3.92.8 \(\int \frac {1+e^{e^{e^4}}+x^4-4 x^4 \log (x)}{x \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {-1-e^{e^{e^4}}-x^4+\log (x)}{\log (x)} \]

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Rubi [A]  time = 0.36, antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6742, 2353, 2302, 30, 2306, 2309, 2178} \begin {gather*} -\frac {x^4}{\log (x)}-\frac {1+e^{e^{e^4}}}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^E^E^4 + x^4 - 4*x^4*Log[x])/(x*Log[x]^2),x]

[Out]

-((1 + E^E^E^4)/Log[x]) - x^4/Log[x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -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 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 {1+e^{e^{e^4}}+x^4}{x \log ^2(x)}-\frac {4 x^3}{\log (x)}\right ) \, dx\\ &=-\left (4 \int \frac {x^3}{\log (x)} \, dx\right )+\int \frac {1+e^{e^{e^4}}+x^4}{x \log ^2(x)} \, dx\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )\right )+\int \left (\frac {1+e^{e^{e^4}}}{x \log ^2(x)}+\frac {x^3}{\log ^2(x)}\right ) \, dx\\ &=-4 \text {Ei}(4 \log (x))+\left (1+e^{e^{e^4}}\right ) \int \frac {1}{x \log ^2(x)} \, dx+\int \frac {x^3}{\log ^2(x)} \, dx\\ &=-4 \text {Ei}(4 \log (x))-\frac {x^4}{\log (x)}+4 \int \frac {x^3}{\log (x)} \, dx+\left (1+e^{e^{e^4}}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=-4 \text {Ei}(4 \log (x))-\frac {1+e^{e^{e^4}}}{\log (x)}-\frac {x^4}{\log (x)}+4 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {1+e^{e^{e^4}}}{\log (x)}-\frac {x^4}{\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 18, normalized size = 0.78 \begin {gather*} -\frac {1+e^{e^{e^4}}+x^4}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^E^E^4 + x^4 - 4*x^4*Log[x])/(x*Log[x]^2),x]

[Out]

-((1 + E^E^E^4 + x^4)/Log[x])

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fricas [A]  time = 0.68, size = 15, normalized size = 0.65 \begin {gather*} -\frac {x^{4} + e^{\left (e^{\left (e^{4}\right )}\right )} + 1}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(exp(exp(4)))-4*x^4*log(x)+x^4+1)/x/log(x)^2,x, algorithm="fricas")

[Out]

-(x^4 + e^(e^(e^4)) + 1)/log(x)

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giac [A]  time = 0.12, size = 15, normalized size = 0.65 \begin {gather*} -\frac {x^{4} + e^{\left (e^{\left (e^{4}\right )}\right )} + 1}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(exp(exp(4)))-4*x^4*log(x)+x^4+1)/x/log(x)^2,x, algorithm="giac")

[Out]

-(x^4 + e^(e^(e^4)) + 1)/log(x)

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maple [A]  time = 0.04, size = 16, normalized size = 0.70

method	result	size
risch	\(-\frac {x^{4}+{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4}}}+1}{\ln \relax (x )}\)	\(16\)
norman	\(\frac {-x^{4}-1-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4}}}}{\ln \relax (x )}\)	\(19\)
default	\(-\frac {x^{4}}{\ln \relax (x )}-\frac {{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4}}}}{\ln \relax (x )}-\frac {1}{\ln \relax (x )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(exp(4)))-4*x^4*ln(x)+x^4+1)/x/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

-(x^4+exp(exp(exp(4)))+1)/ln(x)

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maxima [C]  time = 0.37, size = 32, normalized size = 1.39 \begin {gather*} -\frac {e^{\left (e^{\left (e^{4}\right )}\right )}}{\log \relax (x)} - \frac {1}{\log \relax (x)} - 4 \, {\rm Ei}\left (4 \, \log \relax (x)\right ) + 4 \, \Gamma \left (-1, -4 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(exp(exp(4)))-4*x^4*log(x)+x^4+1)/x/log(x)^2,x, algorithm="maxima")

[Out]

-e^(e^(e^4))/log(x) - 1/log(x) - 4*Ei(4*log(x)) + 4*gamma(-1, -4*log(x))

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mupad [B]  time = 7.64, size = 15, normalized size = 0.65 \begin {gather*} -\frac {x^4+{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^4}}+1}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(exp(4))) - 4*x^4*log(x) + x^4 + 1)/(x*log(x)^2),x)

[Out]

-(exp(exp(exp(4))) + x^4 + 1)/log(x)

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sympy [A]  time = 0.09, size = 15, normalized size = 0.65 \begin {gather*} \frac {- x^{4} - 1 - e^{e^{e^{4}}}}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(exp(exp(4)))-4*x**4*ln(x)+x**4+1)/x/ln(x)**2,x)

[Out]

(-x**4 - 1 - exp(exp(exp(4))))/log(x)

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