3.24.57 \(\int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x} (10+4 x-x^2+e^{2 x} (10 x-2 x^2)+e^{x+x^2} (10 x+18 x^2-4 x^3)+e^{2 x^2} (20 x^2-4 x^3))}{3 x^3} \, dx\)

Optimal. Leaf size=30 \[ \frac {e^{-5-\left (e^x+e^{x^2}\right )^2-x} (-5+x)}{3 x^2} \]

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Rubi [F]  time = 9.25, 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^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x} \left (10+4 x-x^2+e^{2 x} \left (10 x-2 x^2\right )+e^{x+x^2} \left (10 x+18 x^2-4 x^3\right )+e^{2 x^2} \left (20 x^2-4 x^3\right )\right )}{3 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x + x^2) - x)*(10 + 4*x - x^2 + E^(2*x)*(10*x - 2*x^2) + E^(x + x^2)*(
10*x + 18*x^2 - 4*x^3) + E^(2*x^2)*(20*x^2 - 4*x^3)))/(3*x^3),x]

[Out]

(-4*Defer[Int][E^(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x + x^2) + x^2), x])/3 - (4*Defer[Int][E^(-5 - E^(2*x) - E^(
2*x^2) - 2*E^(x + x^2) - x + 2*x^2), x])/3 + (10*Defer[Int][E^(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x + x^2) - x)/x
^3, x])/3 + (4*Defer[Int][E^(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x + x^2) - x)/x^2, x])/3 + (10*Defer[Int][E^(-5 -
E^(2*x) - E^(2*x^2) - 2*E^(x + x^2) + x)/x^2, x])/3 + (10*Defer[Int][E^(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x + x
^2) + x^2)/x^2, x])/3 - Defer[Int][E^(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x + x^2) - x)/x, x]/3 - (2*Defer[Int][E^
(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x + x^2) + x)/x, x])/3 + 6*Defer[Int][E^(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x +
x^2) + x^2)/x, x] + (20*Defer[Int][E^(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x + x^2) - x + 2*x^2)/x, x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x} \left (10+4 x-x^2+e^{2 x} \left (10 x-2 x^2\right )+e^{x+x^2} \left (10 x+18 x^2-4 x^3\right )+e^{2 x^2} \left (20 x^2-4 x^3\right )\right )}{x^3} \, dx\\ &=\frac {1}{3} \int \left (-\frac {4 e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x+2 x^2} (-5+x)}{x}-\frac {2 e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2} (-5+x) (1+2 x)}{x^2}-\frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x} \left (-10-4 x-10 e^{2 x} x+x^2+2 e^{2 x} x^2\right )}{x^3}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x} \left (-10-4 x-10 e^{2 x} x+x^2+2 e^{2 x} x^2\right )}{x^3} \, dx\right )-\frac {2}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2} (-5+x) (1+2 x)}{x^2} \, dx-\frac {4}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x+2 x^2} (-5+x)}{x} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {2 e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x} (-5+x)}{x^2}+\frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x} \left (-10-4 x+x^2\right )}{x^3}\right ) \, dx\right )-\frac {2}{3} \int \left (2 e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2}-\frac {5 e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2}}{x^2}-\frac {9 e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2}}{x}\right ) \, dx-\frac {4}{3} \int \left (e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x+2 x^2}-\frac {5 e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x+2 x^2}}{x}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x} \left (-10-4 x+x^2\right )}{x^3} \, dx\right )-\frac {2}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x} (-5+x)}{x^2} \, dx-\frac {4}{3} \int e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2} \, dx-\frac {4}{3} \int e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x+2 x^2} \, dx+\frac {10}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2}}{x^2} \, dx+6 \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2}}{x} \, dx+\frac {20}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x+2 x^2}}{x} \, dx\\ &=-\left (\frac {1}{3} \int \left (-\frac {10 e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x}}{x^3}-\frac {4 e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x}}{x^2}+\frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x}}{x}\right ) \, dx\right )-\frac {2}{3} \int \left (-\frac {5 e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x}}{x^2}+\frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x}}{x}\right ) \, dx-\frac {4}{3} \int e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2} \, dx-\frac {4}{3} \int e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x+2 x^2} \, dx+\frac {10}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2}}{x^2} \, dx+6 \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2}}{x} \, dx+\frac {20}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x+2 x^2}}{x} \, dx\\ &=-\left (\frac {1}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x}}{x} \, dx\right )-\frac {2}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x}}{x} \, dx-\frac {4}{3} \int e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2} \, dx-\frac {4}{3} \int e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x+2 x^2} \, dx+\frac {4}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x}}{x^2} \, dx+\frac {10}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x}}{x^3} \, dx+\frac {10}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x}}{x^2} \, dx+\frac {10}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2}}{x^2} \, dx+6 \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}+x^2}}{x} \, dx+\frac {20}{3} \int \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x+2 x^2}}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 42, normalized size = 1.40 \begin {gather*} \frac {e^{-5-e^{2 x}-e^{2 x^2}-2 e^{x+x^2}-x} (-5+x)}{3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x + x^2) - x)*(10 + 4*x - x^2 + E^(2*x)*(10*x - 2*x^2) + E^(x +
x^2)*(10*x + 18*x^2 - 4*x^3) + E^(2*x^2)*(20*x^2 - 4*x^3)))/(3*x^3),x]

[Out]

(E^(-5 - E^(2*x) - E^(2*x^2) - 2*E^(x + x^2) - x)*(-5 + x))/(3*x^2)

