\(\int \log ^{-1+2251799813685248 e^{125+5 \log ^2(2)}}(x) (9007199254740992 e^{125+5 \log ^2(2)}+4 \log (x)) \, dx\) [901]

Optimal result

Integrand size = 35, antiderivative size = 18 \[ \int \log ^{-1+2251799813685248 e^{125+5 \log ^2(2)}}(x) \left (9007199254740992 e^{125+5 \log ^2(2)}+4 \log (x)\right ) \, dx=4 x \log ^{2 e^{5 (5+\log (2))^2}}(x) \]

[Out]

4*x*exp(exp(5*(ln(2)+5)^2)*ln(ln(x)))^2

Rubi [C] (verified)

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

Time = 0.08 (sec) , antiderivative size = 173, normalized size of antiderivative = 9.61, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2336, 2212, 2408, 19, 6692} \[ \int \log ^{-1+2251799813685248 e^{125+5 \log ^2(2)}}(x) \left (9007199254740992 e^{125+5 \log ^2(2)}+4 \log (x)\right ) \, dx=4 (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{1+2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x) \Gamma \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right )+4 (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x) \Gamma \left (1+2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right )-4 (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \left (\log (x)+2251799813685248 e^{5 \left (25+\log ^2(2)\right )}\right ) \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x) \Gamma \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right ) \]

[In]

Int[Log[x]^(-1 + 2251799813685248*E^(125 + 5*Log[2]^2))*(9007199254740992*E^(125 + 5*Log[2]^2) + 4*Log[x]),x]

[Out]

(4*Gamma[1 + 2251799813685248*E^(5*(25 + Log[2]^2)), -Log[x]]*Log[x]^(2251799813685248*E^(5*(25 + Log[2]^2))))
/(-Log[x])^(2251799813685248*E^(5*(25 + Log[2]^2))) + (4*Gamma[2251799813685248*E^(5*(25 + Log[2]^2)), -Log[x]
]*Log[x]^(1 + 2251799813685248*E^(5*(25 + Log[2]^2))))/(-Log[x])^(2251799813685248*E^(5*(25 + Log[2]^2))) - (4
*Gamma[2251799813685248*E^(5*(25 + Log[2]^2)), -Log[x]]*Log[x]^(2251799813685248*E^(5*(25 + Log[2]^2)))*(22517
99813685248*E^(5*(25 + Log[2]^2)) + Log[x]))/(-Log[x])^(2251799813685248*E^(5*(25 + Log[2]^2)))

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}& = -4 \Gamma \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right ) (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x) \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )}+\log (x)\right )+4 \int \frac {\Gamma \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right ) (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x)}{x} \, dx \\ & = -4 \Gamma \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right ) (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x) \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )}+\log (x)\right )+\left (4 (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x)\right ) \int \frac {\Gamma \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right )}{x} \, dx \\ & = -4 \Gamma \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right ) (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x) \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )}+\log (x)\right )+\left (4 (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x)\right ) \text {Subst}\left (\int \Gamma \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-x\right ) \, dx,x,\log (x)\right ) \\ & = 4 \Gamma \left (1+2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right ) (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x)+4 \Gamma \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right ) (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{1+2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x)-4 \Gamma \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )},-\log (x)\right ) (-\log (x))^{-2251799813685248 e^{5 \left (25+\log ^2(2)\right )}} \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x) \left (2251799813685248 e^{5 \left (25+\log ^2(2)\right )}+\log (x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \log ^{-1+2251799813685248 e^{125+5 \log ^2(2)}}(x) \left (9007199254740992 e^{125+5 \log ^2(2)}+4 \log (x)\right ) \, dx=4 x \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x) \]

[In]

Integrate[Log[x]^(-1 + 2251799813685248*E^(125 + 5*Log[2]^2))*(9007199254740992*E^(125 + 5*Log[2]^2) + 4*Log[x
]),x]

[Out]

