3.95.46 \(\int \frac {112-288 x+144 x^2-240 x^4-12 \log (x)}{49-434 x+1465 x^2-2400 x^3+2040 x^4-696 x^5-600 x^6+864 x^7-288 x^8+144 x^{10}+(42 x-186 x^2+216 x^3-72 x^4+72 x^6) \log (x)+9 x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {4}{7-x \left (4+3 \left ((3-2 x)^2-4 x^4-\log (x)\right )\right )} \]

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Rubi [A]  time = 0.42, antiderivative size = 29, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 103, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6688, 12, 6686} \begin {gather*} \frac {4}{12 x^5-12 x^3+36 x^2-31 x+3 x \log (x)+7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(112 - 288*x + 144*x^2 - 240*x^4 - 12*Log[x])/(49 - 434*x + 1465*x^2 - 2400*x^3 + 2040*x^4 - 696*x^5 - 600
*x^6 + 864*x^7 - 288*x^8 + 144*x^10 + (42*x - 186*x^2 + 216*x^3 - 72*x^4 + 72*x^6)*Log[x] + 9*x^2*Log[x]^2),x]

[Out]

4/(7 - 31*x + 36*x^2 - 12*x^3 + 12*x^5 + 3*x*Log[x])

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 {4 \left (28-72 x+36 x^2-60 x^4-3 \log (x)\right )}{\left (7-31 x+36 x^2-12 x^3+12 x^5+3 x \log (x)\right )^2} \, dx\\ &=4 \int \frac {28-72 x+36 x^2-60 x^4-3 \log (x)}{\left (7-31 x+36 x^2-12 x^3+12 x^5+3 x \log (x)\right )^2} \, dx\\ &=\frac {4}{7-31 x+36 x^2-12 x^3+12 x^5+3 x \log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 29, normalized size = 0.97 \begin {gather*} \frac {4}{7-31 x+36 x^2-12 x^3+12 x^5+3 x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(112 - 288*x + 144*x^2 - 240*x^4 - 12*Log[x])/(49 - 434*x + 1465*x^2 - 2400*x^3 + 2040*x^4 - 696*x^5
- 600*x^6 + 864*x^7 - 288*x^8 + 144*x^10 + (42*x - 186*x^2 + 216*x^3 - 72*x^4 + 72*x^6)*Log[x] + 9*x^2*Log[x]
^2),x]

[Out]

4/(7 - 31*x + 36*x^2 - 12*x^3 + 12*x^5 + 3*x*Log[x])

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fricas [A]  time = 0.56, size = 29, normalized size = 0.97 \begin {gather*} \frac {4}{12 \, x^{5} - 12 \, x^{3} + 36 \, x^{2} + 3 \, x \log \relax (x) - 31 \, x + 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*log(x)-240*x^4+144*x^2-288*x+112)/(9*x^2*log(x)^2+(72*x^6-72*x^4+216*x^3-186*x^2+42*x)*log(x)+1
44*x^10-288*x^8+864*x^7-600*x^6-696*x^5+2040*x^4-2400*x^3+1465*x^2-434*x+49),x, algorithm="fricas")

[Out]

4/(12*x^5 - 12*x^3 + 36*x^2 + 3*x*log(x) - 31*x + 7)

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giac [A]  time = 0.16, size = 29, normalized size = 0.97 \begin {gather*} \frac {4}{12 \, x^{5} - 12 \, x^{3} + 36 \, x^{2} + 3 \, x \log \relax (x) - 31 \, x + 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*log(x)-240*x^4+144*x^2-288*x+112)/(9*x^2*log(x)^2+(72*x^6-72*x^4+216*x^3-186*x^2+42*x)*log(x)+1
44*x^10-288*x^8+864*x^7-600*x^6-696*x^5+2040*x^4-2400*x^3+1465*x^2-434*x+49),x, algorithm="giac")

[Out]

4/(12*x^5 - 12*x^3 + 36*x^2 + 3*x*log(x) - 31*x + 7)

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maple [A]  time = 0.03, size = 30, normalized size = 1.00

method	result	size
risch	\(\frac {4}{12 x^{5}-12 x^{3}+3 x \ln \relax (x )+36 x^{2}-31 x +7}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12*ln(x)-240*x^4+144*x^2-288*x+112)/(9*x^2*ln(x)^2+(72*x^6-72*x^4+216*x^3-186*x^2+42*x)*ln(x)+144*x^10-2
88*x^8+864*x^7-600*x^6-696*x^5+2040*x^4-2400*x^3+1465*x^2-434*x+49),x,method=_RETURNVERBOSE)

[Out]

4/(12*x^5-12*x^3+3*x*ln(x)+36*x^2-31*x+7)

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maxima [A]  time = 0.37, size = 29, normalized size = 0.97 \begin {gather*} \frac {4}{12 \, x^{5} - 12 \, x^{3} + 36 \, x^{2} + 3 \, x \log \relax (x) - 31 \, x + 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*log(x)-240*x^4+144*x^2-288*x+112)/(9*x^2*log(x)^2+(72*x^6-72*x^4+216*x^3-186*x^2+42*x)*log(x)+1
44*x^10-288*x^8+864*x^7-600*x^6-696*x^5+2040*x^4-2400*x^3+1465*x^2-434*x+49),x, algorithm="maxima")

[Out]

4/(12*x^5 - 12*x^3 + 36*x^2 + 3*x*log(x) - 31*x + 7)

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mupad [B]  time = 7.66, size = 29, normalized size = 0.97 \begin {gather*} \frac {4}{x\,\left (3\,\ln \relax (x)-31\right )+36\,x^2-12\,x^3+12\,x^5+7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(288*x + 12*log(x) - 144*x^2 + 240*x^4 - 112)/(log(x)*(42*x - 186*x^2 + 216*x^3 - 72*x^4 + 72*x^6) - 434*
x + 9*x^2*log(x)^2 + 1465*x^2 - 2400*x^3 + 2040*x^4 - 696*x^5 - 600*x^6 + 864*x^7 - 288*x^8 + 144*x^10 + 49),x
)

[Out]

4/(x*(3*log(x) - 31) + 36*x^2 - 12*x^3 + 12*x^5 + 7)

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sympy [A]  time = 0.22, size = 27, normalized size = 0.90 \begin {gather*} \frac {4}{12 x^{5} - 12 x^{3} + 36 x^{2} + 3 x \log {\relax (x )} - 31 x + 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*ln(x)-240*x**4+144*x**2-288*x+112)/(9*x**2*ln(x)**2+(72*x**6-72*x**4+216*x**3-186*x**2+42*x)*ln
(x)+144*x**10-288*x**8+864*x**7-600*x**6-696*x**5+2040*x**4-2400*x**3+1465*x**2-434*x+49),x)

[Out]

4/(12*x**5 - 12*x**3 + 36*x**2 + 3*x*log(x) - 31*x + 7)

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