3.47.21 \(\int \frac {e^{\frac {8}{9} e^{-2 x} x^2} (-72 e^{2+2 x}+e^2 (64 x^2-64 x^3)+e^{2 x} (-144 e^{2+2 x}+e^2 (64 x-64 x^2)))}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx\)

Optimal. Leaf size=27 \[ \frac {4 e^{2+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2} \]

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Rubi [F]  time = 2.57, 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 {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((8*x^2)/(9*E^(2*x)))*(-72*E^(2 + 2*x) + E^2*(64*x^2 - 64*x^3) + E^(2*x)*(-144*E^(2 + 2*x) + E^2*(64*x
- 64*x^2))))/(9*E^(8*x) + 27*E^(6*x)*x + 27*E^(4*x)*x^2 + 9*E^(2*x)*x^3),x]

[Out]

8*Defer[Int][(E^(2 - 2*x + (8*x^2)/(9*E^(2*x)))*x)/(E^(2*x) + x)^3, x] - 16*Defer[Int][(E^(2 - 2*x + (8*x^2)/(
9*E^(2*x)))*x^2)/(E^(2*x) + x)^3, x] - 8*Defer[Int][E^(2 - 2*x + (8*x^2)/(9*E^(2*x)))/(E^(2*x) + x)^2, x] + (3
52*Defer[Int][(E^(2 - 2*x + (8*x^2)/(9*E^(2*x)))*x)/(E^(2*x) + x)^2, x])/9 - (64*Defer[Int][(E^(2 - 2*x + (8*x
^2)/(9*E^(2*x)))*x^2)/(E^(2*x) + x)^2, x])/9 - 16*Defer[Int][E^(2 - 2*x + (8*x^2)/(9*E^(2*x)))/(E^(2*x) + x),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} \left (-18 e^{4 x}-8 (-1+x) x^2-e^{2 x} \left (9-8 x+8 x^2\right )\right )}{9 \left (e^{2 x}+x\right )^3} \, dx\\ &=\frac {8}{9} \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} \left (-18 e^{4 x}-8 (-1+x) x^2-e^{2 x} \left (9-8 x+8 x^2\right )\right )}{\left (e^{2 x}+x\right )^3} \, dx\\ &=\frac {8}{9} \int \left (-\frac {18 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{e^{2 x}+x}-\frac {9 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x (-1+2 x)}{\left (e^{2 x}+x\right )^3}-\frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} \left (9-44 x+8 x^2\right )}{\left (e^{2 x}+x\right )^2}\right ) \, dx\\ &=-\left (\frac {8}{9} \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} \left (9-44 x+8 x^2\right )}{\left (e^{2 x}+x\right )^2} \, dx\right )-8 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x (-1+2 x)}{\left (e^{2 x}+x\right )^3} \, dx-16 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{e^{2 x}+x} \, dx\\ &=-\left (\frac {8}{9} \int \left (\frac {9 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2}-\frac {44 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x}{\left (e^{2 x}+x\right )^2}+\frac {8 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x^2}{\left (e^{2 x}+x\right )^2}\right ) \, dx\right )-8 \int \left (-\frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x}{\left (e^{2 x}+x\right )^3}+\frac {2 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x^2}{\left (e^{2 x}+x\right )^3}\right ) \, dx-16 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{e^{2 x}+x} \, dx\\ &=-\left (\frac {64}{9} \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x^2}{\left (e^{2 x}+x\right )^2} \, dx\right )+8 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x}{\left (e^{2 x}+x\right )^3} \, dx-8 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2} \, dx-16 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x^2}{\left (e^{2 x}+x\right )^3} \, dx-16 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{e^{2 x}+x} \, dx+\frac {352}{9} \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x}{\left (e^{2 x}+x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.69, size = 27, normalized size = 1.00 \begin {gather*} \frac {4 e^{2+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((8*x^2)/(9*E^(2*x)))*(-72*E^(2 + 2*x) + E^2*(64*x^2 - 64*x^3) + E^(2*x)*(-144*E^(2 + 2*x) + E^2*
(64*x - 64*x^2))))/(9*E^(8*x) + 27*E^(6*x)*x + 27*E^(4*x)*x^2 + 9*E^(2*x)*x^3),x]

[Out]

(4*E^(2 + (8*x^2)/(9*E^(2*x))))/(E^(2*x) + x)^2

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fricas [A]  time = 0.58, size = 38, normalized size = 1.41 \begin {gather*} \frac {4 \, e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )} + 6\right )}}{x^{2} e^{4} + 2 \, x e^{\left (2 \, x + 4\right )} + e^{\left (4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*exp(1)^2*exp(x)^2+(-64*x^3+64*x^2)*exp
(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp(x)^2*exp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(
x)^2*x^3),x, algorithm="fricas")

