3.94.4 \(\int \frac {8-4 x+x \log (x^4)}{-4 x+x \log (x^4)} \, dx\)

Optimal. Leaf size=25 \[ x+\frac {x-\frac {x}{e}}{x}+\log \left (\left (4-\log \left (x^4\right )\right )^2\right ) \]

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Rubi [A]  time = 0.13, antiderivative size = 13, normalized size of antiderivative = 0.52, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2561, 6742, 2302, 29} \begin {gather*} 2 \log \left (4-\log \left (x^4\right )\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + 2*Log[4 - Log[x^4]]

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

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Mathematica [A]  time = 0.03, size = 13, normalized size = 0.52 \begin {gather*} x+2 \log \left (4-\log \left (x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + 2*Log[4 - Log[x^4]]

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fricas [A]  time = 1.26, size = 11, normalized size = 0.44 \begin {gather*} x + 2 \, \log \left (\log \left (x^{4}\right ) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2*log(log(x^4) - 4)

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giac [A]  time = 0.16, size = 11, normalized size = 0.44 \begin {gather*} x + 2 \, \log \left (\log \left (x^{4}\right ) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2*log(log(x^4) - 4)

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maple [A]  time = 0.02, size = 12, normalized size = 0.48

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+2*ln(-4+ln(x^4))

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maxima [A]  time = 1.03, size = 9, normalized size = 0.36 \begin {gather*} x + 2 \, \log \left (\log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2*log(log(x) - 1)

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mupad [B]  time = 8.78, size = 11, normalized size = 0.44 \begin {gather*} x+2\,\ln \left (\ln \left (x^4\right )-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2*log(log(x^4) - 4)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.40 \begin {gather*} x + 2 \log {\left (\log {\left (x^{4} \right )} - 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*ln(x**4)-4*x+8)/(x*ln(x**4)-4*x),x)

[Out]

x + 2*log(log(x**4) - 4)

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