3.8.43 \(\int \frac {e^{-\frac {x}{\log (2 x)}} (e^{-4+x} x^2-e^{-4+x} x^2 \log (2 x)+e^{-4+x} (2 x+x^2) \log ^2(2 x))}{\log ^2(2 x)} \, dx\) [743]

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

[Out]

exp(x-4)*x^2/exp(x/ln(2*x))

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(18)=36\).
time = 0.11, antiderivative size = 59, normalized size of antiderivative = 3.28, number of steps used = 1, number of rules used = 1, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2326} \begin {gather*} \frac {e^{-\frac {x}{\log (2 x)}} \left (e^{x-4} x^2-e^{x-4} x^2 \log (2 x)\right )}{\left (\frac {1}{\log ^2(2 x)}-\frac {1}{\log (2 x)}\right ) \log ^2(2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(-4 + x)*x^2 - E^(-4 + x)*x^2*Log[2*x])/(E^(x/Log[2*x])*(Log[2*x]^(-2) - Log[2*x]^(-1))*Log[2*x]^2)

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

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

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Mathematica [A]
time = 0.13, size = 18, normalized size = 1.00 \begin {gather*} e^{-4+x-\frac {x}{\log (2 x)}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-4 + x - x/Log[2*x])*x^2

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Maple [A]
time = 0.09, size = 29, normalized size = 1.61

method	result	size
risch	\(x^{2} {\mathrm e}^{\frac {x \ln \left (2 x \right )-4 \ln \left (2 x \right )-x}{\ln \left (2 x \right )}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+2*x)*exp(x-4)*ln(2*x)^2-x^2*exp(x-4)*ln(2*x)+x^2*exp(x-4))/ln(2*x)^2/exp(x/ln(2*x)),x,method=_RETURN
VERBOSE)

[Out]

x^2*exp((x*ln(2*x)-4*ln(2*x)-x)/ln(2*x))

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Maxima [A]
time = 0.54, size = 18, normalized size = 1.00 \begin {gather*} x^{2} e^{\left (x - \frac {x}{\log \left (2\right ) + \log \left (x\right )} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x)*exp(x-4)*log(2*x)^2-x^2*exp(x-4)*log(2*x)+x^2*exp(x-4))/log(2*x)^2/exp(x/log(2*x)),x, alg
orithm="maxima")

[Out]

x^2*e^(x - x/(log(2) + log(x)) - 4)

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Fricas [A]
time = 0.45, size = 17, normalized size = 0.94 \begin {gather*} x^{2} e^{\left (x - \frac {x}{\log \left (2 \, x\right )} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x)*exp(x-4)*log(2*x)^2-x^2*exp(x-4)*log(2*x)+x^2*exp(x-4))/log(2*x)^2/exp(x/log(2*x)),x, alg
orithm="fricas")

[Out]

x^2*e^(x - x/log(2*x) - 4)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+2*x)*exp(x-4)*ln(2*x)**2-x**2*exp(x-4)*ln(2*x)+x**2*exp(x-4))/ln(2*x)**2/exp(x/ln(2*x)),x)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (17) = 34\).
time = 0.62, size = 74, normalized size = 4.11 \begin {gather*} {\left (x - 4\right )}^{2} e^{\left (\frac {{\left (x - 4\right )} \log \left (2 \, x\right ) - x}{\log \left (2 \, x\right )}\right )} + 8 \, {\left (x - 4\right )} e^{\left (\frac {{\left (x - 4\right )} \log \left (2 \, x\right ) - x}{\log \left (2 \, x\right )}\right )} + 16 \, e^{\left (\frac {{\left (x - 4\right )} \log \left (2 \, x\right ) - x}{\log \left (2 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x)*exp(x-4)*log(2*x)^2-x^2*exp(x-4)*log(2*x)+x^2*exp(x-4))/log(2*x)^2/exp(x/log(2*x)),x, alg
orithm="giac")

[Out]

(x - 4)^2*e^(((x - 4)*log(2*x) - x)/log(2*x)) + 8*(x - 4)*e^(((x - 4)*log(2*x) - x)/log(2*x)) + 16*e^(((x - 4)
*log(2*x) - x)/log(2*x))

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Mupad [B]
time = 0.59, size = 19, normalized size = 1.06 \begin {gather*} x^2\,{\mathrm {e}}^{-\frac {x}{\ln \left (2\right )+\ln \left (x\right )}}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*exp(-x/(log(2) + log(x)))*exp(-4)*exp(x)

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