\(\int \frac {12+23 e^4-3 e^4 \log (10)+(12+23 e^4-3 e^4 \log (10)) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx\) [3810]

Optimal result

Integrand size = 53, antiderivative size = 23 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {-\frac {23}{3}-\frac {4}{e^4}+\log (10)}{x \log (x \log (\log (2)))} \]

[Out]

(ln(10)-4/exp(4)-23/3)/x/ln(x*ln(ln(2)))

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.14 (sec) , antiderivative size = 190, normalized size of antiderivative = 8.26, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {12, 2343, 2346, 2209, 2413, 14, 6617} \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {\left (12+e^4 (23-3 \log (10))\right ) \log (\log (2)) \log (x \log (\log (2))) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))}{3 e^4}-\frac {\log (\log (2)) \left (\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+12+e^4 (23-3 \log (10))\right ) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))}{3 e^4}+\frac {\left (12+e^4 (23-3 \log (10))\right ) \log (\log (2)) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))}{3 e^4}-\frac {\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+12+e^4 (23-3 \log (10))}{3 e^4 x \log (x \log (\log (2)))}+\frac {12+e^4 (23-\log (1000))}{3 e^4 x} \]

[In]

Int[(12 + 23*E^4 - 3*E^4*Log[10] + (12 + 23*E^4 - 3*E^4*Log[10])*Log[x*Log[Log[2]]])/(3*E^4*x^2*Log[x*Log[Log[
2]]]^2),x]

[Out]

(12 + E^4*(23 - Log[1000]))/(3*E^4*x) + (ExpIntegralEi[-Log[x*Log[Log[2]]]]*(12 + E^4*(23 - 3*Log[10]))*Log[Lo
g[2]])/(3*E^4) + (ExpIntegralEi[-Log[x*Log[Log[2]]]]*(12 + E^4*(23 - 3*Log[10]))*Log[Log[2]]*Log[x*Log[Log[2]]
])/(3*E^4) - (ExpIntegralEi[-Log[x*Log[Log[2]]]]*Log[Log[2]]*(12 + E^4*(23 - 3*Log[10]) + (12 + E^4*(23 - 3*Lo
g[10]))*Log[x*Log[Log[2]]]))/(3*E^4) - (12 + E^4*(23 - 3*Log[10]) + (12 + E^4*(23 - 3*Log[10]))*Log[x*Log[Log[
2]]])/(3*E^4*x*Log[x*Log[Log[2]]])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2209

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

Rule 2343

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 2346

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 2413

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 6617

Int[ExpIntegralEi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(ExpIntegralEi[a + b*x]/b), x] - Simp[E^(a
+ b*x)/b, x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{x^2 \log ^2(x \log (\log (2)))} \, dx}{3 e^4} \\ & = -\frac {\text {Ei}(-\log (x \log (\log (2)))) \log (\log (2)) \left (12+e^4 (23-3 \log (10))+\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))\right )}{3 e^4}-\frac {12+e^4 (23-3 \log (10))+\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))}{3 e^4 x \log (x \log (\log (2)))}-\frac {\left (12+e^4 (23-3 \log (10))\right ) \int \frac {-x \text {Ei}(-\log (x \log (\log (2)))) \log (\log (2))-\frac {1}{\log (x \log (\log (2)))}}{x^2} \, dx}{3 e^4} \\ & = -\frac {\text {Ei}(-\log (x \log (\log (2)))) \log (\log (2)) \left (12+e^4 (23-3 \log (10))+\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))\right )}{3 e^4}-\frac {12+e^4 (23-3 \log (10))+\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))}{3 e^4 x \log (x \log (\log (2)))}-\frac {\left (12+e^4 (23-3 \log (10))\right ) \int \left (-\frac {\text {Ei}(-\log (x \log (\log (2)))) \log (\log (2))}{x}-\frac {1}{x^2 \log (x \log (\log (2)))}\right ) \, dx}{3 e^4} \\ & = -\frac {\text {Ei}(-\log (x \log (\log (2)))) \log (\log (2)) \left (12+e^4 (23-3 \log (10))+\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))\right )}{3 e^4}-\frac {12+e^4 (23-3 \log (10))+\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))}{3 e^4 x \log (x \log (\log (2)))}+\frac {\left (12+e^4 (23-3 \log (10))\right ) \int \frac {1}{x^2 \log (x \log (\log (2)))} \, dx}{3 e^4}+\frac {\left (\left (12+e^4 (23-3 \log (10))\right ) \log (\log (2))\right ) \int \frac {\text {Ei}(-\log (x \log (\log (2))))}{x} \, dx}{3 e^4} \\ & = -\frac {\text {Ei}(-\log (x \log (\log (2)))) \log (\log (2)) \left (12+e^4 (23-3 \log (10))+\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))\right )}{3 e^4}-\frac {12+e^4 (23-3 \log (10))+\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))}{3 e^4 x \log (x \log (\log (2)))}+\frac {\left (\left (12+e^4 (23-3 \log (10))\right ) \log (\log (2))\right ) \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x \log (\log (2)))\right )}{3 e^4}+\frac {\left (\left (12+e^4 (23-3 \log (10))\right ) \log (\log (2))\right ) \text {Subst}(\int \text {Ei}(-x) \, dx,x,\log (x \log (\log (2))))}{3 e^4} \\ & = \frac {12+e^4 (23-3 \log (10))}{3 e^4 x}+\frac {\text {Ei}(-\log (x \log (\log (2)))) \left (12+e^4 (23-3 \log (10))\right ) \log (\log (2))}{3 e^4}+\frac {\text {Ei}(-\log (x \log (\log (2)))) \left (12+e^4 (23-3 \log (10))\right ) \log (\log (2)) \log (x \log (\log (2)))}{3 e^4}-\frac {\text {Ei}(-\log (x \log (\log (2)))) \log (\log (2)) \left (12+e^4 (23-3 \log (10))+\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))\right )}{3 e^4}-\frac {12+e^4 (23-3 \log (10))+\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))}{3 e^4 x \log (x \log (\log (2)))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {-12+e^4 (-23+\log (1000))}{3 e^4 x \log (x \log (\log (2)))} \]

