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

Optimal. Leaf size=18 \[ 4 x \log ^{2 e^{5 (5+\log (2))^2}}(x) \]

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Rubi [C]  time = 0.12, antiderivative size = 173, normalized size of antiderivative = 9.61, number of steps used = 4, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2299, 2181, 2361, 19, 6557} \begin {gather*} 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 ) \end {gather*}

Antiderivative was successfully verified.

[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 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 2299

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 2361

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 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]

Rubi steps

\begin {gather*} \begin {aligned} \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 ) \operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 18, normalized size = 1.00 \begin {gather*} 4 x \log ^{2251799813685248 e^{5 \left (25+\log ^2(2)\right )}}(x) \end {gather*}

Antiderivative was successfully verified.

[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)))

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fricas [A]  time = 0.98, size = 21, normalized size = 1.17 \begin {gather*} 4 \, x \log \relax (x)^{2 \, e^{\left (5 \, \log \relax (2)^{2} + 50 \, \log \relax (2) + 125\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, \log \relax (x)^{2 \, e^{\left (5 \, \log \relax (2)^{2} + 50 \, \log \relax (2) + 125\right )}} {\left (2 \, e^{\left (5 \, \log \relax (2)^{2} + 50 \, \log \relax (2) + 125\right )} + \log \relax (x)\right )}}{\log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 hanged

Verification of antiderivative is not currently implemented for this CAS.

[In]

maple_input

[Out]

maple_input

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maxima [C]  time = 0.71, size = 114, normalized size = 6.33 \begin {gather*} -4 \, \left (-\log \relax (x)\right )^{-2251799813685248 \, e^{\left (5 \, \log \relax (2)^{2} + 125\right )} - 1} \log \relax (x)^{2251799813685248 \, e^{\left (5 \, \log \relax (2)^{2} + 125\right )} + 1} \Gamma \left (2251799813685248 \, e^{\left (5 \, \log \relax (2)^{2} + 125\right )} + 1, -\log \relax (x)\right ) - \frac {9007199254740992 \, \log \relax (x)^{2251799813685248 \, e^{\left (5 \, \log \relax (2)^{2} + 125\right )}} e^{\left (5 \, \log \relax (2)^{2} + 125\right )} \Gamma \left (2251799813685248 \, e^{\left (5 \, \log \relax (2)^{2} + 125\right )}, -\log \relax (x)\right )}{\left (-\log \relax (x)\right )^{2251799813685248 \, e^{\left (5 \, \log \relax (2)^{2} + 125\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 0.78, size = 17, normalized size = 0.94 \begin {gather*} 4\,x\,{\ln \relax (x)}^{2251799813685248\,{\mathrm {e}}^{5\,{\ln \relax (2)}^2+125}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [B]  time = 4.80, size = 128, normalized size = 7.11 \begin {gather*} \frac {9007199254740992 e^{125} e^{5 \log {\relax (2 )}^{2}} \log {\relax (x )}^{-1 + 2251799813685248 e^{125} e^{5 \log {\relax (2 )}^{2}}} \Gamma \left (2251799813685248 e^{125} e^{5 \log {\relax (2 )}^{2}}, - \log {\relax (x )}\right )}{\left (- \log {\relax (x )}\right )^{-1 + 2251799813685248 e^{125} e^{5 \log {\relax (2 )}^{2}}}} + \frac {4 \log {\relax (x )}^{2251799813685248 e^{125} e^{5 \log {\relax (2 )}^{2}}} \Gamma \left (1 + 2251799813685248 e^{125} e^{5 \log {\relax (2 )}^{2}}, - \log {\relax (x )}\right )}{\left (- \log {\relax (x )}\right )^{2251799813685248 e^{125} e^{5 \log {\relax (2 )}^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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