3.59.80 \(\int (4+e^{2 x} (-32+32 e-8 e^2)+e^{2 x+2 x^2} (-8-16 x)+e^{2 x+x^2} (32+e (-16-16 x)+32 x)) \, dx\)

Optimal. Leaf size=24 \[ 2+4 \left (5-e^{2 x} \left (-2+e+e^{x^2}\right )^2+x\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.96, number of steps used = 5, number of rules used = 3, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2194, 2236, 2244} \begin {gather*} 8 (2-e) e^{x^2+2 x}-4 e^{2 x^2+2 x}+4 x-4 (2-e)^2 e^{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4 + E^(2*x)*(-32 + 32*E - 8*E^2) + E^(2*x + 2*x^2)*(-8 - 16*x) + E^(2*x + x^2)*(32 + E*(-16 - 16*x) + 32*x
),x]

[Out]

-4*(2 - E)^2*E^(2*x) + 8*(2 - E)*E^(2*x + x^2) - 4*E^(2*x + 2*x^2) + 4*x

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 x-\left (8 (2-e)^2\right ) \int e^{2 x} \, dx+\int e^{2 x+2 x^2} (-8-16 x) \, dx+\int e^{2 x+x^2} (32+e (-16-16 x)+32 x) \, dx\\ &=-4 (2-e)^2 e^{2 x}-4 e^{2 x+2 x^2}+4 x+\int e^{2 x+x^2} (16 (2-e)+16 (2-e) x) \, dx\\ &=-4 (2-e)^2 e^{2 x}+8 (2-e) e^{2 x+x^2}-4 e^{2 x+2 x^2}+4 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 43, normalized size = 1.79 \begin {gather*} -4 (-2+e)^2 e^{2 x}-8 (-2+e) e^{2 x+x^2}-4 e^{2 x+2 x^2}+4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4 + E^(2*x)*(-32 + 32*E - 8*E^2) + E^(2*x + 2*x^2)*(-8 - 16*x) + E^(2*x + x^2)*(32 + E*(-16 - 16*x)
+ 32*x),x]

[Out]

-4*(-2 + E)^2*E^(2*x) - 8*(-2 + E)*E^(2*x + x^2) - 4*E^(2*x + 2*x^2) + 4*x

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fricas [B]  time = 0.86, size = 71, normalized size = 2.96 \begin {gather*} -4 \, {\left ({\left (e^{2} - 4 \, e + 4\right )} e^{\left (2 \, x^{2} + 4 \, x\right )} + {\left (2 \, {\left (e - 2\right )} e^{\left (x^{2} + 2 \, x\right )} - x\right )} e^{\left (2 \, x^{2} + 2 \, x\right )} + e^{\left (4 \, x^{2} + 4 \, x\right )}\right )} e^{\left (-2 \, x^{2} - 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x-8)*exp(x)^2*exp(x^2)^2+((-16*x-16)*exp(1)+32*x+32)*exp(x)^2*exp(x^2)+(-8*exp(1)^2+32*exp(1)-3
2)*exp(x)^2+4,x, algorithm="fricas")

[Out]

-4*((e^2 - 4*e + 4)*e^(2*x^2 + 4*x) + (2*(e - 2)*e^(x^2 + 2*x) - x)*e^(2*x^2 + 2*x) + e^(4*x^2 + 4*x))*e^(-2*x
^2 - 2*x)

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giac [B]  time = 0.19, size = 44, normalized size = 1.83 \begin {gather*} -8 \, {\left (e - 2\right )} e^{\left (x^{2} + 2 \, x\right )} - 4 \, {\left (e^{2} - 4 \, e + 4\right )} e^{\left (2 \, x\right )} + 4 \, x - 4 \, e^{\left (2 \, x^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x-8)*exp(x)^2*exp(x^2)^2+((-16*x-16)*exp(1)+32*x+32)*exp(x)^2*exp(x^2)+(-8*exp(1)^2+32*exp(1)-3
2)*exp(x)^2+4,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.08, size = 49, normalized size = 2.04

