3.22.11 \(\int -\frac {e^{-x \log (2)+9 \log (2) \log (9+x)} x \log (2)}{9+x} \, dx\)

Optimal. Leaf size=12 \[ 2^{-x+9 \log (9+x)} \]

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Rubi [A]  time = 0.11, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 2274, 2197} \begin {gather*} 2^{-x} (x+9)^{\log (512)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9 + x)^Log[512]/2^x

Rule 12

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

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (\log (2) \int \frac {e^{-x \log (2)+9 \log (2) \log (9+x)} x}{9+x} \, dx\right )\\ &=-\left (\log (2) \int 2^{-x} x (9+x)^{-1+9 \log (2)} \, dx\right )\\ &=2^{-x} (9+x)^{\log (512)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 12, normalized size = 1.00 \begin {gather*} 2^{-x+9 \log (9+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2^(-x + 9*Log[9 + x])

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fricas [A]  time = 0.81, size = 15, normalized size = 1.25 \begin {gather*} e^{\left (-x \log \relax (2) + 9 \, \log \relax (2) \log \left (x + 9\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x*log(2)*exp(9*log(2)*log(x+9)-x*log(2))/(x+9),x, algorithm="fricas")

[Out]

e^(-x*log(2) + 9*log(2)*log(x + 9))

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giac [A]  time = 0.19, size = 15, normalized size = 1.25 \begin {gather*} e^{\left (-x \log \relax (2) + 9 \, \log \relax (2) \log \left (x + 9\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x*log(2)*exp(9*log(2)*log(x+9)-x*log(2))/(x+9),x, algorithm="giac")

[Out]

e^(-x*log(2) + 9*log(2)*log(x + 9))

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maple [A]  time = 0.36, size = 13, normalized size = 1.08

method	result	size
risch	\(\left (x +9\right )^{9 \ln \relax (2)} \left (\frac {1}{2}\right )^{x}\)	\(13\)
gosper	\({\mathrm e}^{9 \ln \relax (2) \ln \left (x +9\right )-x \ln \relax (2)}\)	\(16\)
norman	\({\mathrm e}^{9 \ln \relax (2) \ln \left (x +9\right )-x \ln \relax (2)}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x*ln(2)*exp(9*ln(2)*ln(x+9)-x*ln(2))/(x+9),x,method=_RETURNVERBOSE)

[Out]

(x+9)^(9*ln(2))*(1/2)^x

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maxima [A]  time = 0.43, size = 15, normalized size = 1.25 \begin {gather*} e^{\left (-x \log \relax (2) + 9 \, \log \relax (2) \log \left (x + 9\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x*log(2)*exp(9*log(2)*log(x+9)-x*log(2))/(x+9),x, algorithm="maxima")

[Out]

e^(-x*log(2) + 9*log(2)*log(x + 9))

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mupad [B]  time = 1.23, size = 14, normalized size = 1.17 \begin {gather*} \frac {{\left (x+9\right )}^{9\,\ln \relax (2)}}{2^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*exp(9*log(x + 9)*log(2) - x*log(2))*log(2))/(x + 9),x)

[Out]

(x + 9)^(9*log(2))/2^x

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sympy [A]  time = 0.30, size = 15, normalized size = 1.25 \begin {gather*} e^{- x \log {\relax (2 )} + 9 \log {\relax (2 )} \log {\left (x + 9 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x*ln(2)*exp(9*ln(2)*ln(x+9)-x*ln(2))/(x+9),x)

[Out]

exp(-x*log(2) + 9*log(2)*log(x + 9))

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