3.53.17 \(\int \frac {12 x^3-12 x^4+(-36 x^2+36 x^3) \log (x)+(36 x-36 x^2) \log ^2(x)+(-12+12 x) \log ^3(x)}{16 x-8 x^5+x^9+(32 x^4-8 x^8) \log (x)+(-48 x^3+28 x^7) \log ^2(x)+(32 x^2-56 x^6) \log ^3(x)+(-8 x+70 x^5) \log ^4(x)-56 x^4 \log ^5(x)+28 x^3 \log ^6(x)-8 x^2 \log ^7(x)+x \log ^8(x)} \, dx\)

Optimal. Leaf size=14 \[ \frac {3}{-4+(x-\log (x))^4} \]

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Rubi [B]  time = 0.54, antiderivative size = 40, normalized size of antiderivative = 2.86, number of steps used = 3, number of rules used = 3, integrand size = 157, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6688, 12, 6686} \begin {gather*} -\frac {3}{-x^4+4 x^3 \log (x)-6 x^2 \log ^2(x)-\log ^4(x)+4 x \log ^3(x)+4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12*x^3 - 12*x^4 + (-36*x^2 + 36*x^3)*Log[x] + (36*x - 36*x^2)*Log[x]^2 + (-12 + 12*x)*Log[x]^3)/(16*x - 8
*x^5 + x^9 + (32*x^4 - 8*x^8)*Log[x] + (-48*x^3 + 28*x^7)*Log[x]^2 + (32*x^2 - 56*x^6)*Log[x]^3 + (-8*x + 70*x
^5)*Log[x]^4 - 56*x^4*Log[x]^5 + 28*x^3*Log[x]^6 - 8*x^2*Log[x]^7 + x*Log[x]^8),x]

[Out]

-3/(4 - x^4 + 4*x^3*Log[x] - 6*x^2*Log[x]^2 + 4*x*Log[x]^3 - Log[x]^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12 (1-x) (x-\log (x))^3}{x \left (4-x^4+4 x^3 \log (x)-6 x^2 \log ^2(x)+4 x \log ^3(x)-\log ^4(x)\right )^2} \, dx\\ &=12 \int \frac {(1-x) (x-\log (x))^3}{x \left (4-x^4+4 x^3 \log (x)-6 x^2 \log ^2(x)+4 x \log ^3(x)-\log ^4(x)\right )^2} \, dx\\ &=-\frac {3}{4-x^4+4 x^3 \log (x)-6 x^2 \log ^2(x)+4 x \log ^3(x)-\log ^4(x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 36, normalized size = 2.57 \begin {gather*} \frac {3}{-4+x^4-4 x^3 \log (x)+6 x^2 \log ^2(x)-4 x \log ^3(x)+\log ^4(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12*x^3 - 12*x^4 + (-36*x^2 + 36*x^3)*Log[x] + (36*x - 36*x^2)*Log[x]^2 + (-12 + 12*x)*Log[x]^3)/(16
*x - 8*x^5 + x^9 + (32*x^4 - 8*x^8)*Log[x] + (-48*x^3 + 28*x^7)*Log[x]^2 + (32*x^2 - 56*x^6)*Log[x]^3 + (-8*x
+ 70*x^5)*Log[x]^4 - 56*x^4*Log[x]^5 + 28*x^3*Log[x]^6 - 8*x^2*Log[x]^7 + x*Log[x]^8),x]

[Out]

3/(-4 + x^4 - 4*x^3*Log[x] + 6*x^2*Log[x]^2 - 4*x*Log[x]^3 + Log[x]^4)

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fricas [B]  time = 0.47, size = 36, normalized size = 2.57 \begin {gather*} \frac {3}{x^{4} - 4 \, x^{3} \log \relax (x) + 6 \, x^{2} \log \relax (x)^{2} - 4 \, x \log \relax (x)^{3} + \log \relax (x)^{4} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-12)*log(x)^3+(-36*x^2+36*x)*log(x)^2+(36*x^3-36*x^2)*log(x)-12*x^4+12*x^3)/(x*log(x)^8-8*x^2*
log(x)^7+28*x^3*log(x)^6-56*x^4*log(x)^5+(70*x^5-8*x)*log(x)^4+(-56*x^6+32*x^2)*log(x)^3+(28*x^7-48*x^3)*log(x
)^2+(-8*x^8+32*x^4)*log(x)+x^9-8*x^5+16*x),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.35, size = 36, normalized size = 2.57 \begin {gather*} \frac {3}{x^{4} - 4 \, x^{3} \log \relax (x) + 6 \, x^{2} \log \relax (x)^{2} - 4 \, x \log \relax (x)^{3} + \log \relax (x)^{4} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-12)*log(x)^3+(-36*x^2+36*x)*log(x)^2+(36*x^3-36*x^2)*log(x)-12*x^4+12*x^3)/(x*log(x)^8-8*x^2*
log(x)^7+28*x^3*log(x)^6-56*x^4*log(x)^5+(70*x^5-8*x)*log(x)^4+(-56*x^6+32*x^2)*log(x)^3+(28*x^7-48*x^3)*log(x
)^2+(-8*x^8+32*x^4)*log(x)+x^9-8*x^5+16*x),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.04, size = 37, normalized size = 2.64