[In]

Integrate[(12 + 23*E^4 - 3*E^4*Log[10] + (12 + 23*E^4 - 3*E^4*Log[10])*Log[x*Log[Log[2]]])/(3*E^4*x^2*Log[x*Lo
g[Log[2]]]^2),x]

[Out]

(-12 + E^4*(-23 + Log[1000]))/(3*E^4*x*Log[x*Log[Log[2]]])

Maple [A] (verified)

Time = 1.73 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30

method	result	size
norman	\(\frac {{\mathrm e}^{-4} \left (3 \,{\mathrm e}^{4} \ln \left (10\right )-23 \,{\mathrm e}^{4}-12\right )}{3 x \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}\)	\(30\)
parallelrisch	\(\frac {{\mathrm e}^{-4} \left (3 \,{\mathrm e}^{4} \ln \left (10\right )-23 \,{\mathrm e}^{4}-12\right )}{3 x \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}\)	\(30\)
risch	\(\frac {{\mathrm e}^{-4} \left (3 \,{\mathrm e}^{4} \ln \left (2\right )+3 \,{\mathrm e}^{4} \ln \left (5\right )-23 \,{\mathrm e}^{4}-12\right )}{3 x \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}\)	\(34\)
parts	\(-\frac {{\mathrm e}^{-4} \left (3 \,{\mathrm e}^{4} \ln \left (10\right )-23 \,{\mathrm e}^{4}-12\right ) \ln \left (\ln \left (2\right )\right ) \left (-\frac {1}{x \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}+\operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )\right )}{3}+\frac {{\mathrm e}^{-4} \left (3 \,{\mathrm e}^{4} \ln \left (10\right )-23 \,{\mathrm e}^{4}-12\right ) \ln \left (\ln \left (2\right )\right ) \operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )}{3}\)	\(79\)
derivativedivides	\(\frac {\ln \left (\ln \left (2\right )\right ) {\mathrm e}^{-4} \left (3 \,{\mathrm e}^{4} \ln \left (10\right ) \operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )-3 \,{\mathrm e}^{4} \ln \left (10\right ) \left (-\frac {1}{x \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}+\operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )\right )-23 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )+23 \,{\mathrm e}^{4} \left (-\frac {1}{x \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}+\operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )\right )-\frac {12}{x \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}\right )}{3}\)	\(119\)
default	\(\frac {\ln \left (\ln \left (2\right )\right ) {\mathrm e}^{-4} \left (3 \,{\mathrm e}^{4} \ln \left (10\right ) \operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )-3 \,{\mathrm e}^{4} \ln \left (10\right ) \left (-\frac {1}{x \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}+\operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )\right )-23 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )+23 \,{\mathrm e}^{4} \left (-\frac {1}{x \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}+\operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )\right )-\frac {12}{x \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}\right )}{3}\)	\(119\)

[In]

int(1/3*((-3*exp(4)*ln(10)+23*exp(4)+12)*ln(x*ln(ln(2)))-3*exp(4)*ln(10)+23*exp(4)+12)/x^2/exp(4)/ln(x*ln(ln(2
)))^2,x,method=_RETURNVERBOSE)

