3.20.79 \(\int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} (-2-2 x-32 e^x x+4 e^{2 x} x)}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} (-6 x-2 e^{2+2 x} x)} \, dx\)

Optimal. Leaf size=31 \[ \frac {2}{-3-e^{2+2 x}+e^{3-\left (-8+e^x\right )^2+x} x} \]

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Rubi [F]  time = 24.72, 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 {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*E^(2 + 2*x) + E^(-61 + 16*E^x - E^(2*x) + x)*(-2 - 2*x - 32*E^x*x + 4*E^(2*x)*x))/(9 + 6*E^(2 + 2*x) +
E^(4 + 4*x) + E^(-122 + 32*E^x - 2*E^(2*x) + 2*x)*x^2 + E^(-61 + 16*E^x - E^(2*x) + x)*(-6*x - 2*E^(2 + 2*x)*x
)),x]

[Out]

-12*Defer[Int][E^(2*(61 + E^(2*x)))/(3*E^(61 + E^(2*x)) + E^(63 + E^(2*x) + 2*x) - E^(16*E^x + x)*x)^2, x] - 2
*Defer[Int][E^(61 + 16*E^x + E^(2*x) + x)/(3*E^(61 + E^(2*x)) + E^(63 + E^(2*x) + 2*x) - E^(16*E^x + x)*x)^2,
x] + 96*Defer[Int][(E^(59 + 16*E^x + E^(2*x))*x)/(3*E^(61 + E^(2*x)) + E^(63 + E^(2*x) + 2*x) - E^(16*E^x + x)
*x)^2, x] - 2*(6 - E^2)*Defer[Int][(E^(59 + 16*E^x + E^(2*x) + x)*x)/(3*E^(61 + E^(2*x)) + E^(63 + E^(2*x) + 2
*x) - E^(16*E^x + x)*x)^2, x] - 12*Defer[Int][(E^(4*(-1 + 8*E^x))*x^2)/(3*E^(61 + E^(2*x)) + E^(63 + E^(2*x) +
2*x) - E^(16*E^x + x)*x)^2, x] - 32*Defer[Int][(E^(-2 + 32*E^x + x)*x^2)/(3*E^(61 + E^(2*x)) + E^(63 + E^(2*x
) + 2*x) - E^(16*E^x + x)*x)^2, x] + 4*Defer[Int][(E^(-65 + 48*E^x - E^(2*x) + x)*x^3)/(3*E^(61 + E^(2*x)) + E
^(63 + E^(2*x) + 2*x) - E^(16*E^x + x)*x)^2, x] + 4*Defer[Int][E^(61 + E^(2*x))/(3*E^(61 + E^(2*x)) + E^(63 +
E^(2*x) + 2*x) - E^(16*E^x + x)*x), x] + 4*Defer[Int][(E^(-65 + 32*E^x - E^(2*x))*x^2)/(3*E^(61 + E^(2*x)) + E
^(63 + E^(2*x) + 2*x) - E^(16*E^x + x)*x), x] + 32*Defer[Int][(E^(16*E^x)*x)/(-3*E^(63 + E^(2*x)) - E^(65 + E^
(2*x) + 2*x) + E^(2 + 16*E^x + x)*x), x] - 4*Defer[Int][(E^(16*E^x + x)*x)/(-3*E^(63 + E^(2*x)) - E^(65 + E^(2
*x) + 2*x) + E^(2 + 16*E^x + x)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 \left (61+e^{2 x}\right )} \left (4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )\right )}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx\\ &=\int \left (\frac {4 e^{-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )} \left (e^{126+2 e^{2 x}}-8 e^{63+16 e^x+e^{2 x}} x+e^{63+16 e^x+e^{2 x}+x} x+e^{32 e^x} x^2\right )}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x}+\frac {2 e^{-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )} \left (-6 e^{187+3 e^{2 x}}-e^{126+16 e^x+2 e^{2 x}+x}+48 e^{124+16 e^x+2 e^{2 x}} x-6 e^{124+16 e^x+2 e^{2 x}+x} \left (1-\frac {e^2}{6}\right ) x-6 e^{61+32 e^x+e^{2 x}} x^2-16 e^{63+32 e^x+e^{2 x}+x} x^2+2 e^{48 e^x+x} x^3\right )}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )} \left (-6 e^{187+3 e^{2 x}}-e^{126+16 e^x+2 e^{2 x}+x}+48 e^{124+16 e^x+2 e^{2 x}} x-6 e^{124+16 e^x+2 e^{2 x}+x} \left (1-\frac {e^2}{6}\right ) x-6 e^{61+32 e^x+e^{2 x}} x^2-16 e^{63+32 e^x+e^{2 x}+x} x^2+2 e^{48 e^x+x} x^3\right )}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx+4 \int \frac {e^{-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )} \left (e^{126+2 e^{2 x}}-8 e^{63+16 e^x+e^{2 x}} x+e^{63+16 e^x+e^{2 x}+x} x+e^{32 e^x} x^2\right )}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \, dx\\ &=2 \int \left (-\frac {6 e^{2 \left (61+e^{2 x}\right )}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}-\frac {e^{-61+16 e^x-e^{2 x}+2 \left (61+e^{2 x}\right )+x}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}+\frac {48 \exp \left (-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )+2 \left (62+8 e^x+e^{2 x}\right )\right ) x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}+\frac {e^{-63+16 e^x-e^{2 x}+2 \left (61+e^{2 x}\right )+x} \left (-6+e^2\right ) x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}-\frac {6 e^{-126+32 e^x-2 e^{2 x}+2 \left (61+e^{2 x}\right )} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}-\frac {16 e^{-124+32 e^x-2 e^{2 x}+2 \left (61+e^{2 x}\right )+x} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}+\frac {2 e^{-187+48 e^x-3 e^{2 x}+2 \left (61+e^{2 x}\right )+x} x^3}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}\right ) \, dx+4 \int \frac {e^{61+e^{2 x}}-8 e^{-2+16 e^x} x+e^{-2+16 e^x+x} x+e^{-65+32 e^x-e^{2 x}} x^2}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \, dx\\ &=-\left (2 \int \frac {e^{-61+16 e^x-e^{2 x}+2 \left (61+e^{2 x}\right )+x}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx\right )+4 \int \frac {e^{-187+48 e^x-3 e^{2 x}+2 \left (61+e^{2 x}\right )+x} x^3}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx+4 \int \left (\frac {e^{61+e^{2 x}}}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x}+\frac {e^{-65+32 e^x-e^{2 x}} x^2}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x}+\frac {8 e^{16 e^x} x}{-3 e^{63+e^{2 x}}-e^{65+e^{2 x}+2 x}+e^{2+16 e^x+x} x}-\frac {e^{16 e^x+x} x}{-3 e^{63+e^{2 x}}-e^{65+e^{2 x}+2 x}+e^{2+16 e^x+x} x}\right ) \, dx-12 \int \frac {e^{2 \left (61+e^{2 x}\right )}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-12 \int \frac {e^{-126+32 e^x-2 e^{2 x}+2 \left (61+e^{2 x}\right )} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-32 \int \frac {e^{-124+32 e^x-2 e^{2 x}+2 \left (61+e^{2 x}\right )+x} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx+96 \int \frac {\exp \left (-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )+2 \left (62+8 e^x+e^{2 x}\right )\right ) x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-\left (2 \left (6-e^2\right )\right ) \int \frac {e^{-63+16 e^x-e^{2 x}+2 \left (61+e^{2 x}\right )+x} x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx\\ &=-\left (2 \int \frac {e^{61+16 e^x+e^{2 x}+x}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx\right )+4 \int \frac {e^{-65+48 e^x-e^{2 x}+x} x^3}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx+4 \int \frac {e^{61+e^{2 x}}}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \, dx+4 \int \frac {e^{-65+32 e^x-e^{2 x}} x^2}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \, dx-4 \int \frac {e^{16 e^x+x} x}{-3 e^{63+e^{2 x}}-e^{65+e^{2 x}+2 x}+e^{2+16 e^x+x} x} \, dx-12 \int \frac {e^{2 \left (61+e^{2 x}\right )}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-12 \int \frac {e^{4 \left (-1+8 e^x\right )} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-32 \int \frac {e^{-2+32 e^x+x} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx+32 \int \frac {e^{16 e^x} x}{-3 e^{63+e^{2 x}}-e^{65+e^{2 x}+2 x}+e^{2+16 e^x+x} x} \, dx+96 \int \frac {e^{59+16 e^x+e^{2 x}} x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-\left (2 \left (6-e^2\right )\right ) \int \frac {e^{59+16 e^x+e^{2 x}+x} x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 49, normalized size = 1.58 \begin {gather*} -\frac {2 e^{61+e^{2 x}}}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^(2 + 2*x) + E^(-61 + 16*E^x - E^(2*x) + x)*(-2 - 2*x - 32*E^x*x + 4*E^(2*x)*x))/(9 + 6*E^(2 + 2
*x) + E^(4 + 4*x) + E^(-122 + 32*E^x - 2*E^(2*x) + 2*x)*x^2 + E^(-61 + 16*E^x - E^(2*x) + x)*(-6*x - 2*E^(2 +
2*x)*x)),x]

