3.102.79 \(\int \frac {e^{-x^2} (8 x-x^2+8 x^3-2 x^4+e^5 (4+8 x^2-2 x^3)+e^{x^2} (-2 e^{10}+6 x^2+e^5 (-4 x+2 x^2)+e^x (-8 x+5 x^2-x^3+e^5 (-4+4 x-x^2))))}{e^{10} x^2+2 e^5 x^3+x^4} \, dx\)

Optimal. Leaf size=36 \[ \frac {2-\frac {(4-x) \left (-e^x+e^{-x^2}+2 x\right )}{e^5+x}}{x} \]

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Rubi [F]  time = 4.32, 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^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{e^{10} x^2+2 e^5 x^3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*x - x^2 + 8*x^3 - 2*x^4 + E^5*(4 + 8*x^2 - 2*x^3) + E^x^2*(-2*E^10 + 6*x^2 + E^5*(-4*x + 2*x^2) + E^x*(
-8*x + 5*x^2 - x^3 + E^5*(-4 + 4*x - x^2))))/(E^x^2*(E^10*x^2 + 2*E^5*x^3 + x^4)),x]

[Out]

2/x + (4*E^(-5 + x))/x - (4*E^(-5 - x^2))/x + 1/(E^x^2*(E^5 + x)) + (8*E^(-5 - x^2))/(E^5 + x) + (8*E^(5 - x^2
))/(E^5 + x) + (2*E^(10 - x^2))/(E^5 + x) - (2*(4 + E^5))/(E^5 + x) - (E^(-5 + x)*(4 + E^5))/(E^5 + x) - (2*E^
(-5 - x^2)*(2 + 4*E^10 + E^15))/(E^5 + x) + (4*Sqrt[Pi]*Erf[x])/E^5 + 8*E^5*Sqrt[Pi]*Erf[x] + 2*E^10*Sqrt[Pi]*
Erf[x] - (2*(2 + 4*E^10 + E^15)*Sqrt[Pi]*Erf[x])/E^5 + 8*Defer[Int][1/(E^x^2*(E^5 + x)), x] - 2*E^5*Defer[Int]
[1/(E^x^2*(E^5 + x)), x] - 8*Defer[Int][E^(-10 - x^2)/(E^5 + x), x] + 2*(4 - E^15)*Defer[Int][E^(-10 - x^2)/(E
^5 + x), x] - 16*E^5*Defer[Int][E^(-5 - x^2)/(E^5 + x), x] + 4*E^5*(2 + 4*E^10 + E^15)*Defer[Int][E^(-5 - x^2)
/(E^5 + x), x] + 4*Defer[Int][E^(5 - x^2)/(E^5 + x), x] - 16*E^5*Defer[Int][E^(5 - x^2)/(E^5 + x), x] - 4*E^5*
Defer[Int][E^(10 - x^2)/(E^5 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{x^2 \left (e^{10}+2 e^5 x+x^2\right )} \, dx\\ &=\int \frac {e^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{x^2 \left (e^5+x\right )^2} \, dx\\ &=\int \left (-\frac {e^{-x^2}}{\left (e^5+x\right )^2}+\frac {8 e^{-x^2}}{x \left (e^5+x\right )^2}+\frac {8 e^{-x^2} x}{\left (e^5+x\right )^2}-\frac {2 e^{-x^2} x^2}{\left (e^5+x\right )^2}-\frac {2 e^{5-x^2} \left (-2-4 x^2+x^3\right )}{x^2 \left (e^5+x\right )^2}+\frac {-2 e^{10}-4 e^{5+x}-4 e^5 x-8 e^x \left (1-\frac {e^5}{2}\right ) x+5 e^x \left (1-\frac {e^5}{5}\right ) x^2+6 \left (1+\frac {e^5}{3}\right ) x^2-e^x x^3}{x^2 \left (e^5+x\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{-x^2} x^2}{\left (e^5+x\right )^2} \, dx\right )-2 \int \frac {e^{5-x^2} \left (-2-4 x^2+x^3\right )}{x^2 \left (e^5+x\right )^2} \, dx+8 \int \frac {e^{-x^2}}{x \left (e^5+x\right )^2} \, dx+8 \int \frac {e^{-x^2} x}{\left (e^5+x\right )^2} \, dx-\int \frac {e^{-x^2}}{\left (e^5+x\right )^2} \, dx+\int \frac {-2 e^{10}-4 e^{5+x}-4 e^5 x-8 e^x \left (1-\frac {e^5}{2}\right ) x+5 e^x \left (1-\frac {e^5}{5}\right ) x^2+6 \left (1+\frac {e^5}{3}\right ) x^2-e^x x^3}{x^2 \left (e^5+x\right )^2} \, dx\\ &=\frac {e^{-x^2}}{e^5+x}+2 \int e^{-x^2} \, dx-2 \int \left (e^{-x^2}+\frac {e^{10-x^2}}{\left (e^5+x\right )^2}-\frac {2 e^{5-x^2}}{e^5+x}\right ) \, dx-2 \int \left (-\frac {2 e^{-5-x^2}}{x^2}+\frac {4 e^{-10-x^2}}{x}+\frac {e^{-5-x^2} \left (-2-4 e^{10}-e^{15}\right )}{\left (e^5+x\right )^2}+\frac {e^{-10-x^2} \left (-4+e^{15}\right )}{e^5+x}\right ) \, dx+8 \int \left (-\frac {e^{5-x^2}}{\left (e^5+x\right )^2}+\frac {e^{-x^2}}{e^5+x}\right ) \, dx+8 \int \left (\frac {e^{-10-x^2}}{x}-\frac {e^{-5-x^2}}{\left (e^5+x\right )^2}-\frac {e^{-10-x^2}}{e^5+x}\right ) \, dx-\left (2 e^5\right ) \int \frac {e^{-x^2}}{e^5+x} \, dx+\int \frac {-2 e^{10}-e^{5+x} (-2+x)^2+2 e^5 (-2+x) x+6 x^2-e^x x \left (8-5 x+x^2\right )}{x^2 \left (e^5+x\right )^2} \, dx\\ &=\frac {e^{-x^2}}{e^5+x}+\sqrt {\pi } \text {erf}(x)-2 \int e^{-x^2} \, dx-2 \int \frac {e^{10-x^2}}{\left (e^5+x\right )^2} \, dx+4 \int \frac {e^{-5-x^2}}{x^2} \, dx+4 \int \frac {e^{5-x^2}}{e^5+x} \, dx-8 \int \frac {e^{-5-x^2}}{\left (e^5+x\right )^2} \, dx-8 \int \frac {e^{5-x^2}}{\left (e^5+x\right )^2} \, dx+8 \int \frac {e^{-x^2}}{e^5+x} \, dx-8 \int \frac {e^{-10-x^2}}{e^5+x} \, dx-\left (2 e^5\right ) \int \frac {e^{-x^2}}{e^5+x} \, dx+\left (2 \left (4-e^{15}\right )\right ) \int \frac {e^{-10-x^2}}{e^5+x} \, dx+\left (2 \left (2+4 e^{10}+e^{15}\right )\right ) \int \frac {e^{-5-x^2}}{\left (e^5+x\right )^2} \, dx+\int \left (\frac {2 \left (-e^{10}-2 e^5 x+\left (3+e^5\right ) x^2\right )}{x^2 \left (e^5+x\right )^2}+\frac {e^x \left (-4 e^5-4 \left (2-e^5\right ) x+\left (5-e^5\right ) x^2-x^3\right )}{x^2 \left (e^5+x\right )^2}\right ) \, dx\\ &=-\frac {4 e^{-5-x^2}}{x}+\frac {e^{-x^2}}{e^5+x}+\frac {8 e^{-5-x^2}}{e^5+x}+\frac {8 e^{5-x^2}}{e^5+x}+\frac {2 e^{10-x^2}}{e^5+x}-\frac {2 e^{-5-x^2} \left (2+4 e^{10}+e^{15}\right )}{e^5+x}+2 \int \frac {-e^{10}-2 e^5 x+\left (3+e^5\right ) x^2}{x^2 \left (e^5+x\right )^2} \, dx+4 \int e^{10-x^2} \, dx+4 \int \frac {e^{5-x^2}}{e^5+x} \, dx-8 \int e^{-5-x^2} \, dx+8 \int \frac {e^{-x^2}}{e^5+x} \, dx-8 \int \frac {e^{-10-x^2}}{e^5+x} \, dx+16 \int e^{-5-x^2} \, dx+16 \int e^{5-x^2} \, dx-\left (2 e^5\right ) \int \frac {e^{-x^2}}{e^5+x} \, dx-\left (4 e^5\right ) \int \frac {e^{10-x^2}}{e^5+x} \, dx-\left (16 e^5\right ) \int \frac {e^{-5-x^2}}{e^5+x} \, dx-\left (16 e^5\right ) \int \frac {e^{5-x^2}}{e^5+x} \, dx+\left (2 \left (4-e^{15}\right )\right ) \int \frac {e^{-10-x^2}}{e^5+x} \, dx-\left (4 \left (2+4 