\(\int \frac {(\frac {139}{12})^{\frac {1}{\log (5)}} (1-\log (x)) (-\frac {\log (x)}{x})^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx\) [6444]

Optimal result

Integrand size = 38, antiderivative size = 21 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \]

[Out]

exp(ln(-139/12*ln(x)/x)/ln(5))

Rubi [C] (verified)

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

Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 8.48, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {12, 6851, 2347, 2212, 2413, 6692} \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=-\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \log ^{\frac {1}{\log (5)}-1}(5) x^{\frac {1}{\log (5)}} (1-\log (x)) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )+\left (\frac {139 \log (5)}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \Gamma \left (1+\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )-\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \log ^{\frac {1}{\log (5)}-1}(5) x^{\frac {1}{\log (5)}} \log ^{1-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \]

[In]

Int[((139/12)^Log[5]^(-1)*(1 - Log[x])*(-(Log[x]/x))^Log[5]^(-1))/(x*Log[5]*Log[x]),x]

[Out]

-((139/12)^Log[5]^(-1)*x^Log[5]^(-1)*Gamma[Log[5]^(-1), Log[x]/Log[5]]*Log[5]^(-1 + Log[5]^(-1))*Log[x]^(1 - L
og[5]^(-1))*(-(Log[x]/x))^Log[5]^(-1)) + (x^Log[5]^(-1)*Gamma[1 + Log[5]^(-1), Log[x]/Log[5]]*((139*Log[5])/12
)^Log[5]^(-1)*(-(Log[x]/x))^Log[5]^(-1))/Log[x]^Log[5]^(-1) - ((139/12)^Log[5]^(-1)*x^Log[5]^(-1)*Gamma[Log[5]
^(-1), Log[x]/Log[5]]*Log[5]^(-1 + Log[5]^(-1))*(1 - Log[x])*(-(Log[x]/x))^Log[5]^(-1))/Log[x]^Log[5]^(-1)

Rule 12

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

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

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 6692

Int[Gamma[n_, (a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(Gamma[n, a + b*x]/b), x] - Simp[Gamma[n + 1, a
+ b*x]/b, x] /; FreeQ[{a, b, n}, x]

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \int \frac {(1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (x)} \, dx}{\log (5)} \\ & = \frac {\left (\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}\right ) \int x^{-1-\frac {1}{\log (5)}} (1-\log (x)) \log ^{-1+\frac {1}{\log (5)}}(x) \, dx}{\log (5)} \\ & = -\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \log ^{-1+\frac {1}{\log (5)}}(5) (1-\log (x)) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}-\frac {\left (\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}\right ) \int \frac {\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \sqrt [\log (5)]{\log (5)}}{x} \, dx}{\log (5)} \\ & = -\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \log ^{-1+\frac {1}{\log (5)}}(5) (1-\log (x)) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}-\left (\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-1+\frac {1}{\log (5)}}(5) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}\right ) \int \frac {\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )}{x} \, dx \\ & = -\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \log ^{-1+\frac {1}{\log (5)}}(5) (1-\log (x)) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}-\left (\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-1+\frac {1}{\log (5)}}(5) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}\right ) \text {Subst}\left (\int \Gamma \left (\frac {1}{\log (5)},\frac {x}{\log (5)}\right ) \, dx,x,\log (x)\right ) \\ & = -\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \log ^{-1+\frac {1}{\log (5)}}(5) \log ^{1-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}+x^{\frac {1}{\log (5)}} \Gamma \left (1+\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \left (\frac {139 \log (5)}{12}\right )^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}-\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \log ^{-1+\frac {1}{\log (5)}}(5) (1-\log (x)) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \]

[In]

Integrate[((139/12)^Log[5]^(-1)*(1 - Log[x])*(-(Log[x]/x))^Log[5]^(-1))/(x*Log[5]*Log[x]),x]

[Out]

(139/12)^Log[5]^(-1)*(-(Log[x]/x))^Log[5]^(-1)

