3.30.56 \(\int \frac {-9 e^{-x}-9 e^{-x} x \log (\frac {4 x}{21})}{2 x \log ^2(\frac {4 x}{21})} \, dx\)

Optimal. Leaf size=17 \[ \frac {9 e^{-x}}{2 \log \left (\frac {4 x}{21}\right )} \]

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Rubi [A]  time = 0.33, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {12, 6741, 2202} \begin {gather*} \frac {9 e^{-x}}{2 \log \left (\frac {4 x}{21}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9/E^x - (9*x*Log[(4*x)/21])/E^x)/(2*x*Log[(4*x)/21]^2),x]

[Out]

9/(2*E^x*Log[(4*x)/21])

Rule 12

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

Rule 2202

Int[Log[(d_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(x_)^(m_.)*((e_) + Log[(d_.)*(x_)]*(h_.)*((f_.) +
(g_.)*(x_))), x_Symbol] :> Simp[(e*x^(m + 1)*F^(c*(a + b*x))*Log[d*x]^(n + 1))/(n + 1), x] /; FreeQ[{F, a, b,
c, d, e, f, g, h, m, n}, x] && EqQ[e*(m + 1) - f*h*(n + 1), 0] && EqQ[g*h*(n + 1) - b*c*e*Log[F], 0] && NeQ[n,
-1]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {-9 e^{-x}-9 e^{-x} x \log \left (\frac {4 x}{21}\right )}{x \log ^2\left (\frac {4 x}{21}\right )} \, dx\\ &=\frac {1}{2} \int \frac {9 e^{-x} \left (-1-x \log \left (\frac {4 x}{21}\right )\right )}{x \log ^2\left (\frac {4 x}{21}\right )} \, dx\\ &=\frac {9}{2} \int \frac {e^{-x} \left (-1-x \log \left (\frac {4 x}{21}\right )\right )}{x \log ^2\left (\frac {4 x}{21}\right )} \, dx\\ &=\frac {9 e^{-x}}{2 \log \left (\frac {4 x}{21}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {9 e^{-x}}{2 \log \left (\frac {4 x}{21}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9/E^x - (9*x*Log[(4*x)/21])/E^x)/(2*x*Log[(4*x)/21]^2),x]

[Out]

9/(2*E^x*Log[(4*x)/21])

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fricas [A]  time = 0.57, size = 15, normalized size = 0.88 \begin {gather*} \frac {3 \, e^{\left (-x + \log \relax (3)\right )}}{2 \, \log \left (\frac {4}{21} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-3*x*exp(log(3)-x)*log(4/21*x)-3*exp(log(3)-x))/x/log(4/21*x)^2,x, algorithm="fricas")

[Out]

3/2*e^(-x + log(3))/log(4/21*x)

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giac [A]  time = 0.15, size = 12, normalized size = 0.71 \begin {gather*} \frac {9 \, e^{\left (-x\right )}}{2 \, \log \left (\frac {4}{21} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-3*x*exp(log(3)-x)*log(4/21*x)-3*exp(log(3)-x))/x/log(4/21*x)^2,x, algorithm="giac")

[Out]

9/2*e^(-x)/log(4/21*x)

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maple [A]  time = 0.03, size = 13, normalized size = 0.76

method	result	size
risch	\(\frac {9 \,{\mathrm e}^{-x}}{2 \ln \left (\frac {4 x}{21}\right )}\)	\(13\)
norman	\(\frac {3 \,{\mathrm e}^{\ln \relax (3)-x}}{2 \ln \left (\frac {4 x}{21}\right )}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-3*x*exp(ln(3)-x)*ln(4/21*x)-3*exp(ln(3)-x))/x/ln(4/21*x)^2,x,method=_RETURNVERBOSE)

[Out]

9/2*exp(-x)/ln(4/21*x)

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maxima [A]  time = 0.64, size = 21, normalized size = 1.24 \begin {gather*} -\frac {9 \, e^{\left (-x\right )}}{2 \, {\left (\log \relax (7) + \log \relax (3) - 2 \, \log \relax (2) - \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-3*x*exp(log(3)-x)*log(4/21*x)-3*exp(log(3)-x))/x/log(4/21*x)^2,x, algorithm="maxima")

[Out]

-9/2*e^(-x)/(log(7) + log(3) - 2*log(2) - log(x))

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mupad [B]  time = 1.83, size = 12, normalized size = 0.71 \begin {gather*} \frac {9\,{\mathrm {e}}^{-x}}{2\,\ln \left (\frac {4\,x}{21}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*exp(log(3) - x))/2 + (3*x*log((4*x)/21)*exp(log(3) - x))/2)/(x*log((4*x)/21)^2),x)

[Out]

(9*exp(-x))/(2*log((4*x)/21))

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sympy [A]  time = 0.26, size = 12, normalized size = 0.71 \begin {gather*} \frac {9 e^{- x}}{2 \log {\left (\frac {4 x}{21} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-3*x*exp(ln(3)-x)*ln(4/21*x)-3*exp(ln(3)-x))/x/ln(4/21*x)**2,x)

[Out]

9*exp(-x)/(2*log(4*x/21))

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