[Out]

(-2*E^(61 + E^(2*x)))/(3*E^(61 + E^(2*x)) + E^(63 + E^(2*x) + 2*x) - E^(16*E^x + x)*x)

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fricas [A]  time = 0.69, size = 30, normalized size = 0.97 \begin {gather*} \frac {2}{x e^{\left (x - e^{\left (2 \, x\right )} + 16 \, e^{x} - 61\right )} - e^{\left (2 \, x + 2\right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(x)^2-32*exp(x)*x-2*x-2)*exp(-exp(x)^2+16*exp(x)+x-61)+4*exp(2*x+2))/(x^2*exp(-exp(x)^2+16*
exp(x)+x-61)^2+(-2*x*exp(2*x+2)-6*x)*exp(-exp(x)^2+16*exp(x)+x-61)+exp(2*x+2)^2+6*exp(2*x+2)+9),x, algorithm="
fricas")

[Out]

2/(x*e^(x - e^(2*x) + 16*e^x - 61) - e^(2*x + 2) - 3)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(x)^2-32*exp(x)*x-2*x-2)*exp(-exp(x)^2+16*exp(x)+x-61)+4*exp(2*x+2))/(x^2*exp(-exp(x)^2+16*
exp(x)+x-61)^2+(-2*x*exp(2*x+2)-6*x)*exp(-exp(x)^2+16*exp(x)+x-61)+exp(2*x+2)^2+6*exp(2*x+2)+9),x, algorithm="
giac")

[Out]

Timed out

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maple [A]  time = 0.08, size = 30, normalized size = 0.97

method	result	size
risch	\(-\frac {2}{{\mathrm e}^{2 x +2}-{\mathrm e}^{-{\mathrm e}^{2 x}+16 \,{\mathrm e}^{x}+x -61} x +3}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x*exp(x)^2-32*exp(x)*x-2*x-2)*exp(-exp(x)^2+16*exp(x)+x-61)+4*exp(2*x+2))/(x^2*exp(-exp(x)^2+16*exp(x)
+x-61)^2+(-2*x*exp(2*x+2)-6*x)*exp(-exp(x)^2+16*exp(x)+x-61)+exp(2*x+2)^2+6*exp(2*x+2)+9),x,method=_RETURNVERB
OSE)

[Out]

-2/(exp(2*x+2)-exp(-exp(2*x)+16*exp(x)+x-61)*x+3)