method	result	size
norman	\(\left (-4 \,{\mathrm e}^{2}+16 \,{\mathrm e}-16\right ) {\mathrm e}^{2 x}+\left (-8 \,{\mathrm e}+16\right ) {\mathrm e}^{2 x} {\mathrm e}^{x^{2}}+4 x -4 \,{\mathrm e}^{2 x} {\mathrm e}^{2 x^{2}}\)	\(49\)
risch	\(-4 \,{\mathrm e}^{2 \left (x +1\right ) x}-8 \,{\mathrm e}^{x \left (2+x \right )} {\mathrm e}+16 \,{\mathrm e}^{x \left (2+x \right )}-4 \,{\mathrm e}^{2 x} {\mathrm e}^{2}+16 \,{\mathrm e}^{2 x} {\mathrm e}-16 \,{\mathrm e}^{2 x}+4 x\)	\(54\)
default	\(4 x +\frac {\left (-8 \,{\mathrm e}^{2}+32 \,{\mathrm e}-32\right ) {\mathrm e}^{2 x}}{2}-4 \,{\mathrm e}^{2 x^{2}+2 x}+16 \,{\mathrm e}^{x^{2}+2 x}-8 \,{\mathrm e}^{x^{2}+2 x +1}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*x-8)*exp(x)^2*exp(x^2)^2+((-16*x-16)*exp(1)+32*x+32)*exp(x)^2*exp(x^2)+(-8*exp(1)^2+32*exp(1)-32)*exp
(x)^2+4,x,method=_RETURNVERBOSE)

[Out]

(-4*exp(1)^2+16*exp(1)-16)*exp(x)^2+(-8*exp(1)+16)*exp(x)^2*exp(x^2)+4*x-4*exp(x)^2*exp(x^2)^2

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maxima [B]  time = 0.35, size = 44, normalized size = 1.83 \begin {gather*} -8 \, {\left (e - 2\right )} e^{\left (x^{2} + 2 \, x\right )} - 4 \, {\left (e^{2} - 4 \, e + 4\right )} e^{\left (2 \, x\right )} + 4 \, x - 4 \, e^{\left (2 \, x^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x-8)*exp(x)^2*exp(x^2)^2+((-16*x-16)*exp(1)+32*x+32)*exp(x)^2*exp(x^2)+(-8*exp(1)^2+32*exp(1)-3
2)*exp(x)^2+4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.47, size = 59, normalized size = 2.46 \begin {gather*} 4\,x-16\,{\mathrm {e}}^{2\,x}+16\,{\mathrm {e}}^{x^2+2\,x}-8\,{\mathrm {e}}^{x^2+2\,x+1}+16\,{\mathrm {e}}^{2\,x+1}-4\,{\mathrm {e}}^{2\,x+2}-4\,{\mathrm {e}}^{2\,x^2+2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)*exp(x^2)*(32*x - exp(1)*(16*x + 16) + 32) - exp(2*x)*(8*exp(2) - 32*exp(1) + 32) - exp(2*x)*exp(2
*x^2)*(16*x + 8) + 4,x)

[Out]

4*x - 16*exp(2*x) + 16*exp(2*x + x^2) - 8*exp(2*x + x^2 + 1) + 16*exp(2*x + 1) - 4*exp(2*x + 2) - 4*exp(2*x +
2*x^2)

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sympy [B]  time = 0.26, size = 54, normalized size = 2.25 \begin {gather*} 4 x + \left (- 8 e e^{2 x} + 16 e^{2 x}\right ) e^{x^{2}} - 4 e^{2 x} e^{2 x^{2}} + \left (- 4 e^{2} - 16 + 16 e\right ) e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x-8)*exp(x)**2*exp(x**2)**2+((-16*x-16)*exp(1)+32*x+32)*exp(x)**2*exp(x**2)+(-8*exp(1)**2+32*ex
p(1)-32)*exp(x)**2+4,x)

[Out]

4*x + (-8*E*exp(2*x) + 16*exp(2*x))*exp(x**2) - 4*exp(2*x)*exp(2*x**2) + (-4*exp(2) - 16 + 16*E)*exp(2*x)

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