4*x*Log[x]^(2251799813685248*E^(5*(25 + Log[2]^2)))

Maple [F(-1)]

Timed out.

\[\int \frac {\left (4 \ln \left (x \right )+8 \,{\mathrm e}^{5 \ln \left (2\right )^{2}+50 \ln \left (2\right )+125}\right ) {\mathrm e}^{2251799813685248 \,{\mathrm e}^{5 \ln \left (2\right )^{2}+125} \ln \left (\ln \left (x \right )\right )}}{\ln \left (x \right )}d x\]

[In]

int((4*ln(x)+8*exp(5*ln(2)^2+50*ln(2)+125))*exp(exp(5*ln(2)^2+50*ln(2)+125)*ln(ln(x)))^2/ln(x),x)

[Out]

int((4*ln(x)+8*exp(5*ln(2)^2+50*ln(2)+125))*exp(exp(5*ln(2)^2+50*ln(2)+125)*ln(ln(x)))^2/ln(x),x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \log ^{-1+2251799813685248 e^{125+5 \log ^2(2)}}(x) \left (9007199254740992 e^{125+5 \log ^2(2)}+4 \log (x)\right ) \, dx=4 \, x \log \left (x\right )^{2 \, e^{\left (5 \, \log \left (2\right )^{2} + 50 \, \log \left (2\right ) + 125\right )}} \]

[In]

integrate((4*log(x)+8*exp(5*log(2)^2+50*log(2)+125))*exp(exp(5*log(2)^2+50*log(2)+125)*log(log(x)))^2/log(x),x
, algorithm="fricas")

[Out]

4*x*log(x)^(2*e^(5*log(2)^2 + 50*log(2) + 125))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (17) = 34\).

Time = 1.99 (sec) , antiderivative size = 128, normalized size of antiderivative = 7.11 \[ \int \log ^{-1+2251799813685248 e^{125+5 \log ^2(2)}}(x) \left (9007199254740992 e^{125+5 \log ^2(2)}+4 \log (x)\right ) \, dx=\frac {9007199254740992 e^{125} e^{5 \log {\left (2 \right )}^{2}} \log {\left (x \right )}^{-1 + 2251799813685248 e^{125} e^{5 \log {\left (2 \right )}^{2}}} \Gamma \left (2251799813685248 e^{125} e^{5 \log {\left (2 \right )}^{2}}, - \log {\left (x \right )}\right )}{\left (- \log {\left (x \right )}\right )^{-1 + 2251799813685248 e^{125} e^{5 \log {\left (2 \right )}^{2}}}} + \frac {4 \log {\left (x \right )}^{2251799813685248 e^{125} e^{5 \log {\left (2 \right )}^{2}}} \Gamma \left (1 + 2251799813685248 e^{125} e^{5 \log {\left (2 \right )}^{2}}, - \log {\left (x \right )}\right )}{\left (- \log {\left (x \right )}\right )^{2251799813685248 e^{125} e^{5 \log {\left (2 \right )}^{2}}}} \]

[In]

integrate((4*ln(x)+8*exp(5*ln(2)**2+50*ln(2)+125))*exp(exp(5*ln(2)**2+50*ln(2)+125)*ln(ln(x)))**2/ln(x),x)

[Out]

9007199254740992*(-log(x))**(-2251799813685248*exp(125)*exp(5*log(2)**2) + 1)*exp(125)*exp(5*log(2)**2)*log(x)
**(-1 + 2251799813685248*exp(125)*exp(5*log(2)**2))*uppergamma(2251799813685248*exp(125)*exp(5*log(2)**2), -lo
g(x)) + 4*log(x)**(2251799813685248*exp(125)*exp(5*log(2)**2))*uppergamma(1 + 2251799813685248*exp(125)*exp(5*
log(2)**2), -log(x))/(-log(x))**(2251799813685248*exp(125)*exp(5*log(2)**2))

Maxima [C] (verification not implemented)

