3.21.42 \(\int \frac {4+27 x^4-x^8+(108 x^4-8 x^8) \log (x)}{(-4 x-27 x^5+x^9) \log (x)} \, dx\)

Optimal. Leaf size=22 \[ \log (3)-\log \left (\left (4+x^4 \left (27-x^4\right )\right ) \log (x)\right ) \]

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Rubi [A]  time = 0.35, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1594, 6728, 1468, 628, 2302, 29} \begin {gather*} -\log \left (-x^8+27 x^4+4\right )-\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 27*x^4 - x^8 + (108*x^4 - 8*x^8)*Log[x])/((-4*x - 27*x^5 + x^9)*Log[x]),x]

[Out]

-Log[4 + 27*x^4 - x^8] - Log[Log[x]]

Rule 29

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1468

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
&& EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rule 1594

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

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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4+27 x^4-x^8+\left (108 x^4-8 x^8\right ) \log (x)}{x \left (-4-27 x^4+x^8\right ) \log (x)} \, dx\\ &=\int \left (-\frac {4 x^3 \left (-27+2 x^4\right )}{-4-27 x^4+x^8}-\frac {1}{x \log (x)}\right ) \, dx\\ &=-\left (4 \int \frac {x^3 \left (-27+2 x^4\right )}{-4-27 x^4+x^8} \, dx\right )-\int \frac {1}{x \log (x)} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )-\operatorname {Subst}\left (\int \frac {-27+2 x}{-4-27 x+x^2} \, dx,x,x^4\right )\\ &=-\log \left (4+27 x^4-x^8\right )-\log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 21, normalized size = 0.95 \begin {gather*} -\log \left (4+27 x^4-x^8\right )-\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 27*x^4 - x^8 + (108*x^4 - 8*x^8)*Log[x])/((-4*x - 27*x^5 + x^9)*Log[x]),x]

[Out]

-Log[4 + 27*x^4 - x^8] - Log[Log[x]]

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fricas [A]  time = 0.77, size = 19, normalized size = 0.86 \begin {gather*} -\log \left (x^{8} - 27 \, x^{4} - 4\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^8+108*x^4)*log(x)-x^8+27*x^4+4)/(x^9-27*x^5-4*x)/log(x),x, algorithm="fricas")

[Out]

-log(x^8 - 27*x^4 - 4) - log(log(x))

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giac [A]  time = 0.30, size = 19, normalized size = 0.86 \begin {gather*} -\log \left (x^{8} - 27 \, x^{4} - 4\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^8+108*x^4)*log(x)-x^8+27*x^4+4)/(x^9-27*x^5-4*x)/log(x),x, algorithm="giac")

[Out]

-log(x^8 - 27*x^4 - 4) - log(log(x))

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maple [A]  time = 0.04, size = 20, normalized size = 0.91

method	result	size
default	\(-\ln \left (\ln \relax (x )\right )-\ln \left (x^{8}-27 x^{4}-4\right )\)	\(20\)
norman	\(-\ln \left (\ln \relax (x )\right )-\ln \left (x^{8}-27 x^{4}-4\right )\)	\(20\)
risch	\(-\ln \left (\ln \relax (x )\right )-\ln \left (x^{8}-27 x^{4}-4\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^8+108*x^4)*ln(x)-x^8+27*x^4+4)/(x^9-27*x^5-4*x)/ln(x),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(x))-ln(x^8-27*x^4-4)

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maxima [A]  time = 0.43, size = 19, normalized size = 0.86 \begin {gather*} -\log \left (x^{8} - 27 \, x^{4} - 4\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^8+108*x^4)*log(x)-x^8+27*x^4+4)/(x^9-27*x^5-4*x)/log(x),x, algorithm="maxima")

[Out]

-log(x^8 - 27*x^4 - 4) - log(log(x))

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mupad [B]  time = 1.28, size = 19, normalized size = 0.86 \begin {gather*} -\ln \left (\ln \relax (x)\right )-\ln \left (x^8-27\,x^4-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(108*x^4 - 8*x^8) + 27*x^4 - x^8 + 4)/(log(x)*(4*x + 27*x^5 - x^9)),x)

[Out]

- log(log(x)) - log(x^8 - 27*x^4 - 4)

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sympy [A]  time = 0.17, size = 17, normalized size = 0.77 \begin {gather*} - \log {\left (x^{8} - 27 x^{4} - 4 \right )} - \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**8+108*x**4)*ln(x)-x**8+27*x**4+4)/(x**9-27*x**5-4*x)/ln(x),x)

[Out]

-log(x**8 - 27*x**4 - 4) - log(log(x))

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