3.55.78 \(\int \frac {2-2 x+x^8+(2 x-x^8) \log (x)}{-2 x+2 x \log (x)} \, dx\)

Optimal. Leaf size=16 \[ x-\frac {x^8}{16}+\log (1-\log (x)) \]

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Rubi [A]  time = 0.31, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2561, 6741, 12, 6742, 2302, 29} \begin {gather*} -\frac {x^8}{16}+x+\log (1-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - x^8/16 + Log[1 - 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 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 {2-2 x+x^8+\left (2 x-x^8\right ) \log (x)}{x (-2+2 \log (x))} \, dx\\ &=\int \frac {-2+2 x-x^8-\left (2 x-x^8\right ) \log (x)}{2 x (1-\log (x))} \, dx\\ &=\frac {1}{2} \int \frac {-2+2 x-x^8-\left (2 x-x^8\right ) \log (x)}{x (1-\log (x))} \, dx\\ &=\frac {1}{2} \int \left (2-x^7+\frac {2}{x (-1+\log (x))}\right ) \, dx\\ &=x-\frac {x^8}{16}+\int \frac {1}{x (-1+\log (x))} \, dx\\ &=x-\frac {x^8}{16}+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,-1+\log (x)\right )\\ &=x-\frac {x^8}{16}+\log (1-\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 16, normalized size = 1.00 \begin {gather*} x-\frac {x^8}{16}+\log (1-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - x^8/16 + Log[1 - Log[x]]

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fricas [A]  time = 0.72, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{16} \, x^{8} + x + \log \left (\log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*x^8 + x + log(log(x) - 1)

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giac [A]  time = 0.13, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{16} \, x^{8} + x + \log \left (\log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*x^8 + x + log(log(x) - 1)

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maple [A]  time = 0.03, size = 13, normalized size = 0.81

method	result	size
norman	\(x -\frac {x^{8}}{16}+\ln \left (\ln \relax (x )-1\right )\)	\(13\)
risch	\(x -\frac {x^{8}}{16}+\ln \left (\ln \relax (x )-1\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-1/16*x^8+ln(ln(x)-1)

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maxima [A]  time = 0.42, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{16} \, x^{8} + x + \log \left (\log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*x^8 + x + log(log(x) - 1)

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mupad [B]  time = 3.49, size = 12, normalized size = 0.75 \begin {gather*} x+\ln \left (\ln \relax (x)-1\right )-\frac {x^8}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(log(x) - 1) - x^8/16

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sympy [A]  time = 0.10, size = 12, normalized size = 0.75 \begin {gather*} - \frac {x^{8}}{16} + x + \log {\left (\log {\relax (x )} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**8+2*x)*ln(x)+x**8-2*x+2)/(2*x*ln(x)-2*x),x)

[Out]

-x**8/16 + x + log(log(x) - 1)

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