3.49.59 \(\int \frac {x^4+x^5+4 x^4 \log (3)+(x^4-12 x^5-7 x^6-24 x^5 \log (3)) \log (x)+5 x^4 \log ^2(x)+(5 x^4+6 x^5+20 x^4 \log (3)) \log (x) \log (\log (x))}{\log (x)} \, dx\)

Optimal. Leaf size=25 \[ x^5 (-x+\log (x)+(1+x+4 \log (3)) (-x+\log (\log (x)))) \]

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Rubi [A]  time = 0.83, antiderivative size = 39, normalized size of antiderivative = 1.56, number of steps used = 21, number of rules used = 8, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6, 6688, 6742, 2353, 2309, 2178, 2304, 2522} \begin {gather*} -x^7+x^6 \log (\log (x))-2 x^6 (1+\log (9))+x^5 \log (x)+x^5 (1+\log (81)) \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4 + x^5 + 4*x^4*Log[3] + (x^4 - 12*x^5 - 7*x^6 - 24*x^5*Log[3])*Log[x] + 5*x^4*Log[x]^2 + (5*x^4 + 6*x^
5 + 20*x^4*Log[3])*Log[x]*Log[Log[x]])/Log[x],x]

[Out]

-x^7 - 2*x^6*(1 + Log[9]) + x^5*Log[x] + x^6*Log[Log[x]] + x^5*(1 + Log[81])*Log[Log[x]]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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 {x^5+x^4 (1+4 \log (3))+\left (x^4-12 x^5-7 x^6-24 x^5 \log (3)\right ) \log (x)+5 x^4 \log ^2(x)+\left (5 x^4+6 x^5+20 x^4 \log (3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx\\ &=\int \frac {x^4 \left (1+x+\log (81)+5 \log ^2(x)+\log (x) \left (1-7 x^2-12 x (1+\log (9))+(5+6 x+20 \log (3)) \log (\log (x))\right )\right )}{\log (x)} \, dx\\ &=\int \left (\frac {x^4 \left (1+x+\log (81)+\log (x)-7 x^2 \log (x)-12 x (1+\log (9)) \log (x)+5 \log ^2(x)\right )}{\log (x)}+x^4 (5+6 x+20 \log (3)) \log (\log (x))\right ) \, dx\\ &=\int \frac {x^4 \left (1+x+\log (81)+\log (x)-7 x^2 \log (x)-12 x (1+\log (9)) \log (x)+5 \log ^2(x)\right )}{\log (x)} \, dx+\int x^4 (5+6 x+20 \log (3)) \log (\log (x)) \, dx\\ &=\int \left (x^4-7 x^6-12 x^5 (1+\log (9))+\frac {x^4 (1+x+\log (81))}{\log (x)}+5 x^4 \log (x)\right ) \, dx+\int \left (6 x^5 \log (\log (x))+5 x^4 (1+\log (81)) \log (\log (x))\right ) \, dx\\ &=\frac {x^5}{5}-x^7-2 x^6 (1+\log (9))+5 \int x^4 \log (x) \, dx+6 \int x^5 \log (\log (x)) \, dx+(5 (1+\log (81))) \int x^4 \log (\log (x)) \, dx+\int \frac {x^4 (1+x+\log (81))}{\log (x)} \, dx\\ &=-x^7-2 x^6 (1+\log (9))+x^5 \log (x)+x^6 \log (\log (x))+x^5 (1+\log (81)) \log (\log (x))+(-1-\log (81)) \int \frac {x^4}{\log (x)} \, dx+\int \left (\frac {x^5}{\log (x)}+\frac {x^4 (1+\log (81))}{\log (x)}\right ) \, dx-\int \frac {x^5}{\log (x)} \, dx\\ &=-x^7-2 x^6 (1+\log (9))+x^5 \log (x)+x^6 \log (\log (x))+x^5 (1+\log (81)) \log (\log (x))+(-1-\log (81)) \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )+(1+\log (81)) \int \frac {x^4}{\log (x)} \, dx+\int \frac {x^5}{\log (x)} \, dx-\operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )\\ &=-x^7-\text {Ei}(6 \log (x))-2 x^6 (1+\log (9))-\text {Ei}(5 \log (x)) (1+\log (81))+x^5 \log (x)+x^6 \log (\log (x))+x^5 (1+\log (81)) \log (\log (x))+(1+\log (81)) \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )+\operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )\\ &=-x^7-2 x^6 (1+\log (9))+x^5 \log (x)+x^6 \log (\log (x))+x^5 (1+\log (81)) \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 24, normalized size = 0.96 \begin {gather*} x^5 (-x (2+x+\log (81))+\log (x)+(1+x+\log (81)) \log (\log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4 + x^5 + 4*x^4*Log[3] + (x^4 - 12*x^5 - 7*x^6 - 24*x^5*Log[3])*Log[x] + 5*x^4*Log[x]^2 + (5*x^4
+ 6*x^5 + 20*x^4*Log[3])*Log[x]*Log[Log[x]])/Log[x],x]

[Out]

x^5*(-(x*(2 + x + Log[81])) + Log[x] + (1 + x + Log[81])*Log[Log[x]])

