3.82.67 \(\int \frac {-1-4 x-64 e^{2 x+\frac {8 e^{2 x}}{2+4 x}} x-4 x^2}{1+4 x+4 x^2} \, dx\)

Optimal. Leaf size=23 \[ 2-4 e^{\frac {8 e^{2 x}}{2+4 x}}-x \]

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Rubi [F]  time = 0.95, 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 {-1-4 x-64 e^{2 x+\frac {8 e^{2 x}}{2+4 x}} x-4 x^2}{1+4 x+4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 - 4*x - 64*E^(2*x + (8*E^(2*x))/(2 + 4*x))*x - 4*x^2)/(1 + 4*x + 4*x^2),x]

[Out]

-x + 32*Defer[Int][E^(2*x + (4*E^(2*x))/(1 + 2*x))/(1 + 2*x)^2, x] - 32*Defer[Int][E^(2*x + (4*E^(2*x))/(1 + 2
*x))/(1 + 2*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1-4 x-64 e^{2 x+\frac {8 e^{2 x}}{2+4 x}} x-4 x^2}{(1+2 x)^2} \, dx\\ &=\int \left (-1-\frac {64 e^{2 x+\frac {4 e^{2 x}}{1+2 x}} x}{(1+2 x)^2}\right ) \, dx\\ &=-x-64 \int \frac {e^{2 x+\frac {4 e^{2 x}}{1+2 x}} x}{(1+2 x)^2} \, dx\\ &=-x-64 \int \left (-\frac {e^{2 x+\frac {4 e^{2 x}}{1+2 x}}}{2 (1+2 x)^2}+\frac {e^{2 x+\frac {4 e^{2 x}}{1+2 x}}}{2 (1+2 x)}\right ) \, dx\\ &=-x+32 \int \frac {e^{2 x+\frac {4 e^{2 x}}{1+2 x}}}{(1+2 x)^2} \, dx-32 \int \frac {e^{2 x+\frac {4 e^{2 x}}{1+2 x}}}{1+2 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 22, normalized size = 0.96 \begin {gather*} -4 e^{\frac {4 e^{2 x}}{1+2 x}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 4*x - 64*E^(2*x + (8*E^(2*x))/(2 + 4*x))*x - 4*x^2)/(1 + 4*x + 4*x^2),x]

[Out]

-4*E^((4*E^(2*x))/(1 + 2*x)) - x

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fricas [B]  time = 0.85, size = 61, normalized size = 2.65 \begin {gather*} -{\left (x e^{\left (2 \, x + 2 \, \log \relax (2)\right )} + 4 \, e^{\left (\frac {4 \, x^{2} + 2 \, {\left (2 \, x + 1\right )} \log \relax (2) + 2 \, x + e^{\left (2 \, x + 2 \, \log \relax (2)\right )}}{2 \, x + 1}\right )}\right )} e^{\left (-2 \, x - 2 \, \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x*exp(2*log(2)+2*x)*exp(exp(2*log(2)+2*x)/(4*x+2))^2-4*x^2-4*x-1)/(4*x^2+4*x+1),x, algorithm="f
ricas")

[Out]

-(x*e^(2*x + 2*log(2)) + 4*e^((4*x^2 + 2*(2*x + 1)*log(2) + 2*x + e^(2*x + 2*log(2)))/(2*x + 1)))*e^(-2*x - 2*
log(2))

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giac [A]  time = 0.19, size = 38, normalized size = 1.65 \begin {gather*} -{\left (x e^{\left (2 \, x\right )} + 4 \, e^{\left (\frac {2 \, {\left (2 \, x^{2} + x + 2 \, e^{\left (2 \, x\right )}\right )}}{2 \, x + 1}\right )}\right )} e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x*exp(2*log(2)+2*x)*exp(exp(2*log(2)+2*x)/(4*x+2))^2-4*x^2-4*x-1)/(4*x^2+4*x+1),x, algorithm="g
iac")

[Out]

-(x*e^(2*x) + 4*e^(2*(2*x^2 + x + 2*e^(2*x))/(2*x + 1)))*e^(-2*x)

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maple [A]  time = 0.35, size = 21, normalized size = 0.91

method	result	size
risch	\(-4 \,{\mathrm e}^{\frac {4 \,{\mathrm e}^{2 x}}{2 x +1}}-x\)	\(21\)
norman	\(\frac {-x -2 x^{2}-4 \,{\mathrm e}^{\frac {8 \,{\mathrm e}^{2 x}}{4 x +2}}-8 x \,{\mathrm e}^{\frac {8 \,{\mathrm e}^{2 x}}{4 x +2}}}{2 x +1}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.37, size = 20, normalized size = 0.87 \begin {gather*} -x - 4 \, e^{\left (\frac {4 \, e^{\left (2 \, x\right )}}{2 \, x + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x*exp(2*log(2)+2*x)*exp(exp(2*log(2)+2*x)/(4*x+2))^2-4*x^2-4*x-1)/(4*x^2+4*x+1),x, algorithm="m
axima")

[Out]

-x - 4*e^(4*e^(2*x)/(2*x + 1))

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mupad [B]  time = 5.35, size = 20, normalized size = 0.87 \begin {gather*} -x-4\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{2\,x}}{2\,x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 4*x^2 + 16*x*exp(2*x + 2*log(2))*exp((2*exp(2*x + 2*log(2)))/(4*x + 2)) + 1)/(4*x + 4*x^2 + 1),x)

[Out]

- x - 4*exp((4*exp(2*x))/(2*x + 1))

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sympy [A]  time = 0.19, size = 17, normalized size = 0.74 \begin {gather*} - x - 4 e^{\frac {8 e^{2 x}}{4 x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x*exp(2*ln(2)+2*x)*exp(exp(2*ln(2)+2*x)/(4*x+2))**2-4*x**2-4*x-1)/(4*x**2+4*x+1),x)

[Out]

-x - 4*exp(8*exp(2*x)/(4*x + 2))

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