e^{10}+e^{15}\right )\right ) \int e^{-5-x^2} \, dx+\left (4 e^5 \left (2+4 e^{10}+e^{15}\right )\right ) \int \frac {e^{-5-x^2}}{e^5+x} \, dx+\int \frac {e^x \left (-4 e^5-4 \left (2-e^5\right ) x+\left (5-e^5\right ) x^2-x^3\right )}{x^2 \left (e^5+x\right )^2} \, dx\\ &=-\frac {4 e^{-5-x^2}}{x}+\frac {e^{-x^2}}{e^5+x}+\frac {8 e^{-5-x^2}}{e^5+x}+\frac {8 e^{5-x^2}}{e^5+x}+\frac {2 e^{10-x^2}}{e^5+x}-\frac {2 e^{-5-x^2} \left (2+4 e^{10}+e^{15}\right )}{e^5+x}+\frac {4 \sqrt {\pi } \text {erf}(x)}{e^5}+8 e^5 \sqrt {\pi } \text {erf}(x)+2 e^{10} \sqrt {\pi } \text {erf}(x)-\frac {2 \left (2+4 e^{10}+e^{15}\right ) \sqrt {\pi } \text {erf}(x)}{e^5}+2 \int \left (-\frac {1}{x^2}+\frac {4+e^5}{\left (e^5+x\right )^2}\right ) \, dx+4 \int \frac {e^{5-x^2}}{e^5+x} \, dx+8 \int \frac {e^{-x^2}}{e^5+x} \, dx-8 \int \frac {e^{-10-x^2}}{e^5+x} \, dx-\left (2 e^5\right ) \int \frac {e^{-x^2}}{e^5+x} \, dx-\left (4 e^5\right ) \int \frac {e^{10-x^2}}{e^5+x} \, dx-\left (16 e^5\right ) \int \frac {e^{-5-x^2}}{e^5+x} \, dx-\left (16 e^5\right ) \int \frac {e^{5-x^2}}{e^5+x} \, dx+\left (2 \left (4-e^{15}\right )\right ) \int \frac {e^{-10-x^2}}{e^5+x} \, dx+\left (4 e^5 \left (2+4 e^{10}+e^{15}\right )\right ) \int \frac {e^{-5-x^2}}{e^5+x} \, dx+\int \left (-\frac {4 e^{-5+x}}{x^2}+\frac {4 e^{-5+x}}{x}+\frac {e^{-5+x} \left (4+e^5\right )}{\left (e^5+x\right )^2}+\frac {e^{-5+x} \left (-4-e^5\right )}{e^5+x}\right ) \, dx\\ &=\frac {2}{x}-\frac {4 e^{-5-x^2}}{x}+\frac {e^{-x^2}}{e^5+x}+\frac {8 e^{-5-x^2}}{e^5+x}+\frac {8 e^{5-x^2}}{e^5+x}+\frac {2 e^{10-x^2}}{e^5+x}-\frac {2 \left (4+e^5\right )}{e^5+x}-\frac {2 e^{-5-x^2} \left (2+4 e^{10}+e^{15}\right )}{e^5+x}+\frac {4 \sqrt {\pi } \text {erf}(x)}{e^5}+8 e^5 \sqrt {\pi } \text {erf}(x)+2 e^{10} \sqrt {\pi } \text {erf}(x)-\frac {2 \left (2+4 e^{10}+e^{15}\right ) \sqrt {\pi } \text {erf}(x)}{e^5}-4 \int \frac {e^{-5+x}}{x^2} \, dx+4 \int \frac {e^{-5+x}}{x} \, dx+4 \int \frac {e^{5-x^2}}{e^5+x} \, dx+8 \int \frac {e^{-x^2}}{e^5+x} \, dx-8 \int \frac {e^{-10-x^2}}{e^5+x} \, dx-\left (2 e^5\right ) \int \frac {e^{-x^2}}{e^5+x} \, dx-\left (4 e^5\right ) \int \frac {e^{10-x^2}}{e^5+x} \, dx-\left (16 e^5\right ) \int \frac {e^{-5-x^2}}{e^5+x} \, dx-\left (16 e^5\right ) \int \frac {e^{5-x^2}}{e^5+x} \, dx+\left (-4-e^5\right ) \int \frac {e^{-5+x}}{e^5+x} \, dx+\left (4+e^5\right ) \int \frac {e^{-5+x}}{\left (e^5+x\right )^2} \, dx+\left (2 \left (4-e^{15}\right )\right ) \int \frac {e^{-10-x^2}}{e^5+x} \, dx+\left (4 e^5 \left (2+4 e^{10}+e^{15}\right )\right ) \int \frac {e^{-5-x^2}}{e^5+x} \, dx\\ &=\frac {2}{x}+\frac {4 