[Out]

1/3/exp(4)*(3*exp(4)*ln(10)-23*exp(4)-12)/x/ln(x*ln(ln(2)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {{\left (3 \, e^{4} \log \left (10\right ) - 23 \, e^{4} - 12\right )} e^{\left (-4\right )}}{3 \, x \log \left (x \log \left (\log \left (2\right )\right )\right )} \]

[In]

integrate(1/3*((-3*exp(4)*log(10)+23*exp(4)+12)*log(x*log(log(2)))-3*exp(4)*log(10)+23*exp(4)+12)/x^2/exp(4)/l
og(x*log(log(2)))^2,x, algorithm="fricas")

[Out]

1/3*(3*e^4*log(10) - 23*e^4 - 12)*e^(-4)/(x*log(x*log(log(2))))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {- 23 e^{4} - 12 + 3 e^{4} \log {\left (10 \right )}}{3 x e^{4} \log {\left (x \log {\left (\log {\left (2 \right )} \right )} \right )}} \]

[In]

integrate(1/3*((-3*exp(4)*ln(10)+23*exp(4)+12)*ln(x*ln(ln(2)))-3*exp(4)*ln(10)+23*exp(4)+12)/x**2/exp(4)/ln(x*
ln(ln(2)))**2,x)

[Out]

(-23*exp(4) - 12 + 3*exp(4)*log(10))*exp(-4)/(3*x*log(x*log(log(2))))

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.61 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=-\frac {1}{3} \, {\left (3 \, {\rm Ei}\left (-\log \left (x \log \left (\log \left (2\right )\right )\right )\right ) e^{4} \log \left (10\right ) - 3 \, e^{4} \Gamma \left (-1, \log \left (x \log \left (\log \left (2\right )\right )\right )\right ) \log \left (10\right ) - 23 \, {\rm Ei}\left (-\log \left (x \log \left (\log \left (2\right )\right )\right )\right ) e^{4} + 23 \, e^{4} \Gamma \left (-1, \log \left (x \log \left (\log \left (2\right )\right )\right )\right ) - 12 \, {\rm Ei}\left (-\log \left (x \log \left (\log \left (2\right )\right )\right )\right ) + 12 \, \Gamma \left (-1, \log \left (x \log \left (\log \left (2\right )\right )\right )\right )\right )} e^{\left (-4\right )} \log \left (\log \left (2\right )\right ) \]

[In]

integrate(1/3*((-3*exp(4)*log(10)+23*exp(4)+12)*log(x*log(log(2)))-3*exp(4)*log(10)+23*exp(4)+12)/x^2/exp(4)/l
og(x*log(log(2)))^2,x, algorithm="maxima")

[Out]

-1/3*(3*Ei(-log(x*log(log(2))))*e^4*log(10) - 3*e^4*gamma(-1, log(x*log(log(2))))*log(10) - 23*Ei(-log(x*log(l
og(2))))*e^4 + 23*e^4*gamma(-1, log(x*log(log(2)))) - 12*Ei(-log(x*log(log(2)))) + 12*gamma(-1, log(x*log(log(
2)))))*e^(-4)*log(log(2))

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {{\left (3 \, e^{4} \log \left (5\right ) + 3 \, e^{4} \log \left (2\right ) - 23 \, e^{4} - 12\right )} e^{\left (-4\right )}}{3 \, x \log \left (x \log \left (\log \left (2\right )\right )\right )} \]

[In]

integrate(1/3*((-3*exp(4)*log(10)+23*exp(4)+12)*log(x*log(log(2)))-3*exp(4)*log(10)+23*exp(4)+12)/x^2/exp(4)/l
og(x*log(log(2)))^2,x, algorithm="giac")

[Out]

1/3*(3*e^4*log(5) + 3*e^4*log(2) - 23*e^4 - 12)*e^(-4)/(x*log(x*log(log(2))))

Mupad [B] (verification not implemented)

Time = 9.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=-\frac {4\,{\mathrm {e}}^{-4}-\ln \left (10\right )+\frac {23}{3}}{x\,\ln \left (x\,\ln \left (\ln \left (2\right )\right )\right )} \]

[In]

int((exp(-4)*((23*exp(4))/3 - exp(4)*log(10) + (log(x*log(log(2)))*(23*exp(4) - 3*exp(4)*log(10) + 12))/3 + 4)
)/(x^2*log(x*log(log(2)))^2),x)

[Out]

-(4*exp(-4) - log(10) + 23/3)/(x*log(x*log(log(2))))