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fricas [A]  time = 0.56, size = 42, normalized size = 1.68 \begin {gather*} -x^{7} - 4 \, x^{6} \log \relax (3) - 2 \, x^{6} + x^{5} \log \relax (x) + {\left (x^{6} + 4 \, x^{5} \log \relax (3) + x^{5}\right )} \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^4*log(3)+6*x^5+5*x^4)*log(x)*log(log(x))+5*x^4*log(x)^2+(-24*x^5*log(3)-7*x^6-12*x^5+x^4)*log
(x)+4*x^4*log(3)+x^5+x^4)/log(x),x, algorithm="fricas")

[Out]

-x^7 - 4*x^6*log(3) - 2*x^6 + x^5*log(x) + (x^6 + 4*x^5*log(3) + x^5)*log(log(x))

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giac [A]  time = 0.15, size = 48, normalized size = 1.92 \begin {gather*} -x^{7} - 4 \, x^{6} \log \relax (3) + x^{6} \log \left (\log \relax (x)\right ) + 4 \, x^{5} \log \relax (3) \log \left (\log \relax (x)\right ) - 2 \, x^{6} + x^{5} \log \relax (x) + x^{5} \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^4*log(3)+6*x^5+5*x^4)*log(x)*log(log(x))+5*x^4*log(x)^2+(-24*x^5*log(3)-7*x^6-12*x^5+x^4)*log
(x)+4*x^4*log(3)+x^5+x^4)/log(x),x, algorithm="giac")

[Out]

-x^7 - 4*x^6*log(3) + x^6*log(log(x)) + 4*x^5*log(3)*log(log(x)) - 2*x^6 + x^5*log(x) + x^5*log(log(x))

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maple [A]  time = 0.06, size = 43, normalized size = 1.72

method	result	size
risch	\(\left (4 x^{5} \ln \relax (3)+x^{6}+x^{5}\right ) \ln \left (\ln \relax (x )\right )-4 x^{6} \ln \relax (3)-x^{7}-2 x^{6}+x^{5} \ln \relax (x )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^4*ln(3)+6*x^5+5*x^4)*ln(x)*ln(ln(x))+5*x^4*ln(x)^2+(-24*x^5*ln(3)-7*x^6-12*x^5+x^4)*ln(x)+4*x^4*ln(
3)+x^5+x^4)/ln(x),x,method=_RETURNVERBOSE)

[Out]

(4*x^5*ln(3)+x^6+x^5)*ln(ln(x))-4*x^6*ln(3)-x^7-2*x^6+x^5*ln(x)

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maxima [C]  time = 0.40, size = 66, normalized size = 2.64 \begin {gather*} -x^{7} - 4 \, x^{6} \log \relax (3) + x^{6} \log \left (\log \relax (x)\right ) - 2 \, x^{6} + x^{5} \log \relax (x) + x^{5} \log \left (\log \relax (x)\right ) + 4 \, {\left (x^{5} \log \left (\log \relax (x)\right ) - {\rm Ei}\left (5 \, \log \relax (x)\right )\right )} \log \relax (3) + 4 \, {\rm Ei}\left (5 \, \log \relax (x)\right ) \log \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^4*log(3)+6*x^5+5*x^4)*log(x)*log(log(x))+5*x^4*log(x)^2+(-24*x^5*log(3)-7*x^6-12*x^5+x^4)*log
(x)+4*x^4*log(3)+x^5+x^4)/log(x),x, algorithm="maxima")

[Out]

-x^7 - 4*x^6*log(3) + x^6*log(log(x)) - 2*x^6 + x^5*log(x) + x^5*log(log(x)) + 4*(x^5*log(log(x)) - Ei(5*log(x
)))*log(3) + 4*Ei(5*log(x))*log(3)

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mupad [B]  time = 3.50, size = 41, normalized size = 1.64 \begin {gather*} x^5\,\ln \relax (x)+\ln \left (\ln \relax (x)\right )\,\left (x^6+\left (4\,\ln \relax (3)+1\right )\,x^5\right )-x^6\,\left (4\,\ln \relax (3)+2\right )-x^7 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^4*log(x)^2 + 4*x^4*log(3) + x^4 + x^5 - log(x)*(24*x^5*log(3) - x^4 + 12*x^5 + 7*x^6) + log(log(x))*l
og(x)*(20*x^4*log(3) + 5*x^4 + 6*x^5))/log(x),x)

[Out]

x^5*log(x) + log(log(x))*(x^5*(4*log(3) + 1) + x^6) - x^6*(4*log(3) + 2) - x^7

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sympy [A]  time = 0.45, size = 41, normalized size = 1.64 \begin {gather*} - x^{7} + x^{6} \left (- 4 \log {\relax (3 )} - 2\right ) + x^{5} \log {\relax (x )} + \left (x^{6} + x^{5} + 4 x^{5} \log {\relax (3 )}\right ) \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**4*ln(3)+6*x**5+5*x**4)*ln(x)*ln(ln(x))+5*x**4*ln(x)**2+(-24*x**5*ln(3)-7*x**6-12*x**5+x**4)*
ln(x)+4*x**4*ln(3)+x**5+x**4)/ln(x),x)

[Out]

-x**7 + x**6*(-4*log(3) - 2) + x**5*log(x) + (x**6 + x**5 + 4*x**5*log(3))*log(log(x))

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