e^{-5+x}}{x}-\frac {4 e^{-5-x^2}}{x}+\frac {e^{-x^2}}{e^5+x}+\frac {8 e^{-5-x^2}}{e^5+x}+\frac {8 e^{5-x^2}}{e^5+x}+\frac {2 e^{10-x^2}}{e^5+x}-\frac {2 \left (4+e^5\right )}{e^5+x}-\frac {e^{-5+x} \left (4+e^5\right )}{e^5+x}-\frac {2 e^{-5-x^2} \left (2+4 e^{10}+e^{15}\right )}{e^5+x}+\frac {4 \sqrt {\pi } \text {erf}(x)}{e^5}+8 e^5 \sqrt {\pi } \text {erf}(x)+2 e^{10} \sqrt {\pi } \text {erf}(x)-\frac {2 \left (2+4 e^{10}+e^{15}\right ) \sqrt {\pi } \text {erf}(x)}{e^5}+\frac {4 \text {Ei}(x)}{e^5}-e^{-5-e^5} \left (4+e^5\right ) \text {Ei}\left (e^5+x\right )-4 \int \frac {e^{-5+x}}{x} \, dx+4 \int \frac {e^{5-x^2}}{e^5+x} \, dx+8 \int \frac {e^{-x^2}}{e^5+x} \, dx-8 \int \frac {e^{-10-x^2}}{e^5+x} \, dx-\left (2 e^5\right ) \int \frac {e^{-x^2}}{e^5+x} \, dx-\left (4 e^5\right ) \int \frac {e^{10-x^2}}{e^5+x} \, dx-\left (16 e^5\right ) \int \frac {e^{-5-x^2}}{e^5+x} \, dx-\left (16 e^5\right ) \int \frac {e^{5-x^2}}{e^5+x} \, dx+\left (4+e^5\right ) \int \frac {e^{-5+x}}{e^5+x} \, dx+\left (2 \left (4-e^{15}\right )\right ) \int \frac {e^{-10-x^2}}{e^5+x} \, dx+\left (4 e^5 \left (2+4 e^{10}+e^{15}\right )\right ) \int \frac {e^{-5-x^2}}{e^5+x} \, dx\\ &=\frac {2}{x}+\frac {4 e^{-5+x}}{x}-\frac {4 e^{-5-x^2}}{x}+\frac {e^{-x^2}}{e^5+x}+\frac {8 e^{-5-x^2}}{e^5+x}+\frac {8 e^{5-x^2}}{e^5+x}+\frac {2 e^{10-x^2}}{e^5+x}-\frac {2 \left (4+e^5\right )}{e^5+x}-\frac {e^{-5+x} \left (4+e^5\right )}{e^5+x}-\frac {2 e^{-5-x^2} \left (2+4 e^{10}+e^{15}\right )}{e^5+x}+\frac {4 \sqrt {\pi } \text {erf}(x)}{e^5}+8 e^5 \sqrt {\pi } \text {erf}(x)+2 e^{10} \sqrt {\pi } \text {erf}(x)-\frac {2 \left (2+4 e^{10}+e^{15}\right ) \sqrt {\pi } \text {erf}(x)}{e^5}+4 \int \frac {e^{5-x^2}}{e^5+x} \, dx+8 \int \frac {e^{-x^2}}{e^5+x} \, dx-8 \int \frac {e^{-10-x^2}}{e^5+x} \, dx-\left (2 e^5\right ) \int \frac {e^{-x^2}}{e^5+x} \, dx-\left (4 e^5\right ) \int \frac {e^{10-x^2}}{e^5+x} \, dx-\left (16 e^5\right ) \int \frac {e^{-5-x^2}}{e^5+x} \, dx-\left (16 e^5\right ) \int \frac {e^{5-x^2}}{e^5+x} \, dx+\left (2 \left (4-e^{15}\right )\right ) \int \frac {e^{-10-x^2}}{e^5+x} \, dx+\left (4 e^5 \left (2+4 e^{10}+e^{15}\right )\right ) \int \frac {e^{-5-x^2}}{e^5+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 53, normalized size = 1.47 \begin {gather*} \frac {e^{-x^2} \left (-4-e^{x+x^2} (-4+x)-2 e^{5+x^2} (-1+x)+x-6 e^{x^2} x\right )}{x \left (e^5+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x - x^2 + 8*x^3 - 2*x^4 + E^5*(4 + 8*x^2 - 2*x^3) + E^x^2*(-2*E^10 + 6*x^2 + E^5*(-4*x + 2*x^2) +
E^x*(-8*x + 5*x^2 - x^3 + E^5*(-4 + 4*x - x^2))))/(E^x^2*(E^10*x^2 + 2*E^5*x^3 + x^4)),x]

