3.61.5 \(\int \frac {12 e^{4+x}-\frac {1}{2} e^{8+2 x}}{144+\frac {1}{4} e^{4+x} (96-24 e+48 x)+\frac {1}{16} e^{8+2 x} (16+e^2+e (-8-4 x)+16 x+4 x^2)} \, dx\)

Optimal. Leaf size=21 \[ \frac {4}{4-e+48 e^{-4-x}+2 x} \]

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Rubi [F]  time = 1.71, 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 {12 e^{4+x}-\frac {1}{2} e^{8+2 x}}{144+\frac {1}{4} e^{4+x} (96-24 e+48 x)+\frac {1}{16} e^{8+2 x} \left (16+e^2+e (-8-4 x)+16 x+4 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(12*E^(4 + x) - E^(8 + 2*x)/2)/(144 + (E^(4 + x)*(96 - 24*E + 48*x))/4 + (E^(8 + 2*x)*(16 + E^2 + E*(-8 -
4*x) + 16*x + 4*x^2))/16),x]

[Out]

192*Defer[Int][E^(4 + x)/(48 + 4*(1 - E/4)*E^(4 + x) + 2*E^(4 + x)*x)^2, x] + 384*Defer[Int][E^(4 + x)/((4 - E
+ 2*x)*(48 + 4*(1 - E/4)*E^(4 + x) + 2*E^(4 + x)*x)^2), x] + 8*Defer[Int][E^(4 + x)/((-4 + E - 2*x)*(48 + 4*(
1 - E/4)*E^(4 + x) + 2*E^(4 + x)*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^{4+x} \left (24-e^{4+x}\right )}{\left (48-e^{5+x}+2 e^{4+x} (2+x)\right )^2} \, dx\\ &=8 \int \frac {e^{4+x} \left (24-e^{4+x}\right )}{\left (48-e^{5+x}+2 e^{4+x} (2+x)\right )^2} \, dx\\ &=8 \int \left (\frac {24 e^{4+x} (6-e+2 x)}{(4-e+2 x) \left (48+4 \left (1-\frac {e}{4}\right ) e^{4+x}+2 e^{4+x} x\right )^2}+\frac {e^{4+x}}{(-4+e-2 x) \left (48+4 \left (1-\frac {e}{4}\right ) e^{4+x}+2 e^{4+x} x\right )}\right ) \, dx\\ &=8 \int \frac {e^{4+x}}{(-4+e-2 x) \left (48+4 \left (1-\frac {e}{4}\right ) e^{4+x}+2 e^{4+x} x\right )} \, dx+192 \int \frac {e^{4+x} (6-e+2 x)}{(4-e+2 x) \left (48+4 \left (1-\frac {e}{4}\right ) e^{4+x}+2 e^{4+x} x\right )^2} \, dx\\ &=8 \int \frac {e^{4+x}}{(-4+e-2 x) \left (48+4 \left (1-\frac {e}{4}\right ) e^{4+x}+2 e^{4+x} x\right )} \, dx+192 \int \left (\frac {e^{4+x}}{\left (48+4 \left (1-\frac {e}{4}\right ) e^{4+x}+2 e^{4+x} x\right )^2}+\frac {2 e^{4+x}}{(4-e+2 x) \left (48+4 \left (1-\frac {e}{4}\right ) e^{4+x}+2 e^{4+x} x\right )^2}\right ) \, dx\\ &=8 \int \frac {e^{4+x}}{(-4+e-2 x) \left (48+4 \left (1-\frac {e}{4}\right ) e^{4+x}+2 e^{4+x} x\right )} \, dx+192 \int \frac {e^{4+x}}{\left (48+4 \left (1-\frac {e}{4}\right ) e^{4+x}+2 e^{4+x} x\right )^2} \, dx+384 \int \frac {e^{4+x}}{(4-e+2 x) \left (48+4 \left (1-\frac {e}{4}\right ) e^{4+x}+2 e^{4+x} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.43, size = 26, normalized size = 1.24 \begin {gather*} -\frac {4 e^{4+x}}{-48+e^{5+x}-2 e^{4+x} (2+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12*E^(4 + x) - E^(8 + 2*x)/2)/(144 + (E^(4 + x)*(96 - 24*E + 48*x))/4 + (E^(8 + 2*x)*(16 + E^2 + E*
(-8 - 4*x) + 16*x + 4*x^2))/16),x]

[Out]

(-4*E^(4 + x))/(-48 + E^(5 + x) - 2*E^(4 + x)*(2 + x))

