3.18.65 \(\int \frac {e^{1-x+x^2} (1536-1024 x)+e^x (-512+448 x+32 x^2)}{256 e^{2-4 x+2 x^2}+256 x^2+32 x^3+x^4+e^{1-2 x+x^2} (512 x+32 x^2)} \, dx\)

Optimal. Leaf size=25 \[ \frac {2 e^x}{e^{(1-x)^2}+x+\frac {x^2}{16}} \]

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Rubi [F]  time = 2.76, 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^{1-x+x^2} (1536-1024 x)+e^x \left (-512+448 x+32 x^2\right )}{256 e^{2-4 x+2 x^2}+256 x^2+32 x^3+x^4+e^{1-2 x+x^2} \left (512 x+32 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(1 - x + x^2)*(1536 - 1024*x) + E^x*(-512 + 448*x + 32*x^2))/(256*E^(2 - 4*x + 2*x^2) + 256*x^2 + 32*x^
3 + x^4 + E^(1 - 2*x + x^2)*(512*x + 32*x^2)),x]

[Out]

-512*Defer[Int][E^(5*x)/(16*E^(1 + x^2) + 16*E^(2*x)*x + E^(2*x)*x^2)^2, x] - 1088*Defer[Int][(E^(5*x)*x)/(16*
E^(1 + x^2) + 16*E^(2*x)*x + E^(2*x)*x^2)^2, x] + 960*Defer[Int][(E^(5*x)*x^2)/(16*E^(1 + x^2) + 16*E^(2*x)*x
+ E^(2*x)*x^2)^2, x] + 64*Defer[Int][(E^(5*x)*x^3)/(16*E^(1 + x^2) + 16*E^(2*x)*x + E^(2*x)*x^2)^2, x] + 96*De
fer[Int][E^(3*x)/(16*E^(1 + x^2) + 16*E^(2*x)*x + E^(2*x)*x^2), x] - 64*Defer[Int][(E^(3*x)*x)/(16*E^(1 + x^2)
+ 16*E^(2*x)*x + E^(2*x)*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-512 e^{1+3 x+x^2} (-3+2 x)+32 e^{5 x} \left (-16+14 x+x^2\right )}{\left (16 e^{1+x^2}+e^{2 x} x (16+x)\right )^2} \, dx\\ &=\int \left (-\frac {32 e^{3 x} (-3+2 x)}{16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2}+\frac {64 e^{5 x} \left (-8-17 x+15 x^2+x^3\right )}{\left (16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2\right )^2}\right ) \, dx\\ &=-\left (32 \int \frac {e^{3 x} (-3+2 x)}{16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2} \, dx\right )+64 \int \frac {e^{5 x} \left (-8-17 x+15 x^2+x^3\right )}{\left (16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2\right )^2} \, dx\\ &=-\left (32 \int \left (-\frac {3 e^{3 x}}{16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2}+\frac {2 e^{3 x} x}{16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2}\right ) \, dx\right )+64 \int \left (-\frac {8 e^{5 x}}{\left (16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2\right )^2}-\frac {17 e^{5 x} x}{\left (16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2\right )^2}+\frac {15 e^{5 x} x^2}{\left (16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2\right )^2}+\frac {e^{5 x} x^3}{\left (16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2\right )^2}\right ) \, dx\\ &=64 \int \frac {e^{5 x} x^3}{\left (16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2\right )^2} \, dx-64 \int \frac {e^{3 x} x}{16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2} \, dx+96 \int \frac {e^{3 x}}{16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2} \, dx-512 \int \frac {e^{5 x}}{\left (16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2\right )^2} \, dx+960 \int \frac {e^{5 x} x^2}{\left (16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2\right )^2} \, dx-1088 \int \frac {e^{5 x} x}{\left (16 e^{1+x^2}+16 e^{2 x} x+e^{2 x} x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.33, size = 29, normalized size = 1.16 \begin {gather*} \frac {32 e^{3 x}}{16 e^{1+x^2}+e^{2 x} x (16+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(1 - x + x^2)*(1536 - 1024*x) + E^x*(-512 + 448*x + 32*x^2))/(256*E^(2 - 4*x + 2*x^2) + 256*x^2 +
32*x^3 + x^4 + E^(1 - 2*x + x^2)*(512*x + 32*x^2)),x]

[Out]

(32*E^(3*x))/(16*E^(1 + x^2) + E^(2*x)*x*(16 + x))

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fricas [B]  time = 0.74, size = 44, normalized size = 1.76 \begin {gather*} \frac {32 \, e^{\left (x^{2} - x + 1\right )}}{{\left (x^{2} + 16 \, x\right )} e^{\left (x^{2} - 2 \, x + 1\right )} + 16 \, e^{\left (2 \, x^{2} - 4 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1024*x+1536)*exp(x)*exp(x^2-2*x+1)+(32*x^2+448*x-512)*exp(x))/(256*exp(x^2-2*x+1)^2+(32*x^2+512*x
)*exp(x^2-2*x+1)+x^4+32*x^3+256*x^2),x, algorithm="fricas")

[Out]

32*e^(x^2 - x + 1)/((x^2 + 16*x)*e^(x^2 - 2*x + 1) + 16*e^(2*x^2 - 4*x + 2))

