3.31.29 \(\int \frac {-5 x^4+(1+30 x^5) \log (x)-25 x^4 \log (x) \log (\log (x))}{\log (x)} \, dx\)

Optimal. Leaf size=19 \[ x+5 x^6 \left (2-\frac {x+\log (\log (x))}{x}\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 15, normalized size of antiderivative = 0.79, number of steps used = 9, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {6742, 2309, 2178, 2522} \begin {gather*} 5 x^6-5 x^5 \log (\log (x))+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5*x^4 + (1 + 30*x^5)*Log[x] - 25*x^4*Log[x]*Log[Log[x]])/Log[x],x]

[Out]

x + 5*x^6 - 5*x^5*Log[Log[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 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 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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-5 x^4+\log (x)+30 x^5 \log (x)}{\log (x)}-25 x^4 \log (\log (x))\right ) \, dx\\ &=-\left (25 \int x^4 \log (\log (x)) \, dx\right )+\int \frac {-5 x^4+\log (x)+30 x^5 \log (x)}{\log (x)} \, dx\\ &=-5 x^5 \log (\log (x))+5 \int \frac {x^4}{\log (x)} \, dx+\int \left (1+30 x^5-\frac {5 x^4}{\log (x)}\right ) \, dx\\ &=x+5 x^6-5 x^5 \log (\log (x))-5 \int \frac {x^4}{\log (x)} \, dx+5 \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )\\ &=x+5 x^6+5 \text {Ei}(5 \log (x))-5 x^5 \log (\log (x))-5 \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )\\ &=x+5 x^6-5 x^5 \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 15, normalized size = 0.79 \begin {gather*} x+5 x^6-5 x^5 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*x^4 + (1 + 30*x^5)*Log[x] - 25*x^4*Log[x]*Log[Log[x]])/Log[x],x]

[Out]

x + 5*x^6 - 5*x^5*Log[Log[x]]

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fricas [A]  time = 1.61, size = 15, normalized size = 0.79 \begin {gather*} 5 \, x^{6} - 5 \, x^{5} \log \left (\log \relax (x)\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x^4*log(x)*log(log(x))+(30*x^5+1)*log(x)-5*x^4)/log(x),x, algorithm="fricas")

[Out]

5*x^6 - 5*x^5*log(log(x)) + x

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giac [A]  time = 0.22, size = 15, normalized size = 0.79 \begin {gather*} 5 \, x^{6} - 5 \, x^{5} \log \left (\log \relax (x)\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x^4*log(x)*log(log(x))+(30*x^5+1)*log(x)-5*x^4)/log(x),x, algorithm="giac")

[Out]

5*x^6 - 5*x^5*log(log(x)) + x

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maple [A]  time = 0.03, size = 16, normalized size = 0.84

method	result	size
risch	\(5 x^{6}-5 x^{5} \ln \left (\ln \relax (x )\right )+x\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-25*x^4*ln(x)*ln(ln(x))+(30*x^5+1)*ln(x)-5*x^4)/ln(x),x,method=_RETURNVERBOSE)

[Out]

5*x^6-5*x^5*ln(ln(x))+x

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maxima [A]  time = 0.52, size = 15, normalized size = 0.79 \begin {gather*} 5 \, x^{6} - 5 \, x^{5} \log \left (\log \relax (x)\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x^4*log(x)*log(log(x))+(30*x^5+1)*log(x)-5*x^4)/log(x),x, algorithm="maxima")

[Out]

5*x^6 - 5*x^5*log(log(x)) + x

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mupad [B]  time = 1.77, size = 15, normalized size = 0.79 \begin {gather*} x-5\,x^5\,\ln \left (\ln \relax (x)\right )+5\,x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x^4 - log(x)*(30*x^5 + 1) + 25*x^4*log(log(x))*log(x))/log(x),x)

[Out]

x - 5*x^5*log(log(x)) + 5*x^6

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sympy [A]  time = 0.31, size = 15, normalized size = 0.79 \begin {gather*} 5 x^{6} - 5 x^{5} \log {\left (\log {\relax (x )} \right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x**4*ln(x)*ln(ln(x))+(30*x**5+1)*ln(x)-5*x**4)/ln(x),x)

[Out]

5*x**6 - 5*x**5*log(log(x)) + x

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