[Out]

4*e^(8/9*x^2*e^(-2*x) + 6)/(x^2*e^4 + 2*x*e^(2*x + 4) + e^(4*x + 4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {8 \, {\left (8 \, {\left (x^{3} - x^{2}\right )} e^{2} + 2 \, {\left (4 \, {\left (x^{2} - x\right )} e^{2} + 9 \, e^{\left (2 \, x + 2\right )}\right )} e^{\left (2 \, x\right )} + 9 \, e^{\left (2 \, x + 2\right )}\right )} e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )}\right )}}{9 \, {\left (x^{3} e^{\left (2 \, x\right )} + 3 \, x^{2} e^{\left (4 \, x\right )} + 3 \, x e^{\left (6 \, x\right )} + e^{\left (8 \, x\right )}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*exp(1)^2*exp(x)^2+(-64*x^3+64*x^2)*exp
(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp(x)^2*exp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(
x)^2*x^3),x, algorithm="giac")

[Out]

integrate(-8/9*(8*(x^3 - x^2)*e^2 + 2*(4*(x^2 - x)*e^2 + 9*e^(2*x + 2))*e^(2*x) + 9*e^(2*x + 2))*e^(8/9*x^2*e^
(-2*x))/(x^3*e^(2*x) + 3*x^2*e^(4*x) + 3*x*e^(6*x) + e^(8*x)), x)

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maple [A]  time = 0.06, size = 23, normalized size = 0.85

method	result	size
risch	\(\frac {4 \,{\mathrm e}^{2+\frac {8 x^{2} {\mathrm e}^{-2 x}}{9}}}{\left ({\mathrm e}^{2 x}+x \right )^{2}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*exp(1)^2*exp(x)^2+(-64*x^3+64*x^2)*exp(1)^2)
*exp(4/9*x^2/exp(x)^2)^2/(9*exp(x)^2*exp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(x)^2*x
^3),x,method=_RETURNVERBOSE)

[Out]

4/(exp(2*x)+x)^2*exp(2+8/9*x^2*exp(-2*x))

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maxima [A]  time = 0.50, size = 31, normalized size = 1.15 \begin {gather*} \frac {4 \, e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )} + 2\right )}}{x^{2} + 2 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*exp(1)^2*exp(x)^2+(-64*x^3+64*x^2)*exp
(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp(x)^2*exp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(
x)^2*x^3),x, algorithm="maxima")

[Out]

4*e^(8/9*x^2*e^(-2*x) + 2)/(x^2 + 2*x*e^(2*x) + e^(4*x))

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mupad [B]  time = 3.61, size = 31, normalized size = 1.15 \begin {gather*} \frac {4\,{\mathrm {e}}^2\,{\mathrm {e}}^{\frac {8\,x^2\,{\mathrm {e}}^{-2\,x}}{9}}}{{\mathrm {e}}^{4\,x}+2\,x\,{\mathrm {e}}^{2\,x}+x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((8*x^2*exp(-2*x))/9)*(72*exp(2*x)*exp(2) + exp(2*x)*(144*exp(2*x)*exp(2) - exp(2)*(64*x - 64*x^2)) -
exp(2)*(64*x^2 - 64*x^3)))/(9*exp(8*x) + 27*x*exp(6*x) + 9*x^3*exp(2*x) + 27*x^2*exp(4*x)),x)

[Out]

(4*exp(2)*exp((8*x^2*exp(-2*x))/9))/(exp(4*x) + 2*x*exp(2*x) + x^2)

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sympy [A]  time = 0.27, size = 34, normalized size = 1.26 \begin {gather*} \frac {4 e^{2} e^{\frac {8 x^{2} e^{- 2 x}}{9}}}{x^{2} + 2 x e^{2 x} + e^{4 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*exp(1)**2*exp(x)**2+(-64*x**2+64*x)*exp(1)**2)*exp(2*x)-72*exp(1)**2*exp(x)**2+(-64*x**3+64*x
**2)*exp(1)**2)*exp(4/9*x**2/exp(x)**2)**2/(9*exp(x)**2*exp(2*x)**3+27*x*exp(x)**2*exp(2*x)**2+27*x**2*exp(x)*
*2*exp(2*x)+9*exp(x)**2*x**3),x)

[Out]

4*exp(2)*exp(8*x**2*exp(-2*x)/9)/(x**2 + 2*x*exp(2*x) + exp(4*x))

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