method	result	size
risch	\(\frac {3}{\ln \relax (x )^{4}-4 x \ln \relax (x )^{3}+6 x^{2} \ln \relax (x )^{2}-4 x^{3} \ln \relax (x )+x^{4}-4}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x-12)*ln(x)^3+(-36*x^2+36*x)*ln(x)^2+(36*x^3-36*x^2)*ln(x)-12*x^4+12*x^3)/(x*ln(x)^8-8*x^2*ln(x)^7+28
*x^3*ln(x)^6-56*x^4*ln(x)^5+(70*x^5-8*x)*ln(x)^4+(-56*x^6+32*x^2)*ln(x)^3+(28*x^7-48*x^3)*ln(x)^2+(-8*x^8+32*x
^4)*ln(x)+x^9-8*x^5+16*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.40, size = 36, normalized size = 2.57 \begin {gather*} \frac {3}{x^{4} - 4 \, x^{3} \log \relax (x) + 6 \, x^{2} \log \relax (x)^{2} - 4 \, x \log \relax (x)^{3} + \log \relax (x)^{4} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-12)*log(x)^3+(-36*x^2+36*x)*log(x)^2+(36*x^3-36*x^2)*log(x)-12*x^4+12*x^3)/(x*log(x)^8-8*x^2*
log(x)^7+28*x^3*log(x)^6-56*x^4*log(x)^5+(70*x^5-8*x)*log(x)^4+(-56*x^6+32*x^2)*log(x)^3+(28*x^7-48*x^3)*log(x
)^2+(-8*x^8+32*x^4)*log(x)+x^9-8*x^5+16*x),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {{\ln \relax (x)}^2\,\left (36\,x-36\,x^2\right )-\ln \relax (x)\,\left (36\,x^2-36\,x^3\right )+12\,x^3-12\,x^4+{\ln \relax (x)}^3\,\left (12\,x-12\right )}{16\,x+\ln \relax (x)\,\left (32\,x^4-8\,x^8\right )-{\ln \relax (x)}^4\,\left (8\,x-70\,x^5\right )+x\,{\ln \relax (x)}^8-{\ln \relax (x)}^2\,\left (48\,x^3-28\,x^7\right )+{\ln \relax (x)}^3\,\left (32\,x^2-56\,x^6\right )-8\,x^2\,{\ln \relax (x)}^7+28\,x^3\,{\ln \relax (x)}^6-56\,x^4\,{\ln \relax (x)}^5-8\,x^5+x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(36*x - 36*x^2) - log(x)*(36*x^2 - 36*x^3) + 12*x^3 - 12*x^4 + log(x)^3*(12*x - 12))/(16*x + log
(x)*(32*x^4 - 8*x^8) - log(x)^4*(8*x - 70*x^5) + x*log(x)^8 - log(x)^2*(48*x^3 - 28*x^7) + log(x)^3*(32*x^2 -
56*x^6) - 8*x^2*log(x)^7 + 28*x^3*log(x)^6 - 56*x^4*log(x)^5 - 8*x^5 + x^9),x)

[Out]

int((log(x)^2*(36*x - 36*x^2) - log(x)*(36*x^2 - 36*x^3) + 12*x^3 - 12*x^4 + log(x)^3*(12*x - 12))/(16*x + log
(x)*(32*x^4 - 8*x^8) - log(x)^4*(8*x - 70*x^5) + x*log(x)^8 - log(x)^2*(48*x^3 - 28*x^7) + log(x)^3*(32*x^2 -
56*x^6) - 8*x^2*log(x)^7 + 28*x^3*log(x)^6 - 56*x^4*log(x)^5 - 8*x^5 + x^9), x)

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sympy [B]  time = 0.19, size = 37, normalized size = 2.64 \begin {gather*} \frac {3}{x^{4} - 4 x^{3} \log {\relax (x )} + 6 x^{2} \log {\relax (x )}^{2} - 4 x \log {\relax (x )}^{3} + \log {\relax (x )}^{4} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-12)*ln(x)**3+(-36*x**2+36*x)*ln(x)**2+(36*x**3-36*x**2)*ln(x)-12*x**4+12*x**3)/(x*ln(x)**8-8*
x**2*ln(x)**7+28*x**3*ln(x)**6-56*x**4*ln(x)**5+(70*x**5-8*x)*ln(x)**4+(-56*x**6+32*x**2)*ln(x)**3+(28*x**7-48
*x**3)*ln(x)**2+(-8*x**8+32*x**4)*ln(x)+x**9-8*x**5+16*x),x)

[Out]

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

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