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fricas [A]  time = 0.79, size = 32, normalized size = 1.52 \begin {gather*} \frac {4 \, e^{\left (x - 2 \, \log \relax (2) + 4\right )}}{{\left (2 \, x - e + 4\right )} e^{\left (x - 2 \, \log \relax (2) + 4\right )} + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(-2*log(2)+4+x)^2+48*exp(-2*log(2)+4+x))/((exp(1)^2+(-4*x-8)*exp(1)+4*x^2+16*x+16)*exp(-2*log
(2)+4+x)^2+(-24*exp(1)+48*x+96)*exp(-2*log(2)+4+x)+144),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 29, normalized size = 1.38 \begin {gather*} \frac {4 \, e^{\left (x + 4\right )}}{2 \, x e^{\left (x + 4\right )} - e^{\left (x + 5\right )} + 4 \, e^{\left (x + 4\right )} + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(-2*log(2)+4+x)^2+48*exp(-2*log(2)+4+x))/((exp(1)^2+(-4*x-8)*exp(1)+4*x^2+16*x+16)*exp(-2*log
(2)+4+x)^2+(-24*exp(1)+48*x+96)*exp(-2*log(2)+4+x)+144),x, algorithm="giac")

[Out]

4*e^(x + 4)/(2*x*e^(x + 4) - e^(x + 5) + 4*e^(x + 4) + 48)

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maple [A]  time = 0.34, size = 47, normalized size = 2.24

method	result	size
norman	\(-\frac {4 \,{\mathrm e}^{-2 \ln \relax (2)+4+x}}{{\mathrm e}^{-2 \ln \relax (2)+4+x} {\mathrm e}-2 \,{\mathrm e}^{-2 \ln \relax (2)+4+x} x -4 \,{\mathrm e}^{-2 \ln \relax (2)+4+x}-12}\)	\(47\)
risch	\(-\frac {4}{{\mathrm e}-2 x -4}-\frac {48}{\left ({\mathrm e}-2 x -4\right ) \left (\frac {{\mathrm e}^{5+x}}{4}-\frac {x \,{\mathrm e}^{4+x}}{2}-{\mathrm e}^{4+x}-12\right )}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*exp(-2*ln(2)+4+x)^2+48*exp(-2*ln(2)+4+x))/((exp(1)^2+(-4*x-8)*exp(1)+4*x^2+16*x+16)*exp(-2*ln(2)+4+x)^
2+(-24*exp(1)+48*x+96)*exp(-2*ln(2)+4+x)+144),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.40, size = 27, normalized size = 1.29 \begin {gather*} \frac {4 \, e^{\left (x + 4\right )}}{{\left (2 \, x e^{4} - e^{5} + 4 \, e^{4}\right )} e^{x} + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(-2*log(2)+4+x)^2+48*exp(-2*log(2)+4+x))/((exp(1)^2+(-4*x-8)*exp(1)+4*x^2+16*x+16)*exp(-2*log
(2)+4+x)^2+(-24*exp(1)+48*x+96)*exp(-2*log(2)+4+x)+144),x, algorithm="maxima")

[Out]

4*e^(x + 4)/((2*x*e^4 - e^5 + 4*e^4)*e^x + 48)

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mupad [B]  time = 4.76, size = 27, normalized size = 1.29 \begin {gather*} \frac {{\mathrm {e}}^4\,{\mathrm {e}}^x}{{\mathrm {e}}^4\,{\mathrm {e}}^x-\frac {{\mathrm {e}}^5\,{\mathrm {e}}^x}{4}+\frac {x\,{\mathrm {e}}^4\,{\mathrm {e}}^x}{2}+12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((48*exp(x - 2*log(2) + 4) - 8*exp(2*x - 4*log(2) + 8))/(exp(x - 2*log(2) + 4)*(48*x - 24*exp(1) + 96) + ex
p(2*x - 4*log(2) + 8)*(16*x + exp(2) + 4*x^2 - exp(1)*(4*x + 8) + 16) + 144),x)

[Out]

(exp(4)*exp(x))/(exp(4)*exp(x) - (exp(5)*exp(x))/4 + (x*exp(4)*exp(x))/2 + 12)

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sympy [B]  time = 0.31, size = 53, normalized size = 2.52 \begin {gather*} - \frac {192}{96 x + \left (4 x^{2} - 4 e x + 16 x - 8 e + e^{2} + 16\right ) e^{x + 4} - 48 e + 192} + \frac {8}{4 x - 2 e + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(-2*ln(2)+4+x)**2+48*exp(-2*ln(2)+4+x))/((exp(1)**2+(-4*x-8)*exp(1)+4*x**2+16*x+16)*exp(-2*ln
(2)+4+x)**2+(-24*exp(1)+48*x+96)*exp(-2*ln(2)+4+x)+144),x)

[Out]

-192/(96*x + (4*x**2 - 4*E*x + 16*x - 8*E + exp(2) + 16)*exp(x + 4) - 48*E + 192) + 8/(4*x - 2*E + 8)

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