\(\int \log ^{-\log (e \pi )}(x) \log (\frac {x}{\pi }) \, dx\) [294]

Optimal result

Integrand size = 16, antiderivative size = 9 \[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=x \log ^{-\log (\pi )}(x) \]

[Out]

x/(ln(x)^ln(Pi))

Rubi [C] (verified)

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

Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 9.67, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2336, 2212, 2408, 19, 6692} \[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=(-\log (x))^{1+\log (\pi )} \log \left (\frac {x}{\pi }\right ) \log ^{-\log (e \pi )}(x) \Gamma (-\log (\pi ),-\log (x))+(-\log (x))^{\log (\pi )} \log ^{-\log (\pi )}(x) \Gamma (1-\log (\pi ),-\log (x))+(-\log (x))^{\log (\pi )} \log ^{1-\log (\pi )}(x) \Gamma (-\log (\pi ),-\log (x)) \]

[In]

Int[Log[x/Pi]/Log[x]^Log[E*Pi],x]

[Out]

Gamma[-Log[Pi], -Log[x]]*(-Log[x])^Log[Pi]*Log[x]^(1 - Log[Pi]) + (Gamma[1 - Log[Pi], -Log[x]]*(-Log[x])^Log[P
i])/Log[x]^Log[Pi] + (Gamma[-Log[Pi], -Log[x]]*(-Log[x])^(1 + Log[Pi])*Log[x/Pi])/Log[x]^Log[E*Pi]

Rule 19

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[a^(m + n)*((b*v)^n/(a*v)^n), Int[u*v^(m + n),
x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[m + n]

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 2336

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

Rule 2408

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.)), x_Symbol] :> With[{u =
IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[SimplifyIntegrand[u/x, x], x],
x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x]

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]

Rubi steps \begin{align*} \text {integral}& = \Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{1+\log (\pi )} \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right )+\int \frac {\Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{\log (\pi )} \log ^{-\log (\pi )}(x)}{x} \, dx \\ & = \Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{1+\log (\pi )} \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right )+\left ((-\log (x))^{\log (\pi )} \log ^{-\log (\pi )}(x)\right ) \int \frac {\Gamma (-\log (\pi ),-\log (x))}{x} \, dx \\ & = \Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{1+\log (\pi )} \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right )+\left ((-\log (x))^{\log (\pi )} \log ^{-\log (\pi )}(x)\right ) \text {Subst}(\int \Gamma (-\log (\pi ),-x) \, dx,x,\log (x)) \\ & = \Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{\log (\pi )} \log ^{1-\log (\pi )}(x)+\Gamma (1-\log (\pi ),-\log (x)) (-\log (x))^{\log (\pi )} \log ^{-\log (\pi )}(x)+\Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{1+\log (\pi )} \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \\ \end{align*}

Mathematica [F]

\[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=\int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx \]

[In]

Integrate[Log[x/Pi]/Log[x]^Log[E*Pi],x]

[Out]

Integrate[Log[x/Pi]/Log[x]^Log[E*Pi], x]

Maple [F]

\[\int \ln \left (\frac {x}{\pi }\right ) \ln \left (x \right )^{-\ln \left (\pi \,{\mathrm e}\right )}d x\]

[In]

int(ln(x/Pi)/(ln(x)^ln(Pi*exp(1))),x)

[Out]

int(ln(x/Pi)/(ln(x)^ln(Pi*exp(1))),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (9) = 18\).

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 3.33 \[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=\frac {x \log \left (\pi \right ) + x \log \left (\frac {x}{\pi }\right )}{{\left (\log \left (\pi \right ) + \log \left (\frac {x}{\pi }\right )\right )}^{\log \left (\pi \right ) + 1}} \]

[In]

integrate(log(x/pi)/(log(x)^log(pi*exp(1))),x, algorithm="fricas")

[Out]

(x*log(pi) + x*log(x/pi))/(log(pi) + log(x/pi))^(log(pi) + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (7) = 14\).

Time = 18.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 6.22 \[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=- \frac {\left (- \log {\left (x \right )}\right )^{1 + \log {\left (\pi \right )}} \log {\left (\pi \right )} \Gamma \left (- \log {\left (\pi \right )}, - \log {\left (x \right )}\right )}{\log {\left (x \right )}^{1 + \log {\left (\pi \right )}}} + \frac {\left (- \log {\left (x \right )}\right )^{\log {\left (\pi \right )}} \Gamma \left (1 - \log {\left (\pi \right )}, - \log {\left (x \right )}\right )}{\log {\left (x \right )}^{\log {\left (\pi \right )}}} \]

[In]

integrate(ln(x/pi)/(ln(x)**ln(pi*exp(1))),x)

[Out]

-(-log(x))**(1 + log(pi))*log(pi)*log(x)**(-log(pi) - 1)*uppergamma(-log(pi), -log(x)) + (-log(x))**log(pi)*up
pergamma(1 - log(pi), -log(x))/log(x)**log(pi)

Maxima [F]

\[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=\int { \frac {\log \left (\frac {x}{\pi }\right )}{\log \left (x\right )^{\log \left (\pi e\right )}} \,d x } \]

[In]

integrate(log(x/pi)/(log(x)^log(pi*exp(1))),x, algorithm="maxima")

[Out]

integrate(log(x)^(-log(pi*e))*log(x/pi), x)

Giac [F]

\[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=\int { \frac {\log \left (\frac {x}{\pi }\right )}{\log \left (x\right )^{\log \left (\pi e\right )}} \,d x } \]

[In]

integrate(log(x/pi)/(log(x)^log(pi*exp(1))),x, algorithm="giac")

[Out]

integrate(log(x/pi)/log(x)^log(pi*e), x)

Mupad [F(-1)]

Timed out. \[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=\int \frac {\ln \left (\frac {x}{\Pi }\right )}{{\ln \left (x\right )}^{\ln \left (\Pi \,\mathrm {e}\right )}} \,d x \]

[In]

int(log(x/Pi)/log(x)^log(Pi*exp(1)),x)

[Out]

int(log(x/Pi)/log(x)^log(Pi*exp(1)), x)