3.1.2 \(\int \frac {32+60 x+110 x^2+32 x^3+6 x^4+(5 x+5 x^2+5 x^3) \log (x^2+2 x^3+3 x^4+2 x^5+x^6)}{-64 x-96 x^2-100 x^3-36 x^4-4 x^5+(16 x+24 x^2+25 x^3+9 x^4+x^5) \log (x^2+2 x^3+3 x^4+2 x^5+x^6)} \, dx\)

Optimal. Leaf size=27 \[ 3-\frac {5}{4+x}+\log \left (4-\log \left (x^2 \left (1+x+x^2\right )^2\right )\right ) \]

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Rubi [A]  time = 2.16, antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, integrand size = 131, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6688, 6728, 6684} \begin {gather*} \log \left (4-\log \left (x^2 \left (x^2+x+1\right )^2\right )\right )-\frac {5}{x+4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(32 + 60*x + 110*x^2 + 32*x^3 + 6*x^4 + (5*x + 5*x^2 + 5*x^3)*Log[x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6])/(-64
*x - 96*x^2 - 100*x^3 - 36*x^4 - 4*x^5 + (16*x + 24*x^2 + 25*x^3 + 9*x^4 + x^5)*Log[x^2 + 2*x^3 + 3*x^4 + 2*x^
5 + x^6]),x]

[Out]

-5/(4 + x) + Log[4 - Log[x^2*(1 + x + x^2)^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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-32-60 x-110 x^2-32 x^3-6 x^4-5 x \left (1+x+x^2\right ) \log \left (x^2 \left (1+x+x^2\right )^2\right )}{x (4+x)^2 \left (1+x+x^2\right ) \left (4-\log \left (x^2 \left (1+x+x^2\right )^2\right )\right )} \, dx\\ &=\int \left (\frac {5}{(4+x)^2}+\frac {2 \left (1+2 x+3 x^2\right )}{x \left (1+x+x^2\right ) \left (-4+\log \left (x^2 \left (1+x+x^2\right )^2\right )\right )}\right ) \, dx\\ &=-\frac {5}{4+x}+2 \int \frac {1+2 x+3 x^2}{x \left (1+x+x^2\right ) \left (-4+\log \left (x^2 \left (1+x+x^2\right )^2\right )\right )} \, dx\\ &=-\frac {5}{4+x}+\log \left (4-\log \left (x^2 \left (1+x+x^2\right )^2\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 26, normalized size = 0.96 \begin {gather*} -\frac {5}{4+x}+\log \left (4-\log \left (x^2 \left (1+x+x^2\right )^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 + 60*x + 110*x^2 + 32*x^3 + 6*x^4 + (5*x + 5*x^2 + 5*x^3)*Log[x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6]
)/(-64*x - 96*x^2 - 100*x^3 - 36*x^4 - 4*x^5 + (16*x + 24*x^2 + 25*x^3 + 9*x^4 + x^5)*Log[x^2 + 2*x^3 + 3*x^4
+ 2*x^5 + x^6]),x]

[Out]

-5/(4 + x) + Log[4 - Log[x^2*(1 + x + x^2)^2]]

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fricas [A]  time = 0.51, size = 38, normalized size = 1.41 \begin {gather*} \frac {{\left (x + 4\right )} \log \left (\log \left (x^{6} + 2 \, x^{5} + 3 \, x^{4} + 2 \, x^{3} + x^{2}\right ) - 4\right ) - 5}{x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3+5*x^2+5*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)+6*x^4+32*x^3+110*x^2+60*x+32)/((x^5+9*x^4+25*x^3+2
4*x^2+16*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)-4*x^5-36*x^4-100*x^3-96*x^2-64*x),x, algorithm="fricas")

[Out]

((x + 4)*log(log(x^6 + 2*x^5 + 3*x^4 + 2*x^3 + x^2) - 4) - 5)/(x + 4)

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giac [A]  time = 0.20, size = 34, normalized size = 1.26 \begin {gather*} -\frac {5}{x + 4} + \log \left (\log \left (x^{6} + 2 \, x^{5} + 3 \, x^{4} + 2 \, x^{3} + x^{2}\right ) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3+5*x^2+5*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)+6*x^4+32*x^3+110*x^2+60*x+32)/((x^5+9*x^4+25*x^3+2
4*x^2+16*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)-4*x^5-36*x^4-100*x^3-96*x^2-64*x),x, algorithm="giac")

[Out]

-5/(x + 4) + log(log(x^6 + 2*x^5 + 3*x^4 + 2*x^3 + x^2) - 4)

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maple [A]  time = 0.09, size = 35, normalized size = 1.30

method	result	size
norman	\(-\frac {5}{4+x}+\ln \left (\ln \left (x^{6}+2 x^{5}+3 x^{4}+2 x^{3}+x^{2}\right )-4\right )\)	\(35\)
risch	\(-\frac {5}{4+x}+\ln \left (\ln \left (x^{6}+2 x^{5}+3 x^{4}+2 x^{3}+x^{2}\right )-4\right )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^3+5*x^2+5*x)*ln(x^6+2*x^5+3*x^4+2*x^3+x^2)+6*x^4+32*x^3+110*x^2+60*x+32)/((x^5+9*x^4+25*x^3+24*x^2+1
6*x)*ln(x^6+2*x^5+3*x^4+2*x^3+x^2)-4*x^5-36*x^4-100*x^3-96*x^2-64*x),x,method=_RETURNVERBOSE)

[Out]

-5/(4+x)+ln(ln(x^6+2*x^5+3*x^4+2*x^3+x^2)-4)

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maxima [A]  time = 0.51, size = 20, normalized size = 0.74 \begin {gather*} -\frac {5}{x + 4} + \log \left (\log \left (x^{2} + x + 1\right ) + \log \relax (x) - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3+5*x^2+5*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)+6*x^4+32*x^3+110*x^2+60*x+32)/((x^5+9*x^4+25*x^3+2
4*x^2+16*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)-4*x^5-36*x^4-100*x^3-96*x^2-64*x),x, algorithm="maxima")

[Out]

-5/(x + 4) + log(log(x^2 + x + 1) + log(x) - 2)

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mupad [B]  time = 0.86, size = 34, normalized size = 1.26 \begin {gather*} \ln \left (\ln \left (x^6+2\,x^5+3\,x^4+2\,x^3+x^2\right )-4\right )-\frac {5}{x+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(60*x + log(x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6)*(5*x + 5*x^2 + 5*x^3) + 110*x^2 + 32*x^3 + 6*x^4 + 32)/(64
*x - log(x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6)*(16*x + 24*x^2 + 25*x^3 + 9*x^4 + x^5) + 96*x^2 + 100*x^3 + 36*x^4
+ 4*x^5),x)

[Out]

log(log(x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6) - 4) - 5/(x + 4)

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sympy [A]  time = 0.28, size = 31, normalized size = 1.15 \begin {gather*} \log {\left (\log {\left (x^{6} + 2 x^{5} + 3 x^{4} + 2 x^{3} + x^{2} \right )} - 4 \right )} - \frac {5}{x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**3+5*x**2+5*x)*ln(x**6+2*x**5+3*x**4+2*x**3+x**2)+6*x**4+32*x**3+110*x**2+60*x+32)/((x**5+9*x*
*4+25*x**3+24*x**2+16*x)*ln(x**6+2*x**5+3*x**4+2*x**3+x**2)-4*x**5-36*x**4-100*x**3-96*x**2-64*x),x)

[Out]

log(log(x**6 + 2*x**5 + 3*x**4 + 2*x**3 + x**2) - 4) - 5/(x + 4)

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