3.86.91 \(\int \frac {20 x^2+24 x^3+9 x^4+x^5+(12+20 x+221 x^2+208 x^3+71 x^4+8 x^5) \log (x)+(60 x^2+76 x^3+31 x^4+4 x^5) \log (x) \log (\log (x))}{(4+4 x+x^2) \log (x)} \, dx\)

Optimal. Leaf size=24 \[ x \left (3+x+x^2 (5+x) \left (2+\frac {3}{2+x}+\log (\log (x))\right )\right ) \]

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Rubi [A]  time = 0.57, antiderivative size = 41, normalized size of antiderivative = 1.71, number of steps used = 29, number of rules used = 8, integrand size = 89, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.090, Rules used = {27, 6688, 2353, 2309, 2178, 6742, 43, 2522} \begin {gather*} 2 x^4+x^4 \log (\log (x))+13 x^3+5 x^3 \log (\log (x))+10 x^2-15 x-\frac {72}{x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20*x^2 + 24*x^3 + 9*x^4 + x^5 + (12 + 20*x + 221*x^2 + 208*x^3 + 71*x^4 + 8*x^5)*Log[x] + (60*x^2 + 76*x^
3 + 31*x^4 + 4*x^5)*Log[x]*Log[Log[x]])/((4 + 4*x + x^2)*Log[x]),x]

[Out]

-15*x + 10*x^2 + 13*x^3 + 2*x^4 - 72/(2 + x) + 5*x^3*Log[Log[x]] + x^4*Log[Log[x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2522

Int[((a_.) + Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)]*(b_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((e*x)^(m + 1
)*(a + b*Log[c*Log[d*x^n]^p]))/(e*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(e*x)^m/Log[d*x^n], x], x] /; FreeQ
[{a, b, c, d, e, m, n, p}, x] && NeQ[m, -1]

Rule 6688

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

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 {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{(2+x)^2 \log (x)} \, dx\\ &=\int \left (\frac {x^2 (5+x)}{\log (x)}+\frac {12+20 x+221 x^2+208 x^3+71 x^4+8 x^5+x^2 (2+x)^2 (15+4 x) \log (\log (x))}{(2+x)^2}\right ) \, dx\\ &=\int \frac {x^2 (5+x)}{\log (x)} \, dx+\int \frac {12+20 x+221 x^2+208 x^3+71 x^4+8 x^5+x^2 (2+x)^2 (15+4 x) \log (\log (x))}{(2+x)^2} \, dx\\ &=\int \left (\frac {5 x^2}{\log (x)}+\frac {x^3}{\log (x)}\right ) \, dx+\int \left (\frac {12}{(2+x)^2}+\frac {20 x}{(2+x)^2}+\frac {221 x^2}{(2+x)^2}+\frac {208 x^3}{(2+x)^2}+\frac {71 x^4}{(2+x)^2}+\frac {8 x^5}{(2+x)^2}+x^2 (15+4 x) \log (\log (x))\right ) \, dx\\ &=-\frac {12}{2+x}+5 \int \frac {x^2}{\log (x)} \, dx+8 \int \frac {x^5}{(2+x)^2} \, dx+20 \int \frac {x}{(2+x)^2} \, dx+71 \int \frac {x^4}{(2+x)^2} \, dx+208 \int \frac {x^3}{(2+x)^2} \, dx+221 \int \frac {x^2}{(2+x)^2} \, dx+\int \frac {x^3}{\log (x)} \, dx+\int x^2 (15+4 x) \log (\log (x)) \, dx\\ &=-\frac {12}{2+x}+5 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+8 \int \left (-32+12 x-4 x^2+x^3-\frac {32}{(2+x)^2}+\frac {80}{2+x}\right ) \, dx+20 \int \left (-\frac {2}{(2+x)^2}+\frac {1}{2+x}\right ) \, dx+71 \int \left (12-4 x+x^2+\frac {16}{(2+x)^2}-\frac {32}{2+x}\right ) \, dx+208 \int \left (-4+x-\frac {8}{(2+x)^2}+\frac {12}{2+x}\right ) \, dx+221 \int \left (1+\frac {4}{(2+x)^2}-\frac {4}{2+x}\right ) \, dx+\int \left (15 x^2 \log (\log (x))+4 x^3 \log (\log (x))\right ) \, dx+\operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )\\ &=-15 x+10 x^2+13 x^3+2 x^4-\frac {72}{2+x}+5 \text {Ei}(3 \log (x))+\text {Ei}(4 \log (x))+4 \int x^3 \log (\log (x)) \, dx+15 \int x^2 \log (\log (x)) \, dx\\ &=-15 x+10 x^2+13 x^3+2 x^4-\frac {72}{2+x}+5 \text {Ei}(3 \log (x))+\text {Ei}(4 \log (x))+5 x^3 \log (\log (x))+x^4 \log (\log (x))-5 \int \frac {x^2}{\log (x)} \, dx-\int \frac {x^3}{\log (x)} \, dx\\ &=-15 x+10 x^2+13 x^3+2 x^4-\frac {72}{2+x}+5 \text {Ei}(3 \log (x))+\text {Ei}(4 \log (x))+5 x^3 \log (\log (x))+x^4 \log (\log (x))-5 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )-\operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )\\ &=-15 x+10 x^2+13 x^3+2 x^4-\frac {72}{2+x}+5 x^3 \log (\log (x))+x^4 \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 36, normalized size = 1.50 \begin {gather*} -15 x+10 x^2+13 x^3+2 x^4-\frac {72}{2+x}+x^3 (5+x) \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20*x^2 + 24*x^3 + 9*x^4 + x^5 + (12 + 20*x + 221*x^2 + 208*x^3 + 71*x^4 + 8*x^5)*Log[x] + (60*x^2 +
76*x^3 + 31*x^4 + 4*x^5)*Log[x]*Log[Log[x]])/((4 + 4*x + x^2)*Log[x]),x]

