∫ 3−6e−11x log(5)+e−11x(−6+66x) log(5) log(x)________________________________

Optimal. Leaf size=20 \[ x \left (\frac {3}{2 \log (5)}-3 e^{-11 x} \log (x)\right ) \]

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Rubi [A]
time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.85, number of steps used = 6, number of rules used = 4, integrand size = 34, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.118, Rules used = {12, 2225, 2207, 2634} \begin {gather*} \frac {3 x}{2 \log (5)}-\frac {3}{11} e^{-11 x} \log (x)+\frac {3}{11} e^{-11 x} (1-11 x) \log (x) \end {gather*}

Int[(3 - (6*Log[5])/E^(11*x) + ((-6 + 66*x)*Log[5]*Log[x])/E^(11*x))/(2*Log[5]),x]

(3*x)/(2*Log[5]) - (3*Log[x])/(11*E^(11*x)) + (3*(1 - 11*x)*Log[x])/(11*E^(11*x))

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

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*

((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x

)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] && !TrueQ[$UseGamma]

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]

/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)\right ) \, dx}{2 \log (5)}\\ &=\frac {3 x}{2 \log (5)}+\frac {1}{2} \int e^{-11 x} (-6+66 x) \log (x) \, dx-3 \int e^{-11 x} \, dx\\ &=\frac {3 e^{-11 x}}{11}+\frac {3 x}{2 \log (5)}-\frac {3}{11} e^{-11 x} \log (x)+\frac {3}{11} e^{-11 x} (1-11 x) \log (x)-\frac {1}{2} \int -6 e^{-11 x} \, dx\\ &=\frac {3 e^{-11 x}}{11}+\frac {3 x}{2 \log (5)}-\frac {3}{11} e^{-11 x} \log (x)+\frac {3}{11} e^{-11 x} (1-11 x) \log (x)+3 \int e^{-11 x} \, dx\\ &=\frac {3 x}{2 \log (5)}-\frac {3}{11} e^{-11 x} \log (x)+\frac {3}{11} e^{-11 x} (1-11 x) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.06, size = 20, normalized size = 1.00 \begin {gather*} \frac {3 \left (x-2 e^{-11 x} x \log (5) \log (x)\right )}{\log (25)} \end {gather*}

Integrate[(3 - (6*Log[5])/E^(11*x) + ((-6 + 66*x)*Log[5]*Log[x])/E^(11*x))/(2*Log[5]),x]

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method	result	size

norman	$\frac {3 x}{2 \ln \left (5\right )}-3 x \,{\mathrm e}^{-11 x} \ln \left (x \right )$	$18$
risch	$\frac {3 x}{2 \ln \left (5\right )}-3 x \,{\mathrm e}^{-11 x} \ln \left (x \right )$	$18$
default	$\frac {3 x -6 \ln \left (5\right ) {\mathrm e}^{-11 x} \ln \left (x \right ) x}{2 \ln \left (5\right )}$	$22$

int(1/2*((66*x-6)*ln(5)*exp(-11*x)*ln(x)-6*ln(5)*exp(-11*x)+3)/ln(5),x,method=_RETURNVERBOSE)

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Maxima [A]
time = 0.26, size = 21, normalized size = 1.05 \begin {gather*} -\frac {3 \, {\left (2 \, x e^{\left (-11 \, x\right )} \log \left (5\right ) \log \left (x\right ) - x\right )}}{2 \, \log \left (5\right )} \end {gather*}

integrate(1/2*((66*x-6)*log(5)*exp(-11*x)*log(x)-6*log(5)*exp(-11*x)+3)/log(5),x, algorithm="maxima")

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Fricas [A]
time = 0.36, size = 21, normalized size = 1.05 \begin {gather*} -\frac {3 \, {\left (2 \, x e^{\left (-11 \, x\right )} \log \left (5\right ) \log \left (x\right ) - x\right )}}{2 \, \log \left (5\right )} \end {gather*}

integrate(1/2*((66*x-6)*log(5)*exp(-11*x)*log(x)-6*log(5)*exp(-11*x)+3)/log(5),x, algorithm="fricas")

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Sympy [A]
time = 0.09, size = 19, normalized size = 0.95 \begin {gather*} \frac {3 x}{2 \log {\left (5 \right )}} - 3 x e^{- 11 x} \log {\left (x \right )} \end {gather*}

integrate(1/2*((66*x-6)*ln(5)*exp(-11*x)*ln(x)-6*ln(5)*exp(-11*x)+3)/ln(5),x)

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Giac [A]
time = 0.40, size = 36, normalized size = 1.80 \begin {gather*} -\frac {3 \, {\left (2 \, {\left (11 \, x e^{\left (-11 \, x\right )} \log \left (x\right ) + e^{\left (-11 \, x\right )}\right )} \log \left (5\right ) - 2 \, e^{\left (-11 \, x\right )} \log \left (5\right ) - 11 \, x\right )}}{22 \, \log \left (5\right )} \end {gather*}

integrate(1/2*((66*x-6)*log(5)*exp(-11*x)*log(x)-6*log(5)*exp(-11*x)+3)/log(5),x, algorithm="giac")

-3/22*(2*(11*x*e^(-11*x)*log(x) + e^(-11*x))*log(5) - 2*e^(-11*x)*log(5) - 11*x)/log(5)

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Mupad [B]
time = 5.10, size = 17, normalized size = 0.85 \begin {gather*} \frac {3\,x}{2\,\ln \left (5\right )}-3\,x\,{\mathrm {e}}^{-11\,x}\,\ln \left (x\right ) \end {gather*}

int(((exp(-11*x)*log(5)*log(x)*(66*x - 6))/2 - 3*exp(-11*x)*log(5) + 3/2)/log(5),x)

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3.83.1 \(\int \frac {3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)}{2 \log (5)} \, dx\) [8201]