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71

method	result	size
norman	\({\mathrm e}^{\frac {\ln \left (-\frac {139 \ln \left (x \right )}{12 x}\right )}{\ln \left (5\right )}}\)	\(15\)
parallelrisch	\({\mathrm e}^{\frac {\ln \left (-\frac {139 \ln \left (x \right )}{12 x}\right )}{\ln \left (5\right )}}\)	\(15\)
risch	\(\left (\frac {1}{3}\right )^{\frac {1}{\ln \left (5\right )}} x^{-\frac {1}{\ln \left (5\right )}} \left (\frac {1}{4}\right )^{\frac {1}{\ln \left (5\right )}} \ln \left (x \right )^{\frac {1}{\ln \left (5\right )}} 139^{\frac {1}{\ln \left (5\right )}} {\mathrm e}^{\frac {i \pi \left (\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right )-\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right )+\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{3}+\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-2 \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{2}+2\right )}{2 \ln \left (5\right )}}\)	\(131\)

[In]

int((1-ln(x))*exp(ln(-139/12*ln(x)/x)/ln(5))/x/ln(5)/ln(x),x,method=_RETURNVERBOSE)

[Out]

exp(ln(-139/12*ln(x)/x)/ln(5))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\left (-\frac {139 \, \log \left (x\right )}{12 \, x}\right )^{\left (\frac {1}{\log \left (5\right )}\right )} \]

[In]

integrate((1-log(x))*exp(log(-139/12*log(x)/x)/log(5))/x/log(5)/log(x),x, algorithm="fricas")

[Out]

(-139/12*log(x)/x)^(1/log(5))

Sympy [A] (verification not implemented)

Time = 18.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\frac {139^{\frac {1}{\log {\left (5 \right )}}} \left (- \frac {\log {\left (x \right )}}{x}\right )^{\frac {1}{\log {\left (5 \right )}}}}{12^{\frac {1}{\log {\left (5 \right )}}}} \]

[In]

integrate((1-ln(x))*exp(ln(-139/12*ln(x)/x)/ln(5))/x/ln(5)/ln(x),x)

[Out]

139**(1/log(5))*(-log(x)/x)**(1/log(5))/12**(1/log(5))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (12) = 24\).

Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.14 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\frac {139^{\left (\frac {1}{\log \left (5\right )}\right )} e^{\left (-\frac {\log \left (x\right )}{\log \left (5\right )} + \frac {\log \left (-\log \left (x\right )\right )}{\log \left (5\right )}\right )}}{3^{\left (\frac {1}{\log \left (5\right )}\right )} 2^{\frac {2}{\log \left (5\right )}}} \]

[In]

integrate((1-log(x))*exp(log(-139/12*log(x)/x)/log(5))/x/log(5)/log(x),x, algorithm="maxima")

[Out]

139^(1/log(5))*e^(-log(x)/log(5) + log(-log(x))/log(5))/(3^(1/log(5))*2^(2/log(5)))

Giac [F]

\[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\int { -\frac {\left (-\frac {139 \, \log \left (x\right )}{12 \, x}\right )^{\left (\frac {1}{\log \left (5\right )}\right )} {\left (\log \left (x\right ) - 1\right )}}{x \log \left (5\right ) \log \left (x\right )} \,d x } \]

[In]

integrate((1-log(x))*exp(log(-139/12*log(x)/x)/log(5))/x/log(5)/log(x),x, algorithm="giac")

[Out]

integrate(-(-139/12*log(x)/x)^(1/log(5))*(log(x) - 1)/(x*log(5)*log(x)), x)

Mupad [B] (verification not implemented)

Time = 11.71 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx={\left (-\frac {139\,\ln \left (x\right )}{12\,x}\right )}^{\frac {1}{\ln \left (5\right )}} \]

[In]

int(-((log(x) - 1)*(-(139*log(x))/(12*x))^(1/log(5)))/(x*log(5)*log(x)),x)

[Out]

(-(139*log(x))/(12*x))^(1/log(5))