[Out]

(-4 - E^(x + x^2)*(-4 + x) - 2*E^(5 + x^2)*(-1 + x) + x - 6*E^x^2*x)/(E^x^2*x*(E^5 + x))

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fricas [A]  time = 0.65, size = 45, normalized size = 1.25 \begin {gather*} -\frac {{\left ({\left (2 \, {\left (x - 1\right )} e^{5} + {\left (x - 4\right )} e^{x} + 6 \, x\right )} e^{\left (x^{2}\right )} - x + 4\right )} e^{\left (-x^{2}\right )}}{x^{2} + x e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x^2+4*x-4)*exp(5)-x^3+5*x^2-8*x)*exp(x)-2*exp(5)^2+(2*x^2-4*x)*exp(5)+6*x^2)*exp(x^2)+(-2*x^3+8
*x^2+4)*exp(5)-2*x^4+8*x^3-x^2+8*x)/(x^2*exp(5)^2+2*x^3*exp(5)+x^4)/exp(x^2),x, algorithm="fricas")

[Out]

-((2*(x - 1)*e^5 + (x - 4)*e^x + 6*x)*e^(x^2) - x + 4)*e^(-x^2)/(x^2 + x*e^5)

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giac [B]  time = 0.21, size = 66, normalized size = 1.83 \begin {gather*} -\frac {x e^{\left (x^{2} + x\right )} + 2 \, x e^{\left (x^{2} + 5\right )} + 6 \, x e^{\left (x^{2}\right )} - x - 4 \, e^{\left (x^{2} + x\right )} - 2 \, e^{\left (x^{2} + 5\right )} + 4}{x^{2} e^{\left (x^{2}\right )} + x e^{\left (x^{2} + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x^2+4*x-4)*exp(5)-x^3+5*x^2-8*x)*exp(x)-2*exp(5)^2+(2*x^2-4*x)*exp(5)+6*x^2)*exp(x^2)+(-2*x^3+8
*x^2+4)*exp(5)-2*x^4+8*x^3-x^2+8*x)/(x^2*exp(5)^2+2*x^3*exp(5)+x^4)/exp(x^2),x, algorithm="giac")

[Out]

-(x*e^(x^2 + x) + 2*x*e^(x^2 + 5) + 6*x*e^(x^2) - x - 4*e^(x^2 + x) - 2*e^(x^2 + 5) + 4)/(x^2*e^(x^2) + x*e^(x
^2 + 5))