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giac [B]  time = 0.31, size = 255, normalized size = 10.20 \begin {gather*} \frac {32 \, {\left (x^{5} e^{\left (2 \, x\right )} + 31 \, x^{4} e^{\left (2 \, x\right )} + 16 \, x^{3} e^{\left (x^{2} + 1\right )} + 223 \, x^{3} e^{\left (2 \, x\right )} + 240 \, x^{2} e^{\left (x^{2} + 1\right )} - 280 \, x^{2} e^{\left (2 \, x\right )} - 272 \, x e^{\left (x^{2} + 1\right )} - 128 \, x e^{\left (2 \, x\right )} - 128 \, e^{\left (x^{2} + 1\right )}\right )}}{x^{7} e^{x} + 47 \, x^{6} e^{x} + 32 \, x^{5} e^{\left (x^{2} - x + 1\right )} + 719 \, x^{5} e^{x} + 992 \, x^{4} e^{\left (x^{2} - x + 1\right )} + 3288 \, x^{4} e^{x} + 256 \, x^{3} e^{\left (2 \, x^{2} - 3 \, x + 2\right )} + 7136 \, x^{3} e^{\left (x^{2} - x + 1\right )} - 4608 \, x^{3} e^{x} + 3840 \, x^{2} e^{\left (2 \, x^{2} - 3 \, x + 2\right )} - 8960 \, x^{2} e^{\left (x^{2} - x + 1\right )} - 2048 \, x^{2} e^{x} - 4352 \, x e^{\left (2 \, x^{2} - 3 \, x + 2\right )} - 4096 \, x e^{\left (x^{2} - x + 1\right )} - 2048 \, e^{\left (2 \, x^{2} - 3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1024*x+1536)*exp(x)*exp(x^2-2*x+1)+(32*x^2+448*x-512)*exp(x))/(256*exp(x^2-2*x+1)^2+(32*x^2+512*x
)*exp(x^2-2*x+1)+x^4+32*x^3+256*x^2),x, algorithm="giac")

[Out]

32*(x^5*e^(2*x) + 31*x^4*e^(2*x) + 16*x^3*e^(x^2 + 1) + 223*x^3*e^(2*x) + 240*x^2*e^(x^2 + 1) - 280*x^2*e^(2*x
) - 272*x*e^(x^2 + 1) - 128*x*e^(2*x) - 128*e^(x^2 + 1))/(x^7*e^x + 47*x^6*e^x + 32*x^5*e^(x^2 - x + 1) + 719*
x^5*e^x + 992*x^4*e^(x^2 - x + 1) + 3288*x^4*e^x + 256*x^3*e^(2*x^2 - 3*x + 2) + 7136*x^3*e^(x^2 - x + 1) - 46
08*x^3*e^x + 3840*x^2*e^(2*x^2 - 3*x + 2) - 8960*x^2*e^(x^2 - x + 1) - 2048*x^2*e^x - 4352*x*e^(2*x^2 - 3*x +
2) - 4096*x*e^(x^2 - x + 1) - 2048*e^(2*x^2 - 3*x + 2))

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maple [A]  time = 0.07, size = 22, normalized size = 0.88

method	result	size
risch	\(\frac {32 \,{\mathrm e}^{x}}{x^{2}+16 \,{\mathrm e}^{\left (x -1\right )^{2}}+16 x}\)	\(22\)
norman	\(\frac {32 \,{\mathrm e}^{x}}{x^{2}+16 \,{\mathrm e}^{x^{2}-2 x +1}+16 x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-1024*x+1536)*exp(x)*exp(x^2-2*x+1)+(32*x^2+448*x-512)*exp(x))/(256*exp(x^2-2*x+1)^2+(32*x^2+512*x)*exp(
x^2-2*x+1)+x^4+32*x^3+256*x^2),x,method=_RETURNVERBOSE)

[Out]

32*exp(x)/(x^2+16*exp((x-1)^2)+16*x)

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maxima [A]  time = 0.50, size = 29, normalized size = 1.16 \begin {gather*} \frac {32 \, e^{\left (3 \, x\right )}}{{\left (x^{2} + 16 \, x\right )} e^{\left (2 \, x\right )} + 16 \, e^{\left (x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1024*x+1536)*exp(x)*exp(x^2-2*x+1)+(32*x^2+448*x-512)*exp(x))/(256*exp(x^2-2*x+1)^2+(32*x^2+512*x
)*exp(x^2-2*x+1)+x^4+32*x^3+256*x^2),x, algorithm="maxima")

[Out]

32*e^(3*x)/((x^2 + 16*x)*e^(2*x) + 16*e^(x^2 + 1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{x^2-x+1}\,\left (1024\,x-1536\right )-{\mathrm {e}}^x\,\left (32\,x^2+448\,x-512\right )}{256\,{\mathrm {e}}^{2\,x^2-4\,x+2}+{\mathrm {e}}^{x^2-2\,x+1}\,\left (32\,x^2+512\,x\right )+256\,x^2+32\,x^3+x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(448*x + 32*x^2 - 512) - exp(x^2 - 2*x + 1)*exp(x)*(1024*x - 1536))/(256*exp(2*x^2 - 4*x + 2) + ex
p(x^2 - 2*x + 1)*(512*x + 32*x^2) + 256*x^2 + 32*x^3 + x^4),x)

[Out]

int(-(exp(x^2 - x + 1)*(1024*x - 1536) - exp(x)*(448*x + 32*x^2 - 512))/(256*exp(2*x^2 - 4*x + 2) + exp(x^2 -
2*x + 1)*(512*x + 32*x^2) + 256*x^2 + 32*x^3 + x^4), x)

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sympy [A]  time = 0.17, size = 22, normalized size = 0.88 \begin {gather*} \frac {32 e^{x}}{x^{2} + 16 x + 16 e^{x^{2} - 2 x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1024*x+1536)*exp(x)*exp(x**2-2*x+1)+(32*x**2+448*x-512)*exp(x))/(256*exp(x**2-2*x+1)**2+(32*x**2+
512*x)*exp(x**2-2*x+1)+x**4+32*x**3+256*x**2),x)

[Out]

32*exp(x)/(x**2 + 16*x + 16*exp(x**2 - 2*x + 1))

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