3.7.22 \(\int \frac {4+x^4+(4 x+4 x^4+x^5) \log (x)}{(4 x+x^5) \log (x)} \, dx\)

Optimal. Leaf size=11 \[ x+\log \left (\left (4+x^4\right ) \log (x)\right ) \]

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Rubi [A]  time = 0.26, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1593, 6725, 1885, 260, 2302, 29} \begin {gather*} \log \left (x^4+4\right )+x+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + x^4 + (4*x + 4*x^4 + x^5)*Log[x])/((4*x + x^5)*Log[x]),x]

[Out]

x + Log[4 + x^4] + Log[Log[x]]

Rule 29

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1593

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

Rule 1885

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[P
q, x, j + (k*n)/2]*x^((k*n)/2), {k, 0, (2*(q - j))/n + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b
, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

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 6725

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4+x^4+\left (4 x+4 x^4+x^5\right ) \log (x)}{x \left (4+x^4\right ) \log (x)} \, dx\\ &=\int \left (\frac {4+4 x^3+x^4}{4+x^4}+\frac {1}{x \log (x)}\right ) \, dx\\ &=\int \frac {4+4 x^3+x^4}{4+x^4} \, dx+\int \frac {1}{x \log (x)} \, dx\\ &=\int \left (1+\frac {4 x^3}{4+x^4}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=x+\log (\log (x))+4 \int \frac {x^3}{4+x^4} \, dx\\ &=x+\log \left (4+x^4\right )+\log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 11, normalized size = 1.00 \begin {gather*} x+\log \left (4+x^4\right )+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + x^4 + (4*x + 4*x^4 + x^5)*Log[x])/((4*x + x^5)*Log[x]),x]

[Out]

x + Log[4 + x^4] + Log[Log[x]]

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fricas [A]  time = 0.72, size = 11, normalized size = 1.00 \begin {gather*} x + \log \left (x^{4} + 4\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+4*x^4+4*x)*log(x)+x^4+4)/(x^5+4*x)/log(x),x, algorithm="fricas")

[Out]

x + log(x^4 + 4) + log(log(x))

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giac [A]  time = 0.30, size = 11, normalized size = 1.00 \begin {gather*} x + \log \left (x^{4} + 4\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+4*x^4+4*x)*log(x)+x^4+4)/(x^5+4*x)/log(x),x, algorithm="giac")

[Out]

x + log(x^4 + 4) + log(log(x))

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maple [A]  time = 0.14, size = 12, normalized size = 1.09

method	result	size
risch	\(x +\ln \left (x^{4}+4\right )+\ln \left (\ln \relax (x )\right )\)	\(12\)
default	\(\ln \left (\ln \relax (x )\right )+x +\ln \left (x^{2}+2 x +2\right )+\ln \left (x^{2}-2 x +2\right )\)	\(24\)
norman	\(\ln \left (\ln \relax (x )\right )+x +\ln \left (x^{2}+2 x +2\right )+\ln \left (x^{2}-2 x +2\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5+4*x^4+4*x)*ln(x)+x^4+4)/(x^5+4*x)/ln(x),x,method=_RETURNVERBOSE)

[Out]

x+ln(x^4+4)+ln(ln(x))

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maxima [B]  time = 0.91, size = 23, normalized size = 2.09 \begin {gather*} x + \log \left (x^{2} + 2 \, x + 2\right ) + \log \left (x^{2} - 2 \, x + 2\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+4*x^4+4*x)*log(x)+x^4+4)/(x^5+4*x)/log(x),x, algorithm="maxima")

[Out]

x + log(x^2 + 2*x + 2) + log(x^2 - 2*x + 2) + log(log(x))

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mupad [B]  time = 0.71, size = 11, normalized size = 1.00 \begin {gather*} x+\ln \left (\ln \relax (x)\,\left (x^4+4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(4*x + 4*x^4 + x^5) + x^4 + 4)/(log(x)*(4*x + x^5)),x)

[Out]

x + log(log(x)*(x^4 + 4))

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sympy [A]  time = 0.13, size = 12, normalized size = 1.09 \begin {gather*} x + \log {\left (x^{4} + 4 \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**5+4*x**4+4*x)*ln(x)+x**4+4)/(x**5+4*x)/ln(x),x)

[Out]

x + log(x**4 + 4) + log(log(x))

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