3.37.4 \(\int (-3 e^{3 x}+e^{2 x} (-2+4 e^4-4 x)+e^x (-e^8-2 x-x^2+e^4 (2+2 x))) \, dx\)

Optimal. Leaf size=19 \[ -e^x \left (e^4-e^x-x\right )^2 \]

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Rubi [B]  time = 0.08, antiderivative size = 62, normalized size of antiderivative = 3.26, number of steps used = 14, number of rules used = 3, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2194, 2176, 2196} \begin {gather*} -e^x x^2+e^{2 x}-e^{3 x}-2 e^{x+4}-e^{x+8}+2 e^{x+4} (x+1)-e^{2 x} \left (2 x-2 e^4+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*x) - E^(3*x) - 2*E^(4 + x) - E^(8 + x) - E^x*x^2 + 2*E^(4 + x)*(1 + x) - E^(2*x)*(1 - 2*E^4 + 2*x)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (3 \int e^{3 x} \, dx\right )+\int e^{2 x} \left (-2+4 e^4-4 x\right ) \, dx+\int e^x \left (-e^8-2 x-x^2+e^4 (2+2 x)\right ) \, dx\\ &=-e^{3 x}-e^{2 x} \left (1-2 e^4+2 x\right )+2 \int e^{2 x} \, dx+\int \left (-e^{8+x}-2 e^x x-e^x x^2+2 e^{4+x} (1+x)\right ) \, dx\\ &=e^{2 x}-e^{3 x}-e^{2 x} \left (1-2 e^4+2 x\right )-2 \int e^x x \, dx+2 \int e^{4+x} (1+x) \, dx-\int e^{8+x} \, dx-\int e^x x^2 \, dx\\ &=e^{2 x}-e^{3 x}-e^{8+x}-2 e^x x-e^x x^2+2 e^{4+x} (1+x)-e^{2 x} \left (1-2 e^4+2 x\right )+2 \int e^x \, dx-2 \int e^{4+x} \, dx+2 \int e^x x \, dx\\ &=2 e^x+e^{2 x}-e^{3 x}-2 e^{4+x}-e^{8+x}-e^x x^2+2 e^{4+x} (1+x)-e^{2 x} \left (1-2 e^4+2 x\right )-2 \int e^x \, dx\\ &=e^{2 x}-e^{3 x}-2 e^{4+x}-e^{8+x}-e^x x^2+2 e^{4+x} (1+x)-e^{2 x} \left (1-2 e^4+2 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 17, normalized size = 0.89 \begin {gather*} -e^x \left (-e^4+e^x+x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^x*(-E^4 + E^x + x)^2)

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fricas [B]  time = 0.66, size = 34, normalized size = 1.79 \begin {gather*} -2 \, {\left (x - e^{4}\right )} e^{\left (2 \, x\right )} - {\left (x^{2} - 2 \, x e^{4} + e^{8}\right )} e^{x} - e^{\left (3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(x - e^4)*e^(2*x) - (x^2 - 2*x*e^4 + e^8)*e^x - e^(3*x)

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giac [B]  time = 0.14, size = 42, normalized size = 2.21 \begin {gather*} -x^{2} e^{x} - 2 \, x e^{\left (2 \, x\right )} + 2 \, x e^{\left (x + 4\right )} - e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x + 4\right )} - e^{\left (x + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2*e^x - 2*x*e^(2*x) + 2*x*e^(x + 4) - e^(3*x) + 2*e^(2*x + 4) - e^(x + 8)

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maple [B]  time = 0.05, size = 39, normalized size = 2.05

method	result	size
risch	\(-{\mathrm e}^{3 x}+\left (2 \,{\mathrm e}^{4}-2 x \right ) {\mathrm e}^{2 x}+\left (-{\mathrm e}^{8}+2 x \,{\mathrm e}^{4}-x^{2}\right ) {\mathrm e}^{x}\)	\(39\)
norman	\(-{\mathrm e}^{3 x}-2 x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{4} {\mathrm e}^{2 x}-{\mathrm e}^{8} {\mathrm e}^{x}-{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{4} {\mathrm e}^{x}\)	\(45\)
default	\(-2 x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{4} {\mathrm e}^{2 x}-{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{4} {\mathrm e}^{x}-{\mathrm e}^{8} {\mathrm e}^{x}+2 \,{\mathrm e}^{4} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )-{\mathrm e}^{3 x}\)	\(57\)
meijerg	\(2-\frac {\left (3 x^{2}-6 x +6\right ) {\mathrm e}^{x}}{3}+\frac {\left (-4 x +2\right ) {\mathrm e}^{2 x}}{2}-\left (-2 \,{\mathrm e}^{4}+2\right ) \left (1-\frac {\left (-2 x +2\right ) {\mathrm e}^{x}}{2}\right )-{\mathrm e}^{3 x}-\left (2 \,{\mathrm e}^{4}-1\right ) \left (1-{\mathrm e}^{2 x}\right )-\left (-{\mathrm e}^{8}+2 \,{\mathrm e}^{4}\right ) \left (1-{\mathrm e}^{x}\right )\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(3*x)+(2*exp(4)-2*x)*exp(2*x)+(-exp(8)+2*x*exp(4)-x^2)*exp(x)

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maxima [B]  time = 0.34, size = 63, normalized size = 3.32 \begin {gather*} -2 \, {\left (x - e^{4}\right )} e^{\left (2 \, x\right )} - {\left (x^{2} - 2 \, x + 2\right )} e^{x} + 2 \, {\left (x e^{4} - e^{4}\right )} e^{x} - 2 \, {\left (x - 1\right )} e^{x} - e^{\left (3 \, x\right )} - e^{\left (x + 8\right )} + 2 \, e^{\left (x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(x - e^4)*e^(2*x) - (x^2 - 2*x + 2)*e^x + 2*(x*e^4 - e^4)*e^x - 2*(x - 1)*e^x - e^(3*x) - e^(x + 8) + 2*e^(
x + 4)

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mupad [B]  time = 0.12, size = 14, normalized size = 0.74 \begin {gather*} -{\mathrm {e}}^x\,{\left (x-{\mathrm {e}}^4+{\mathrm {e}}^x\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- 3*exp(3*x) - exp(2*x)*(4*x - 4*exp(4) + 2) - exp(x)*(2*x + exp(8) + x^2 - exp(4)*(2*x + 2)),x)

[Out]

-exp(x)*(x - exp(4) + exp(x))^2

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sympy [B]  time = 0.16, size = 34, normalized size = 1.79 \begin {gather*} \left (- 2 x + 2 e^{4}\right ) e^{2 x} + \left (- x^{2} + 2 x e^{4} - e^{8}\right ) e^{x} - e^{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-3*exp(x)**3+(4*exp(4)-4*x-2)*exp(x)**2+(-exp(4)**2+(2*x+2)*exp(4)-x**2-2*x)*exp(x),x)

[Out]

(-2*x + 2*exp(4))*exp(2*x) + (-x**2 + 2*x*exp(4) - exp(8))*exp(x) - exp(3*x)

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