3.74.29 \(\int \frac {96 x-16 x^2+e^{-e+2 x} (12+20 x)}{e^{-e+2 x} x+4 x^2} \, dx\)

Optimal. Leaf size=27 \[ 4 \left (9-x+3 \log \left (x \left (\frac {1}{4} e^{-e+2 x}+x\right )\right )\right ) \]

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Rubi [F]  time = 0.71, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {96 x-16 x^2+e^{-e+2 x} (12+20 x)}{e^{-e+2 x} x+4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(96*x - 16*x^2 + E^(-E + 2*x)*(12 + 20*x))/(E^(-E + 2*x)*x + 4*x^2),x]

[Out]

20*x + 12*Log[x] + 24*E^E*Defer[Subst][Defer[Int][(E^x + 2*E^E*x)^(-1), x], x, 2*x] - 24*E^E*Defer[Subst][Defe
r[Int][x/(E^x + 2*E^E*x), x], x, 2*x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^e \left (96 x-16 x^2+e^{-e+2 x} (12+20 x)\right )}{x \left (e^{2 x}+4 e^e x\right )} \, dx\\ &=e^e \int \frac {96 x-16 x^2+e^{-e+2 x} (12+20 x)}{x \left (e^{2 x}+4 e^e x\right )} \, dx\\ &=e^e \int \left (\frac {4 e^{-e} (3+5 x)}{x}-\frac {48 (-1+2 x)}{e^{2 x}+4 e^e x}\right ) \, dx\\ &=4 \int \frac {3+5 x}{x} \, dx-\left (48 e^e\right ) \int \frac {-1+2 x}{e^{2 x}+4 e^e x} \, dx\\ &=4 \int \left (5+\frac {3}{x}\right ) \, dx-\left (24 e^e\right ) \operatorname {Subst}\left (\int \frac {-1+x}{e^x+2 e^e x} \, dx,x,2 x\right )\\ &=20 x+12 \log (x)-\left (24 e^e\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{e^x+2 e^e x}+\frac {x}{e^x+2 e^e x}\right ) \, dx,x,2 x\right )\\ &=20 x+12 \log (x)+\left (24 e^e\right ) \operatorname {Subst}\left (\int \frac {1}{e^x+2 e^e x} \, dx,x,2 x\right )-\left (24 e^e\right ) \operatorname {Subst}\left (\int \frac {x}{e^x+2 e^e x} \, dx,x,2 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 25, normalized size = 0.93 \begin {gather*} 4 \left (-x+3 \log (x)+3 \log \left (e^{2 x}+4 e^e x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(96*x - 16*x^2 + E^(-E + 2*x)*(12 + 20*x))/(E^(-E + 2*x)*x + 4*x^2),x]

[Out]

4*(-x + 3*Log[x] + 3*Log[E^(2*x) + 4*E^E*x])

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fricas [A]  time = 0.64, size = 24, normalized size = 0.89 \begin {gather*} -4 \, x + 12 \, \log \left (4 \, x + e^{\left (2 \, x - e\right )}\right ) + 12 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x+12)*exp(x-exp(1))*exp(x)-16*x^2+96*x)/(x*exp(x-exp(1))*exp(x)+4*x^2),x, algorithm="fricas")

[Out]

-4*x + 12*log(4*x + e^(2*x - e)) + 12*log(x)

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giac [A]  time = 0.13, size = 26, normalized size = 0.96 \begin {gather*} -4 \, x + 12 \, \log \relax (x) + 12 \, \log \left (-4 \, x - e^{\left (2 \, x - e\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x+12)*exp(x-exp(1))*exp(x)-16*x^2+96*x)/(x*exp(x-exp(1))*exp(x)+4*x^2),x, algorithm="giac")

[Out]

-4*x + 12*log(x) + 12*log(-4*x - e^(2*x - e))

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maple [A]  time = 0.10, size = 23, normalized size = 0.85

method	result	size
norman	\(-4 x +12 \ln \relax (x )+12 \ln \left ({\mathrm e}^{2 x}+4 x \,{\mathrm e}^{{\mathrm e}}\right )\)	\(23\)
risch	\(-4 x +12 \ln \relax (x )+12 \ln \left ({\mathrm e}^{2 x}+4 x \,{\mathrm e}^{{\mathrm e}}\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x+12)*exp(x-exp(1))*exp(x)-16*x^2+96*x)/(x*exp(x-exp(1))*exp(x)+4*x^2),x,method=_RETURNVERBOSE)

[Out]

-4*x+12*ln(x)+12*ln(exp(x)^2+4*x*exp(exp(1)))

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maxima [A]  time = 0.38, size = 22, normalized size = 0.81 \begin {gather*} -4 \, x + 12 \, \log \left (4 \, x e^{e} + e^{\left (2 \, x\right )}\right ) + 12 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x+12)*exp(x-exp(1))*exp(x)-16*x^2+96*x)/(x*exp(x-exp(1))*exp(x)+4*x^2),x, algorithm="maxima")

[Out]

-4*x + 12*log(4*x*e^e + e^(2*x)) + 12*log(x)

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mupad [B]  time = 4.66, size = 24, normalized size = 0.89 \begin {gather*} 12\,\ln \left (x+\frac {{\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^{2\,x}}{4}\right )-4\,x+12\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((96*x - 16*x^2 + exp(x - exp(1))*exp(x)*(20*x + 12))/(4*x^2 + x*exp(x - exp(1))*exp(x)),x)

[Out]

12*log(x + (exp(-exp(1))*exp(2*x))/4) - 4*x + 12*log(x)

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sympy [A]  time = 0.18, size = 24, normalized size = 0.89 \begin {gather*} - 4 x + 12 \log {\relax (x )} + 12 \log {\left (4 x e^{e} + e^{2 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x+12)*exp(x-exp(1))*exp(x)-16*x**2+96*x)/(x*exp(x-exp(1))*exp(x)+4*x**2),x)

[Out]

-4*x + 12*log(x) + 12*log(4*x*exp(E) + exp(2*x))

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