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

Optimal. Leaf size=21 \[ \left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \]

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Rubi [C]  time = 0.39, antiderivative size = 178, normalized size of antiderivative = 8.48, number of steps used = 6, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {12, 6719, 2310, 2181, 2366, 6557} \begin {gather*} -\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 ) \end {gather*}

Antiderivative was successfully verified.

[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 2181

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

Rule 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2366

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 6557

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 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[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 {gather*} \begin {aligned} \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 ) \operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 21, normalized size = 1.00 \begin {gather*} \left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.60, size = 12, normalized size = 0.57 \begin {gather*} \left (-\frac {139 \, \log \relax (x)}{12 \, x}\right )^{\left (\frac {1}{\log \relax (5)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {\left (-\frac {139 \, \log \relax (x)}{12 \, x}\right )^{\left (\frac {1}{\log \relax (5)}\right )} {\left (\log \relax (x) - 1\right )}}{x \log \relax (5) \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 1.60, size = 15, normalized size = 0.71

method	result	size
norman	\({\mathrm e}^{\frac {\ln \left (-\frac {139 \ln \relax (x )}{12 x}\right )}{\ln \relax (5)}}\)	\(15\)
risch	\(x^{-\frac {1}{\ln \relax (5)}} \ln \relax (x )^{\frac {1}{\ln \relax (5)}} \left (\frac {1}{4}\right )^{\frac {1}{\ln \relax (5)}} \left (\frac {1}{3}\right )^{\frac {1}{\ln \relax (5)}} 139^{\frac {1}{\ln \relax (5)}} {\mathrm e}^{-\frac {i \pi \left (-\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}-\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )-\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right )+2 \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}-2\right )}{2 \ln \relax (5)}}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [B]  time = 0.51, size = 45, normalized size = 2.14 \begin {gather*} \frac {139^{\left (\frac {1}{\log \relax (5)}\right )} e^{\left (-\frac {\log \relax (x)}{\log \relax (5)} + \frac {\log \left (-\log \relax (x)\right )}{\log \relax (5)}\right )}}{3^{\left (\frac {1}{\log \relax (5)}\right )} 2^{\frac {2}{\log \relax (5)}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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mupad [B]  time = 4.13, size = 12, normalized size = 0.57 \begin {gather*} {\left (-\frac {139\,\ln \relax (x)}{12\,x}\right )}^{\frac {1}{\ln \relax (5)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 28.75, size = 24, normalized size = 1.14 \begin {gather*} \frac {139^{\frac {1}{\log {\relax (5 )}}} \left (- \frac {\log {\relax (x )}}{x}\right )^{\frac {1}{\log {\relax (5 )}}}}{12^{\frac {1}{\log {\relax (5 )}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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