3.8.73 \(\int \frac {1}{4} (4-2 e x-16 x^3+e^2 (2 x+x^3)) \, dx\)

Optimal. Leaf size=27 \[ -3+x-x^4+\frac {1}{16} e^2 \left (2-\frac {2}{e}+x^2\right )^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.30, number of steps used = 3, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12} \begin {gather*} \frac {e^2 x^4}{16}-x^4+\frac {e^2 x^2}{4}-\frac {e x^2}{4}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 2*E*x - 16*x^3 + E^2*(2*x + x^3))/4,x]

[Out]

x - (E*x^2)/4 + (E^2*x^2)/4 - x^4 + (E^2*x^4)/16

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (4-2 e x-16 x^3+e^2 \left (2 x+x^3\right )\right ) \, dx\\ &=x-\frac {e x^2}{4}-x^4+\frac {1}{4} e^2 \int \left (2 x+x^3\right ) \, dx\\ &=x-\frac {e x^2}{4}+\frac {e^2 x^2}{4}-x^4+\frac {e^2 x^4}{16}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 36, normalized size = 1.33 \begin {gather*} \frac {1}{4} \left (4 x-e x^2+e^2 x^2-4 x^4+\frac {e^2 x^4}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 2*E*x - 16*x^3 + E^2*(2*x + x^3))/4,x]

[Out]

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

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fricas [A]  time = 0.78, size = 27, normalized size = 1.00 \begin {gather*} -x^{4} - \frac {1}{4} \, x^{2} e + \frac {1}{16} \, {\left (x^{4} + 4 \, x^{2}\right )} e^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.46, size = 27, normalized size = 1.00 \begin {gather*} -x^{4} - \frac {1}{4} \, x^{2} e + \frac {1}{16} \, {\left (x^{4} + 4 \, x^{2}\right )} e^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 29, normalized size = 1.07

method	result	size
risch	\(\frac {x^{4} {\mathrm e}^{2}}{16}+\frac {x^{2} {\mathrm e}^{2}}{4}-\frac {x^{2} {\mathrm e}}{4}-x^{4}+x\)	\(29\)
default	\(\frac {{\mathrm e}^{2} \left (\frac {1}{4} x^{4}+x^{2}\right )}{4}-\frac {x^{2} {\mathrm e}}{4}-x^{4}+x\)	\(30\)
norman	\(x +\left (\frac {{\mathrm e}^{2}}{4}-\frac {{\mathrm e}}{4}\right ) x^{2}+\left (\frac {{\mathrm e}^{2}}{16}-1\right ) x^{4}\)	\(30\)
gosper	\(\frac {x \left (x^{3} {\mathrm e}^{2}+4 \,{\mathrm e}^{2} x -16 x^{3}-4 x \,{\mathrm e}+16\right )}{16}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 27, normalized size = 1.00 \begin {gather*} -x^{4} - \frac {1}{4} \, x^{2} e + \frac {1}{16} \, {\left (x^{4} + 4 \, x^{2}\right )} e^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.57, size = 26, normalized size = 0.96 \begin {gather*} \left (\frac {{\mathrm {e}}^2}{16}-1\right )\,x^4+\left (\frac {{\mathrm {e}}^2}{4}-\frac {\mathrm {e}}{4}\right )\,x^2+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2)*(2*x + x^3))/4 - (x*exp(1))/2 - 4*x^3 + 1,x)

[Out]

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

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sympy [A]  time = 0.06, size = 24, normalized size = 0.89 \begin {gather*} x^{4} \left (-1 + \frac {e^{2}}{16}\right ) + x^{2} \left (- \frac {e}{4} + \frac {e^{2}}{4}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(x**3+2*x)*exp(1)**2-1/2*x*exp(1)-4*x**3+1,x)

[Out]

x**4*(-1 + exp(2)/16) + x**2*(-E/4 + exp(2)/4) + x

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