Integrand size = 53, antiderivative size = 23 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {-\frac {23}{3}-\frac {4}{e^4}+\log (10)}{x \log (x \log (\log (2)))} \]
Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {-12+e^4 (-23+\log (1000))}{3 e^4 x \log (x \log (\log (2)))} \]
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*Log[Log[2]]]^2),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.43 (sec) , antiderivative size = 146, normalized size of antiderivative = 6.35, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {27, 2813, 25, 2010, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))+23 e^4+12-3 e^4 \log (10)}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+e^4 (23-3 \log (10))+12}{x^2 \log ^2(x \log (\log (2)))}dx}{3 e^4}\) |
\(\Big \downarrow \) 2813 |
\(\displaystyle \frac {-\left (12+e^4 (23-3 \log (10))\right ) \int -\frac {x \log (\log (2)) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))+\frac {1}{\log (x \log (\log (2)))}}{x^2}dx-\log (\log (2)) \left (\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+12+e^4 (23-3 \log (10))\right ) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))-\frac {\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+12+e^4 (23-3 \log (10))}{x \log (x \log (\log (2)))}}{3 e^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\left (12+e^4 (23-3 \log (10))\right ) \int \frac {x \log (\log (2)) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))+\frac {1}{\log (x \log (\log (2)))}}{x^2}dx-\log (\log (2)) \left (\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+12+e^4 (23-3 \log (10))\right ) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))-\frac {\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+12+e^4 (23-3 \log (10))}{x \log (x \log (\log (2)))}}{3 e^4}\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {\left (12+e^4 (23-3 \log (10))\right ) \int \left (\frac {\log (\log (2)) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))}{x}+\frac {1}{x^2 \log (x \log (\log (2)))}\right )dx-\log (\log (2)) \left (\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+12+e^4 (23-3 \log (10))\right ) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))-\frac {\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+12+e^4 (23-3 \log (10))}{x \log (x \log (\log (2)))}}{3 e^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\log (\log (2)) \left (\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+12+e^4 (23-3 \log (10))\right ) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))+\left (12+e^4 (23-3 \log (10))\right ) \left (\log (\log (2)) \log (x \log (\log (2))) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))+\log (\log (2)) \operatorname {ExpIntegralEi}(-\log (x \log (\log (2))))+\frac {1}{x}\right )-\frac {\left (12+e^4 (23-3 \log (10))\right ) \log (x \log (\log (2)))+12+e^4 (23-3 \log (10))}{x \log (x \log (\log (2)))}}{3 e^4}\) |
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]
(-(ExpIntegralEi[-Log[x*Log[Log[2]]]]*Log[Log[2]]*(12 + E^4*(23 - 3*Log[10 ]) + (12 + E^4*(23 - 3*Log[10]))*Log[x*Log[Log[2]]])) - (12 + E^4*(23 - 3* Log[10]) + (12 + E^4*(23 - 3*Log[10]))*Log[x*Log[Log[2]]])/(x*Log[x*Log[Lo g[2]]]) + (12 + E^4*(23 - 3*Log[10]))*(x^(-1) + ExpIntegralEi[-Log[x*Log[L og[2]]]]*Log[Log[2]] + ExpIntegralEi[-Log[x*Log[Log[2]]]]*Log[Log[2]]*Log[ x*Log[Log[2]]]))/(3*E^4)
3.9.10.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_ .)]*(e_.))*((g_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Simp[(d + e*Log[f*x^r]) u, x] - Simp[e*r Int[Simp lifyIntegrand[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])
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 | \(\frac {\ln \left (\ln \left (2\right )\right ) {\mathrm e}^{-4} \left (3 \,{\mathrm e}^{4} \ln \left (10\right ) \operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )-3 \,{\mathrm e}^{4} \ln \left (10\right ) \left (-\frac {1}{x \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}+\operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )\right )-23 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )+23 \,{\mathrm e}^{4} \left (-\frac {1}{x \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}+\operatorname {Ei}_{1}\left (\ln \left (x \ln \left (\ln \left (2\right )\right )\right )\right )\right )-\frac {12}{x \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )}\right )}{3}\) | \(119\) |
int(1/3*((-3*exp(4)*ln(10)+23*exp(4)+12)*ln(x*ln(ln(2)))-3*exp(4)*ln(10)+2 3*exp(4)+12)/x^2/exp(4)/ln(x*ln(ln(2)))^2,x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {{\left (3 \, e^{4} \log \left (10\right ) - 23 \, e^{4} - 12\right )} e^{\left (-4\right )}}{3 \, x \log \left (x \log \left (\log \left (2\right )\right )\right )} \]
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)/log(x*log(log(2)))^2,x, algorithm=\
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {- 23 e^{4} - 12 + 3 e^{4} \log {\left (10 \right )}}{3 x e^{4} \log {\left (x \log {\left (\log {\left (2 \right )} \right )} \right )}} \]
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)
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 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=-\frac {1}{3} \, {\left (3 \, {\rm Ei}\left (-\log \left (x \log \left (\log \left (2\right )\right )\right )\right ) e^{4} \log \left (10\right ) - 3 \, e^{4} \Gamma \left (-1, \log \left (x \log \left (\log \left (2\right )\right )\right )\right ) \log \left (10\right ) - 23 \, {\rm Ei}\left (-\log \left (x \log \left (\log \left (2\right )\right )\right )\right ) e^{4} + 23 \, e^{4} \Gamma \left (-1, \log \left (x \log \left (\log \left (2\right )\right )\right )\right ) - 12 \, {\rm Ei}\left (-\log \left (x \log \left (\log \left (2\right )\right )\right )\right ) + 12 \, \Gamma \left (-1, \log \left (x \log \left (\log \left (2\right )\right )\right )\right )\right )} e^{\left (-4\right )} \log \left (\log \left (2\right )\right ) \]
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)/log(x*log(log(2)))^2,x, algorithm=\
-1/3*(3*Ei(-log(x*log(log(2))))*e^4*log(10) - 3*e^4*gamma(-1, log(x*log(lo g(2))))*log(10) - 23*Ei(-log(x*log(log(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))
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=\frac {{\left (3 \, e^{4} \log \left (5\right ) + 3 \, e^{4} \log \left (2\right ) - 23 \, e^{4} - 12\right )} e^{\left (-4\right )}}{3 \, x \log \left (x \log \left (\log \left (2\right )\right )\right )} \]
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)/log(x*log(log(2)))^2,x, algorithm=\
Time = 9.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {12+23 e^4-3 e^4 \log (10)+\left (12+23 e^4-3 e^4 \log (10)\right ) \log (x \log (\log (2)))}{3 e^4 x^2 \log ^2(x \log (\log (2)))} \, dx=-\frac {4\,{\mathrm {e}}^{-4}-\ln \left (10\right )+\frac {23}{3}}{x\,\ln \left (x\,\ln \left (\ln \left (2\right )\right )\right )} \]