3.72.52 \(\int \frac {e^{\log ^2(\frac {x^2 \log (5)}{\log (x^3)})} (-30+20 \log (x^3)) \log (\frac {x^2 \log (5)}{\log (x^3)})}{x \log (x^3)} \, dx\)

Optimal. Leaf size=19 \[ 5 e^{\log ^2\left (\frac {x^2 \log (5)}{\log \left (x^3\right )}\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.88, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 4, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6741, 12, 6742, 6706} \begin {gather*} 5 e^{\log ^2\left (\frac {x^2 \log (5)}{\log \left (x^3\right )}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^Log[(x^2*Log[5])/Log[x^3]]^2*(-30 + 20*Log[x^3])*Log[(x^2*Log[5])/Log[x^3]])/(x*Log[x^3]),x]

[Out]

5*E^Log[(x^2*Log[5])/Log[x^3]]^2

Rule 12

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=5 e^{\log ^2\left (\frac {x^2 \log (5)}{\log \left (x^3\right )}\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 19, normalized size = 1.00 \begin {gather*} 5 e^{\log ^2\left (\frac {x^2 \log (5)}{\log \left (x^3\right )}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^Log[(x^2*Log[5])/Log[x^3]]^2*(-30 + 20*Log[x^3])*Log[(x^2*Log[5])/Log[x^3]])/(x*Log[x^3]),x]

[Out]

5*E^Log[(x^2*Log[5])/Log[x^3]]^2

________________________________________________________________________________________

fricas [A]  time = 1.13, size = 18, normalized size = 0.95 \begin {gather*} 5 \, e^{\left (\log \left (\frac {x^{2} \log \relax (5)}{\log \left (x^{3}\right )}\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*log(x^3)-30)*log(x^2*log(5)/log(x^3))/x/log(x^3)/exp(-log(x^2*log(5)/log(x^3))^2),x, algorithm="
fricas")

[Out]

5*e^(log(x^2*log(5)/log(x^3))^2)

________________________________________________________________________________________

giac [A]  time = 0.51, size = 18, normalized size = 0.95 \begin {gather*} 5 \, e^{\left (\log \left (\frac {x^{2} \log \relax (5)}{\log \left (x^{3}\right )}\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*log(x^3)-30)*log(x^2*log(5)/log(x^3))/x/log(x^3)/exp(-log(x^2*log(5)/log(x^3))^2),x, algorithm="
giac")

[Out]

5*e^(log(x^2*log(5)/log(x^3))^2)

________________________________________________________________________________________

maple [F]  time = 0.70, size = 0, normalized size = 0.00 \[\int \frac {\left (20 \ln \left (x^{3}\right )-30\right ) \ln \left (\frac {x^{2} \ln \relax (5)}{\ln \left (x^{3}\right )}\right ) {\mathrm e}^{\ln \left (\frac {x^{2} \ln \relax (5)}{\ln \left (x^{3}\right )}\right )^{2}}}{x \ln \left (x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*ln(x^3)-30)*ln(x^2*ln(5)/ln(x^3))/x/ln(x^3)/exp(-ln(x^2*ln(5)/ln(x^3))^2),x)

[Out]

int((20*ln(x^3)-30)*ln(x^2*ln(5)/ln(x^3))/x/ln(x^3)/exp(-ln(x^2*ln(5)/ln(x^3))^2),x)

________________________________________________________________________________________

maxima [B]  time = 0.79, size = 68, normalized size = 3.58 \begin {gather*} \frac {5 \, e^{\left (\log \relax (3)^{2} - 4 \, \log \relax (3) \log \relax (x) + 4 \, \log \relax (x)^{2} + 4 \, \log \relax (x) \log \left (\log \relax (5)\right ) + \log \left (\log \relax (5)\right )^{2} + 2 \, \log \relax (3) \log \left (\log \relax (x)\right ) - 4 \, \log \relax (x) \log \left (\log \relax (x)\right ) - 2 \, \log \left (\log \relax (5)\right ) \log \left (\log \relax (x)\right ) + \log \left (\log \relax (x)\right )^{2}\right )}}{3^{2 \, \log \left (\log \relax (5)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*log(x^3)-30)*log(x^2*log(5)/log(x^3))/x/log(x^3)/exp(-log(x^2*log(5)/log(x^3))^2),x, algorithm="
maxima")

[Out]

5*e^(log(3)^2 - 4*log(3)*log(x) + 4*log(x)^2 + 4*log(x)*log(log(5)) + log(log(5))^2 + 2*log(3)*log(log(x)) - 4
*log(x)*log(log(x)) - 2*log(log(5))*log(log(x)) + log(log(x))^2)/3^(2*log(log(5)))

________________________________________________________________________________________

mupad [B]  time = 4.26, size = 51, normalized size = 2.68 \begin {gather*} 5\,{\mathrm {e}}^{{\ln \left (x^2\right )}^2}\,{\mathrm {e}}^{{\ln \left (\ln \relax (5)\right )}^2}\,{\mathrm {e}}^{{\ln \left (\frac {1}{\ln \left (x^3\right )}\right )}^2}\,{\left (\frac {1}{\ln \left (x^3\right )}\right )}^{2\,\ln \left (x^2\right )+2\,\ln \left (\ln \relax (5)\right )}\,{\left (x^4\right )}^{\ln \left (\ln \relax (5)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log((x^2*log(5))/log(x^3))^2)*log((x^2*log(5))/log(x^3))*(20*log(x^3) - 30))/(x*log(x^3)),x)

[Out]

5*exp(log(x^2)^2)*exp(log(log(5))^2)*exp(log(1/log(x^3))^2)*(1/log(x^3))^(2*log(x^2) + 2*log(log(5)))*(x^4)^lo
g(log(5))

________________________________________________________________________________________

sympy [A]  time = 0.54, size = 17, normalized size = 0.89 \begin {gather*} 5 e^{\log {\left (\frac {x^{2} \log {\relax (5 )}}{\log {\left (x^{3} \right )}} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*ln(x**3)-30)*ln(x**2*ln(5)/ln(x**3))/x/ln(x**3)/exp(-ln(x**2*ln(5)/ln(x**3))**2),x)

[Out]

5*exp(log(x**2*log(5)/log(x**3))**2)

________________________________________________________________________________________