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

Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 6.33 \[ \int \log ^{-1+2251799813685248 e^{125+5 \log ^2(2)}}(x) \left (9007199254740992 e^{125+5 \log ^2(2)}+4 \log (x)\right ) \, dx=-4 \, \left (-\log \left (x\right )\right )^{-2251799813685248 \, e^{\left (5 \, \log \left (2\right )^{2} + 125\right )} - 1} \log \left (x\right )^{2251799813685248 \, e^{\left (5 \, \log \left (2\right )^{2} + 125\right )} + 1} \Gamma \left (2251799813685248 \, e^{\left (5 \, \log \left (2\right )^{2} + 125\right )} + 1, -\log \left (x\right )\right ) - \frac {9007199254740992 \, \log \left (x\right )^{2251799813685248 \, e^{\left (5 \, \log \left (2\right )^{2} + 125\right )}} e^{\left (5 \, \log \left (2\right )^{2} + 125\right )} \Gamma \left (2251799813685248 \, e^{\left (5 \, \log \left (2\right )^{2} + 125\right )}, -\log \left (x\right )\right )}{\left (-\log \left (x\right )\right )^{2251799813685248 \, e^{\left (5 \, \log \left (2\right )^{2} + 125\right )}}} \]

[In]

integrate((4*log(x)+8*exp(5*log(2)^2+50*log(2)+125))*exp(exp(5*log(2)^2+50*log(2)+125)*log(log(x)))^2/log(x),x
, algorithm="maxima")

[Out]

-4*(-log(x))^(-2251799813685248*e^(5*log(2)^2 + 125) - 1)*log(x)^(2251799813685248*e^(5*log(2)^2 + 125) + 1)*g
amma(2251799813685248*e^(5*log(2)^2 + 125) + 1, -log(x)) - 9007199254740992*log(x)^(2251799813685248*e^(5*log(
2)^2 + 125))*e^(5*log(2)^2 + 125)*gamma(2251799813685248*e^(5*log(2)^2 + 125), -log(x))/(-log(x))^(22517998136
85248*e^(5*log(2)^2 + 125))

Giac [F]

\[ \int \log ^{-1+2251799813685248 e^{125+5 \log ^2(2)}}(x) \left (9007199254740992 e^{125+5 \log ^2(2)}+4 \log (x)\right ) \, dx=\int { \frac {4 \, \log \left (x\right )^{2 \, e^{\left (5 \, \log \left (2\right )^{2} + 50 \, \log \left (2\right ) + 125\right )}} {\left (2 \, e^{\left (5 \, \log \left (2\right )^{2} + 50 \, \log \left (2\right ) + 125\right )} + \log \left (x\right )\right )}}{\log \left (x\right )} \,d x } \]

[In]

integrate((4*log(x)+8*exp(5*log(2)^2+50*log(2)+125))*exp(exp(5*log(2)^2+50*log(2)+125)*log(log(x)))^2/log(x),x
, algorithm="giac")

[Out]

integrate(4*log(x)^(2*e^(5*log(2)^2 + 50*log(2) + 125))*(2*e^(5*log(2)^2 + 50*log(2) + 125) + log(x))/log(x),
x)

Mupad [B] (verification not implemented)

Time = 8.42 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \log ^{-1+2251799813685248 e^{125+5 \log ^2(2)}}(x) \left (9007199254740992 e^{125+5 \log ^2(2)}+4 \log (x)\right ) \, dx=4\,x\,{\ln \left (x\right )}^{2251799813685248\,{\mathrm {e}}^{5\,{\ln \left (2\right )}^2+125}} \]

[In]

int((log(x)^(2*exp(50*log(2) + 5*log(2)^2 + 125))*(8*exp(50*log(2) + 5*log(2)^2 + 125) + 4*log(x)))/log(x),x)

[Out]

4*x*log(x)^(2251799813685248*exp(5*log(2)^2 + 125))