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maxima [A]  time = 0.60, size = 39, normalized size = 1.26 \begin {gather*} \frac {2 \, e^{\left (e^{\left (2 \, x\right )} + 61\right )}}{x e^{\left (x + 16 \, e^{x}\right )} - {\left (3 \, e^{61} + e^{\left (2 \, x + 63\right )}\right )} e^{\left (e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(x)^2-32*exp(x)*x-2*x-2)*exp(-exp(x)^2+16*exp(x)+x-61)+4*exp(2*x+2))/(x^2*exp(-exp(x)^2+16*
exp(x)+x-61)^2+(-2*x*exp(2*x+2)-6*x)*exp(-exp(x)^2+16*exp(x)+x-61)+exp(2*x+2)^2+6*exp(2*x+2)+9),x, algorithm="
maxima")

[Out]

2*e^(e^(2*x) + 61)/(x*e^(x + 16*e^x) - (3*e^61 + e^(2*x + 63))*e^(e^(2*x)))

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mupad [B]  time = 1.48, size = 166, normalized size = 5.35 \begin {gather*} \frac {6\,x+96\,x^2\,{\mathrm {e}}^x+{\mathrm {e}}^{2\,x+2}\,\left (2\,x-2\,x^2\right )-12\,x^2\,{\mathrm {e}}^{2\,x}+32\,x^2\,{\mathrm {e}}^{3\,x+2}-4\,x^2\,{\mathrm {e}}^{4\,x+2}+6\,x^2}{\left ({\mathrm {e}}^{x-{\mathrm {e}}^{2\,x}+16\,{\mathrm {e}}^x-61}-\frac {{\mathrm {e}}^{2\,x+2}+3}{x}\right )\,\left (48\,x^3\,{\mathrm {e}}^x-6\,x^3\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^{2\,x+2}-x^3\,{\mathrm {e}}^{2\,x+2}+16\,x^3\,{\mathrm {e}}^{3\,x+2}-2\,x^3\,{\mathrm {e}}^{4\,x+2}+3\,x^2+3\,x^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(2*x + 2) - exp(x - exp(2*x) + 16*exp(x) - 61)*(2*x - 4*x*exp(2*x) + 32*x*exp(x) + 2))/(6*exp(2*x +
2) + exp(4*x + 4) - exp(x - exp(2*x) + 16*exp(x) - 61)*(6*x + 2*x*exp(2*x + 2)) + x^2*exp(2*x - 2*exp(2*x) + 3
2*exp(x) - 122) + 9),x)

[Out]

(6*x + 96*x^2*exp(x) + exp(2*x + 2)*(2*x - 2*x^2) - 12*x^2*exp(2*x) + 32*x^2*exp(3*x + 2) - 4*x^2*exp(4*x + 2)
+ 6*x^2)/((exp(x - exp(2*x) + 16*exp(x) - 61) - (exp(2*x + 2) + 3)/x)*(48*x^3*exp(x) - 6*x^3*exp(2*x) + x^2*e
xp(2*x + 2) - x^3*exp(2*x + 2) + 16*x^3*exp(3*x + 2) - 2*x^3*exp(4*x + 2) + 3*x^2 + 3*x^3))

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sympy [A]  time = 0.29, size = 27, normalized size = 0.87 \begin {gather*} \frac {2}{x e^{x - e^{2 x} + 16 e^{x} - 61} - e^{2} e^{2 x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(x)**2-32*exp(x)*x-2*x-2)*exp(-exp(x)**2+16*exp(x)+x-61)+4*exp(2*x+2))/(x**2*exp(-exp(x)**2
+16*exp(x)+x-61)**2+(-2*x*exp(2*x+2)-6*x)*exp(-exp(x)**2+16*exp(x)+x-61)+exp(2*x+2)**2+6*exp(2*x+2)+9),x)

[Out]

2/(x*exp(x - exp(2*x) + 16*exp(x) - 61) - exp(2)*exp(2*x) - 3)

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