[Out]

-15*x + 10*x^2 + 13*x^3 + 2*x^4 - 72/(2 + x) + x^3*(5 + x)*Log[Log[x]]

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fricas [B]  time = 0.72, size = 49, normalized size = 2.04 \begin {gather*} \frac {2 \, x^{5} + 17 \, x^{4} + 36 \, x^{3} + 5 \, x^{2} + {\left (x^{5} + 7 \, x^{4} + 10 \, x^{3}\right )} \log \left (\log \relax (x)\right ) - 30 \, x - 72}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+31*x^4+76*x^3+60*x^2)*log(x)*log(log(x))+(8*x^5+71*x^4+208*x^3+221*x^2+20*x+12)*log(x)+x^5+9
*x^4+24*x^3+20*x^2)/(x^2+4*x+4)/log(x),x, algorithm="fricas")

[Out]

(2*x^5 + 17*x^4 + 36*x^3 + 5*x^2 + (x^5 + 7*x^4 + 10*x^3)*log(log(x)) - 30*x - 72)/(x + 2)

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giac [A]  time = 0.24, size = 39, normalized size = 1.62 \begin {gather*} 2 \, x^{4} + 13 \, x^{3} + 10 \, x^{2} + {\left (x^{4} + 5 \, x^{3}\right )} \log \left (\log \relax (x)\right ) - 15 \, x - \frac {72}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+31*x^4+76*x^3+60*x^2)*log(x)*log(log(x))+(8*x^5+71*x^4+208*x^3+221*x^2+20*x+12)*log(x)+x^5+9
*x^4+24*x^3+20*x^2)/(x^2+4*x+4)/log(x),x, algorithm="giac")

[Out]

2*x^4 + 13*x^3 + 10*x^2 + (x^4 + 5*x^3)*log(log(x)) - 15*x - 72/(x + 2)

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maple [A]  time = 0.42, size = 46, normalized size = 1.92

method	result	size
risch	\(\left (x^{4}+5 x^{3}\right ) \ln \left (\ln \relax (x )\right )+\frac {2 x^{5}+17 x^{4}+36 x^{3}+5 x^{2}-30 x -72}{2+x}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^5+31*x^4+76*x^3+60*x^2)*ln(x)*ln(ln(x))+(8*x^5+71*x^4+208*x^3+221*x^2+20*x+12)*ln(x)+x^5+9*x^4+24*x^
3+20*x^2)/(x^2+4*x+4)/ln(x),x,method=_RETURNVERBOSE)

[Out]

(x^4+5*x^3)*ln(ln(x))+(2*x^5+17*x^4+36*x^3+5*x^2-30*x-72)/(2+x)

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maxima [B]  time = 0.38, size = 49, normalized size = 2.04 \begin {gather*} \frac {2 \, x^{5} + 17 \, x^{4} + 36 \, x^{3} + 5 \, x^{2} + {\left (x^{5} + 7 \, x^{4} + 10 \, x^{3}\right )} \log \left (\log \relax (x)\right ) - 30 \, x - 72}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+31*x^4+76*x^3+60*x^2)*log(x)*log(log(x))+(8*x^5+71*x^4+208*x^3+221*x^2+20*x+12)*log(x)+x^5+9
*x^4+24*x^3+20*x^2)/(x^2+4*x+4)/log(x),x, algorithm="maxima")

[Out]

(2*x^5 + 17*x^4 + 36*x^3 + 5*x^2 + (x^5 + 7*x^4 + 10*x^3)*log(log(x)) - 30*x - 72)/(x + 2)

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mupad [B]  time = 5.46, size = 39, normalized size = 1.62 \begin {gather*} \ln \left (\ln \relax (x)\right )\,\left (x^4+5\,x^3\right )-15\,x-\frac {72}{x+2}+10\,x^2+13\,x^3+2\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(20*x + 221*x^2 + 208*x^3 + 71*x^4 + 8*x^5 + 12) + 20*x^2 + 24*x^3 + 9*x^4 + x^5 + log(log(x))*log
(x)*(60*x^2 + 76*x^3 + 31*x^4 + 4*x^5))/(log(x)*(4*x + x^2 + 4)),x)

[Out]

log(log(x))*(5*x^3 + x^4) - 15*x - 72/(x + 2) + 10*x^2 + 13*x^3 + 2*x^4

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sympy [A]  time = 0.38, size = 36, normalized size = 1.50 \begin {gather*} 2 x^{4} + 13 x^{3} + 10 x^{2} - 15 x + \left (x^{4} + 5 x^{3}\right ) \log {\left (\log {\relax (x )} \right )} - \frac {72}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**5+31*x**4+76*x**3+60*x**2)*ln(x)*ln(ln(x))+(8*x**5+71*x**4+208*x**3+221*x**2+20*x+12)*ln(x)+x
**5+9*x**4+24*x**3+20*x**2)/(x**2+4*x+4)/ln(x),x)

[Out]

2*x**4 + 13*x**3 + 10*x**2 - 15*x + (x**4 + 5*x**3)*log(log(x)) - 72/(x + 2)

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