {4 x}{\log ^2(3+x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x + (-3 - x + 24*x^2 + 8*x^3)*Log[3 + x] + (-6*x^2 - 2*x^3)*Log[3 + x]*Log[(4*x)/Log[3 + x]^2])/(
(-12*x - 4*x^2)*Log[3 + x] + (3*x + x^2)*Log[3 + x]*Log[(4*x)/Log[3 + x]^2]),x]

[Out]

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

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fricas [A]  time = 0.93, size = 21, normalized size = 1.00 \begin {gather*} -x^{2} - \log \left (\log \left (\frac {4 \, x}{\log \left (x + 3\right )^{2}}\right ) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-6*x^2)*log(3+x)*log(4*x/log(3+x)^2)+(8*x^3+24*x^2-x-3)*log(3+x)+2*x)/((x^2+3*x)*log(3+x)*lo
g(4*x/log(3+x)^2)+(-4*x^2-12*x)*log(3+x)),x, algorithm="fricas")

[Out]

-x^2 - log(log(4*x/log(x + 3)^2) - 4)

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giac [A]  time = 0.34, size = 24, normalized size = 1.14 \begin {gather*} -x^{2} - \log \left (\log \left (\log \left (x + 3\right )^{2}\right ) - \log \left (4 \, x\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-6*x^2)*log(3+x)*log(4*x/log(3+x)^2)+(8*x^3+24*x^2-x-3)*log(3+x)+2*x)/((x^2+3*x)*log(3+x)*lo
g(4*x/log(3+x)^2)+(-4*x^2-12*x)*log(3+x)),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.32, size = 183, normalized size = 8.71

method	result	size
risch	\(-x^{2}-\ln \left (\ln \left (\ln \left (3+x \right )\right )+\frac {i \left (\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \left (3+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (3+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i \ln \left (3+x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \ln \left (3+x \right )\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )^{3}+4 i \ln \relax (2)+2 i \ln \relax (x )-8 i\right )}{4}\right )\)	\(183\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3-6*x^2)*ln(3+x)*ln(4*x/ln(3+x)^2)+(8*x^3+24*x^2-x-3)*ln(3+x)+2*x)/((x^2+3*x)*ln(3+x)*ln(4*x/ln(3+x
)^2)+(-4*x^2-12*x)*ln(3+x)),x,method=_RETURNVERBOSE)

[Out]

-x^2-ln(ln(ln(3+x))+1/4*I*(Pi*csgn(I*x)*csgn(I/ln(3+x)^2)*csgn(I*x/ln(3+x)^2)-Pi*csgn(I*x)*csgn(I*x/ln(3+x)^2)
^2-Pi*csgn(I/ln(3+x)^2)*csgn(I*x/ln(3+x)^2)^2-Pi*csgn(I*ln(3+x))^2*csgn(I*ln(3+x)^2)+2*Pi*csgn(I*ln(3+x))*csgn
(I*ln(3+x)^2)^2-Pi*csgn(I*ln(3+x)^2)^3+Pi*csgn(I*x/ln(3+x)^2)^3+4*I*ln(2)+2*I*ln(x)-8*I))

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maxima [A]  time = 0.47, size = 24, normalized size = 1.14 \begin {gather*} -x^{2} - \log \left (-\log \relax (2) - \frac {1}{2} \, \log \relax (x) + \log \left (\log \left (x + 3\right )\right ) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-6*x^2)*log(3+x)*log(4*x/log(3+x)^2)+(8*x^3+24*x^2-x-3)*log(3+x)+2*x)/((x^2+3*x)*log(3+x)*lo
g(4*x/log(3+x)^2)+(-4*x^2-12*x)*log(3+x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.98, size = 21, normalized size = 1.00 \begin {gather*} -\ln \left (\ln \left (\frac {4\,x}{{\ln \left (x+3\right )}^2}\right )-4\right )-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 3)*(x - 24*x^2 - 8*x^3 + 3) - 2*x + log(x + 3)*log((4*x)/log(x + 3)^2)*(6*x^2 + 2*x^3))/(log(x +
3)*(12*x + 4*x^2) - log(x + 3)*log((4*x)/log(x + 3)^2)*(3*x + x^2)),x)

[Out]

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

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sympy [A]  time = 0.49, size = 19, normalized size = 0.90 \begin {gather*} - x^{2} - \log {\left (\log {\left (\frac {4 x}{\log {\left (x + 3 \right )}^{2}} \right )} - 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3-6*x**2)*ln(3+x)*ln(4*x/ln(3+x)**2)+(8*x**3+24*x**2-x-3)*ln(3+x)+2*x)/((x**2+3*x)*ln(3+x)*l
n(4*x/ln(3+x)**2)+(-4*x**2-12*x)*ln(3+x)),x)

[Out]

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

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