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maple [A]  time = 0.39, size = 57, normalized size = 1.58

method	result	size
norman	\(\frac {\left (-4+x +\left (-2 \,{\mathrm e}^{5}-6\right ) x \,{\mathrm e}^{x^{2}}+2 \,{\mathrm e}^{5} {\mathrm e}^{x^{2}}+4 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}-x \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}\right ) {\mathrm e}^{-x^{2}}}{x \left ({\mathrm e}^{5}+x \right )}\)	\(57\)
risch	\(\frac {\left (-2 \,{\mathrm e}^{5}-6\right ) x +2 \,{\mathrm e}^{5}}{\left ({\mathrm e}^{5}+x \right ) x}-\frac {\left (x -4\right ) {\mathrm e}^{x}}{x \left ({\mathrm e}^{5}+x \right )}+\frac {\left (x -4\right ) {\mathrm e}^{-x^{2}}}{x \left ({\mathrm e}^{5}+x \right )}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-x^2+4*x-4)*exp(5)-x^3+5*x^2-8*x)*exp(x)-2*exp(5)^2+(2*x^2-4*x)*exp(5)+6*x^2)*exp(x^2)+(-2*x^3+8*x^2+4
)*exp(5)-2*x^4+8*x^3-x^2+8*x)/(x^2*exp(5)^2+2*x^3*exp(5)+x^4)/exp(x^2),x,method=_RETURNVERBOSE)

[Out]

(-4+x+(-2*exp(5)-6)*x*exp(x^2)+2*exp(5)*exp(x^2)+4*exp(x)*exp(x^2)-x*exp(x)*exp(x^2))/x/(exp(5)+x)/exp(x^2)

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maxima [A]  time = 0.42, size = 41, normalized size = 1.14 \begin {gather*} -\frac {2 \, x {\left (e^{5} + 3\right )} - {\left (x - 4\right )} e^{\left (-x^{2}\right )} + {\left (x - 4\right )} e^{x} - 2 \, e^{5}}{x^{2} + x e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x^2+4*x-4)*exp(5)-x^3+5*x^2-8*x)*exp(x)-2*exp(5)^2+(2*x^2-4*x)*exp(5)+6*x^2)*exp(x^2)+(-2*x^3+8
*x^2+4)*exp(5)-2*x^4+8*x^3-x^2+8*x)/(x^2*exp(5)^2+2*x^3*exp(5)+x^4)/exp(x^2),x, algorithm="maxima")

[Out]

-(2*x*(e^5 + 3) - (x - 4)*e^(-x^2) + (x - 4)*e^x - 2*e^5)/(x^2 + x*e^5)

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mupad [B]  time = 7.75, size = 49, normalized size = 1.36 \begin {gather*} -\frac {6\,x-2\,{\mathrm {e}}^5+4\,{\mathrm {e}}^{-x^2}-4\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^5-x\,{\mathrm {e}}^{-x^2}+x\,{\mathrm {e}}^x}{x\,\left (x+{\mathrm {e}}^5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x^2)*(8*x - exp(x^2)*(2*exp(10) + exp(5)*(4*x - 2*x^2) + exp(x)*(8*x + exp(5)*(x^2 - 4*x + 4) - 5*x^
2 + x^3) - 6*x^2) + exp(5)*(8*x^2 - 2*x^3 + 4) - x^2 + 8*x^3 - 2*x^4))/(2*x^3*exp(5) + x^2*exp(10) + x^4),x)

[Out]

-(6*x - 2*exp(5) + 4*exp(-x^2) - 4*exp(x) + 2*x*exp(5) - x*exp(-x^2) + x*exp(x))/(x*(x + exp(5)))

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sympy [B]  time = 0.58, size = 54, normalized size = 1.50 \begin {gather*} \frac {\left (4 - x\right ) e^{x}}{x^{2} + x e^{5}} + \frac {\left (x - 4\right ) e^{- x^{2}}}{x^{2} + x e^{5}} + \frac {x \left (- 2 e^{5} - 6\right ) + 2 e^{5}}{x^{2} + x e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x**2+4*x-4)*exp(5)-x**3+5*x**2-8*x)*exp(x)-2*exp(5)**2+(2*x**2-4*x)*exp(5)+6*x**2)*exp(x**2)+(-
2*x**3+8*x**2+4)*exp(5)-2*x**4+8*x**3-x**2+8*x)/(x**2*exp(5)**2+2*x**3*exp(5)+x**4)/exp(x**2),x)

[Out]

(4 - x)*exp(x)/(x**2 + x*exp(5)) + (x - 4)*exp(-x**2)/(x**2 + x*exp(5)) + (x*(-2*exp(5) - 6) + 2*exp(5))/(x**2
+ x*exp(5))

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