3.49.64 \(\int \frac {4-98 x^2+8 x^2 \log (\frac {x}{4})}{-49 x+4 x \log (\frac {x}{4})} \, dx\)

Optimal. Leaf size=15 \[ x^2+\log \left (-\frac {49}{4}+\log \left (\frac {x}{4}\right )\right ) \]

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Rubi [A]  time = 0.37, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2561, 6741, 12, 6742, 2302, 29} \begin {gather*} x^2+\log \left (49-4 \log \left (\frac {x}{4}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 98*x^2 + 8*x^2*Log[x/4])/(-49*x + 4*x*Log[x/4]),x]

[Out]

x^2 + Log[49 - 4*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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6741

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

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 {4-98 x^2+8 x^2 \log \left (\frac {x}{4}\right )}{x \left (-49+4 \log \left (\frac {x}{4}\right )\right )} \, dx\\ &=\int \frac {2 \left (-2+49 x^2-4 x^2 \log \left (\frac {x}{4}\right )\right )}{x \left (49-4 \log \left (\frac {x}{4}\right )\right )} \, dx\\ &=2 \int \frac {-2+49 x^2-4 x^2 \log \left (\frac {x}{4}\right )}{x \left (49-4 \log \left (\frac {x}{4}\right )\right )} \, dx\\ &=2 \int \left (x+\frac {2}{x \left (-49+4 \log \left (\frac {x}{4}\right )\right )}\right ) \, dx\\ &=x^2+4 \int \frac {1}{x \left (-49+4 \log \left (\frac {x}{4}\right )\right )} \, dx\\ &=x^2+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,-49+4 \log \left (\frac {x}{4}\right )\right )\\ &=x^2+\log \left (49-4 \log \left (\frac {x}{4}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 15, normalized size = 1.00 \begin {gather*} x^2+\log \left (49-4 \log \left (\frac {x}{4}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 98*x^2 + 8*x^2*Log[x/4])/(-49*x + 4*x*Log[x/4]),x]

[Out]

x^2 + Log[49 - 4*Log[x/4]]

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fricas [A]  time = 0.63, size = 13, normalized size = 0.87 \begin {gather*} x^{2} + \log \left (4 \, \log \left (\frac {1}{4} \, x\right ) - 49\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*log(1/4*x)*x^2-98*x^2+4)/(4*x*log(1/4*x)-49*x),x, algorithm="fricas")

[Out]

x^2 + log(4*log(1/4*x) - 49)

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giac [A]  time = 0.21, size = 13, normalized size = 0.87 \begin {gather*} x^{2} + \log \left (4 \, \log \left (\frac {1}{4} \, x\right ) - 49\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*log(1/4*x)*x^2-98*x^2+4)/(4*x*log(1/4*x)-49*x),x, algorithm="giac")

[Out]

x^2 + log(4*log(1/4*x) - 49)

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maple [A]  time = 0.05, size = 12, normalized size = 0.80

method	result	size
risch	\(x^{2}+\ln \left (\ln \left (\frac {x}{4}\right )-\frac {49}{4}\right )\)	\(12\)
norman	\(x^{2}+\ln \left (4 \ln \left (\frac {x}{4}\right )-49\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*ln(1/4*x)*x^2-98*x^2+4)/(4*x*ln(1/4*x)-49*x),x,method=_RETURNVERBOSE)

[Out]

x^2+ln(ln(1/4*x)-49/4)

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maxima [A]  time = 0.47, size = 13, normalized size = 0.87 \begin {gather*} x^{2} + \log \left (-2 \, \log \relax (2) + \log \relax (x) - \frac {49}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*log(1/4*x)*x^2-98*x^2+4)/(4*x*log(1/4*x)-49*x),x, algorithm="maxima")

[Out]

x^2 + log(-2*log(2) + log(x) - 49/4)

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mupad [B]  time = 3.37, size = 13, normalized size = 0.87 \begin {gather*} \ln \left (4\,\ln \left (\frac {x}{4}\right )-49\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x^2*log(x/4) - 98*x^2 + 4)/(49*x - 4*x*log(x/4)),x)

[Out]

log(4*log(x/4) - 49) + x^2

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sympy [A]  time = 0.12, size = 12, normalized size = 0.80 \begin {gather*} x^{2} + \log {\left (\log {\left (\frac {x}{4} \right )} - \frac {49}{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*ln(1/4*x)*x**2-98*x**2+4)/(4*x*ln(1/4*x)-49*x),x)

[Out]

x**2 + log(log(x/4) - 49/4)

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