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fricas [B]  time = 0.51, size = 52, normalized size = 1.73 \begin {gather*} \frac {{\left (x - 5\right )} e^{\left (-{\left ({\left (x + 2 \, e^{\left (x^{2} + x\right )} + 5\right )} e^{\left (2 \, x^{2}\right )} + e^{\left (4 \, x^{2}\right )} + e^{\left (2 \, x^{2} + 2 \, x\right )}\right )} e^{\left (-2 \, x^{2}\right )}\right )}}{3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x^3+20*x^2)*exp(x^2)^2+(-4*x^3+18*x^2+10*x)*exp(x)*exp(x^2)+(-2*x^2+10*x)*exp(x)^2-x^2+4*x+
10)/x^3/exp(exp(x^2)^2+2*exp(x)*exp(x^2)+exp(x)^2+5+x),x, algorithm="fricas")

[Out]

1/3*(x - 5)*e^(-((x + 2*e^(x^2 + x) + 5)*e^(2*x^2) + e^(4*x^2) + e^(2*x^2 + 2*x))*e^(-2*x^2))/x^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{2} + 4 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (2 \, x^{3} - 9 \, x^{2} - 5 \, x\right )} e^{\left (x^{2} + x\right )} + 2 \, {\left (x^{2} - 5 \, x\right )} e^{\left (2 \, x\right )} - 4 \, x - 10\right )} e^{\left (-x - e^{\left (2 \, x^{2}\right )} - 2 \, e^{\left (x^{2} + x\right )} - e^{\left (2 \, x\right )} - 5\right )}}{3 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x^3+20*x^2)*exp(x^2)^2+(-4*x^3+18*x^2+10*x)*exp(x)*exp(x^2)+(-2*x^2+10*x)*exp(x)^2-x^2+4*x+
10)/x^3/exp(exp(x^2)^2+2*exp(x)*exp(x^2)+exp(x)^2+5+x),x, algorithm="giac")

[Out]

integrate(-1/3*(x^2 + 4*(x^3 - 5*x^2)*e^(2*x^2) + 2*(2*x^3 - 9*x^2 - 5*x)*e^(x^2 + x) + 2*(x^2 - 5*x)*e^(2*x)
- 4*x - 10)*e^(-x - e^(2*x^2) - 2*e^(x^2 + x) - e^(2*x) - 5)/x^3, x)

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maple [A]  time = 0.10, size = 37, normalized size = 1.23

method	result	size
risch	\(\frac {\left (x -5\right ) {\mathrm e}^{-{\mathrm e}^{2 x^{2}}-2 \,{\mathrm e}^{\left (x +1\right ) x}-{\mathrm e}^{2 x}-5-x}}{3 x^{2}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((-4*x^3+20*x^2)*exp(x^2)^2+(-4*x^3+18*x^2+10*x)*exp(x)*exp(x^2)+(-2*x^2+10*x)*exp(x)^2-x^2+4*x+10)/x^
3/exp(exp(x^2)^2+2*exp(x)*exp(x^2)+exp(x)^2+5+x),x,method=_RETURNVERBOSE)

[Out]

1/3/x^2*(x-5)*exp(-exp(2*x^2)-2*exp((x+1)*x)-exp(2*x)-5-x)

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maxima [A]  time = 0.64, size = 36, normalized size = 1.20 \begin {gather*} \frac {{\left (x - 5\right )} e^{\left (-x - e^{\left (2 \, x^{2}\right )} - 2 \, e^{\left (x^{2} + x\right )} - e^{\left (2 \, x\right )} - 5\right )}}{3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x^3+20*x^2)*exp(x^2)^2+(-4*x^3+18*x^2+10*x)*exp(x)*exp(x^2)+(-2*x^2+10*x)*exp(x)^2-x^2+4*x+
10)/x^3/exp(exp(x^2)^2+2*exp(x)*exp(x^2)+exp(x)^2+5+x),x, algorithm="maxima")

[Out]

1/3*(x - 5)*e^(-x - e^(2*x^2) - 2*e^(x^2 + x) - e^(2*x) - 5)/x^2

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mupad [B]  time = 1.50, size = 39, normalized size = 1.30 \begin {gather*} \frac {{\mathrm {e}}^{-{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-{\mathrm {e}}^{2\,x^2}}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{-2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x}\,\left (x-5\right )}{3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- x - exp(2*x) - exp(2*x^2) - 2*exp(x^2)*exp(x) - 5)*((4*x)/3 + (exp(2*x)*(10*x - 2*x^2))/3 + (exp(2*
x^2)*(20*x^2 - 4*x^3))/3 - x^2/3 + (exp(x^2)*exp(x)*(10*x + 18*x^2 - 4*x^3))/3 + 10/3))/x^3,x)

[Out]

(exp(-exp(2*x))*exp(-x)*exp(-exp(2*x^2))*exp(-5)*exp(-2*exp(x^2)*exp(x))*(x - 5))/(3*x^2)

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sympy [A]  time = 0.32, size = 36, normalized size = 1.20 \begin {gather*} \frac {\left (x - 5\right ) e^{- x - e^{2 x} - 2 e^{x} e^{x^{2}} - e^{2 x^{2}} - 5}}{3 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x**3+20*x**2)*exp(x**2)**2+(-4*x**3+18*x**2+10*x)*exp(x)*exp(x**2)+(-2*x**2+10*x)*exp(x)**2
-x**2+4*x+10)/x**3/exp(exp(x**2)**2+2*exp(x)*exp(x**2)+exp(x)**2+5+x),x)

[Out]

(x - 5)*exp(-x - exp(2*x) - 2*exp(x)*exp(x**2) - exp(2*x**2) - 5)/(3*x**2)

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