3.83.98 \(\int \frac {16 x^3+e^2 (256 x-64 x^3)+e^4 (1-512 x+64 x^3)}{e^4} \, dx\)

Optimal. Leaf size=19 \[ x+4 \left (5+\left (16+\left (-2+\frac {1}{e^2}\right ) x^2\right )^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.89, number of steps used = 4, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {12} \begin {gather*} -\frac {16 x^4}{e^2}+\frac {4 x^4}{e^4}+16 x^4+\frac {128 x^2}{e^2}-256 x^2+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(16*x^3 + E^2*(256*x - 64*x^3) + E^4*(1 - 512*x + 64*x^3))/E^4,x]

[Out]

x - 256*x^2 + (128*x^2)/E^2 + 16*x^4 + (4*x^4)/E^4 - (16*x^4)/E^2

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 {\int \left (16 x^3+e^2 \left (256 x-64 x^3\right )+e^4 \left (1-512 x+64 x^3\right )\right ) \, dx}{e^4}\\ &=\frac {4 x^4}{e^4}+\frac {\int \left (256 x-64 x^3\right ) \, dx}{e^2}+\int \left (1-512 x+64 x^3\right ) \, dx\\ &=x-256 x^2+\frac {128 x^2}{e^2}+16 x^4+\frac {4 x^4}{e^4}-\frac {16 x^4}{e^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 34, normalized size = 1.79 \begin {gather*} x-\frac {128 \left (-1+2 e^2\right ) x^2}{e^2}+\frac {4 \left (-1+2 e^2\right )^2 x^4}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*x^3 + E^2*(256*x - 64*x^3) + E^4*(1 - 512*x + 64*x^3))/E^4,x]

[Out]

x - (128*(-1 + 2*E^2)*x^2)/E^2 + (4*(-1 + 2*E^2)^2*x^4)/E^4

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fricas [B]  time = 0.53, size = 37, normalized size = 1.95 \begin {gather*} {\left (4 \, x^{4} + {\left (16 \, x^{4} - 256 \, x^{2} + x\right )} e^{4} - 16 \, {\left (x^{4} - 8 \, x^{2}\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^3-512*x+1)*exp(2)^2+(-64*x^3+256*x)*exp(2)+16*x^3)/exp(2)^2,x, algorithm="fricas")

[Out]

(4*x^4 + (16*x^4 - 256*x^2 + x)*e^4 - 16*(x^4 - 8*x^2)*e^2)*e^(-4)

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giac [B]  time = 0.18, size = 37, normalized size = 1.95 \begin {gather*} {\left (4 \, x^{4} + {\left (16 \, x^{4} - 256 \, x^{2} + x\right )} e^{4} - 16 \, {\left (x^{4} - 8 \, x^{2}\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^3-512*x+1)*exp(2)^2+(-64*x^3+256*x)*exp(2)+16*x^3)/exp(2)^2,x, algorithm="giac")

[Out]

(4*x^4 + (16*x^4 - 256*x^2 + x)*e^4 - 16*(x^4 - 8*x^2)*e^2)*e^(-4)

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maple [A]  time = 0.03, size = 34, normalized size = 1.79

method	result	size
risch	\(16 x^{4}-16 \,{\mathrm e}^{-2} x^{4}-256 x^{2}+4 \,{\mathrm e}^{-4} x^{4}+x +128 x^{2} {\mathrm e}^{-2}\)	\(34\)
norman	\(\left (\left (-256 \,{\mathrm e}^{2}+128\right ) x^{2}+{\mathrm e}^{2} x +4 \left (4 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{2}+1\right ) {\mathrm e}^{-2} x^{4}\right ) {\mathrm e}^{-2}\)	\(42\)
default	\({\mathrm e}^{-4} \left ({\mathrm e}^{4} \left (16 x^{4}-256 x^{2}+x \right )+{\mathrm e}^{2} \left (-16 x^{4}+128 x^{2}\right )+4 x^{4}\right )\)	\(43\)
gosper	\(x \left (16 x^{3} {\mathrm e}^{4}-16 x^{3} {\mathrm e}^{2}-256 x \,{\mathrm e}^{4}+4 x^{3}+{\mathrm e}^{4}+128 \,{\mathrm e}^{2} x \right ) {\mathrm e}^{-4}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((64*x^3-512*x+1)*exp(2)^2+(-64*x^3+256*x)*exp(2)+16*x^3)/exp(2)^2,x,method=_RETURNVERBOSE)

[Out]

16*x^4-16*exp(-2)*x^4-256*x^2+4*exp(-4)*x^4+x+128*x^2*exp(-2)

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maxima [B]  time = 0.36, size = 37, normalized size = 1.95 \begin {gather*} {\left (4 \, x^{4} + {\left (16 \, x^{4} - 256 \, x^{2} + x\right )} e^{4} - 16 \, {\left (x^{4} - 8 \, x^{2}\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^3-512*x+1)*exp(2)^2+(-64*x^3+256*x)*exp(2)+16*x^3)/exp(2)^2,x, algorithm="maxima")

[Out]

(4*x^4 + (16*x^4 - 256*x^2 + x)*e^4 - 16*(x^4 - 8*x^2)*e^2)*e^(-4)

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mupad [B]  time = 0.10, size = 30, normalized size = 1.58 \begin {gather*} 4\,{\mathrm {e}}^{-4}\,{\left (2\,{\mathrm {e}}^2-1\right )}^2\,x^4-128\,{\mathrm {e}}^{-2}\,\left (2\,{\mathrm {e}}^2-1\right )\,x^2+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(exp(2)*(256*x - 64*x^3) + exp(4)*(64*x^3 - 512*x + 1) + 16*x^3),x)

[Out]

x + 4*x^4*exp(-4)*(2*exp(2) - 1)^2 - 128*x^2*exp(-2)*(2*exp(2) - 1)

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sympy [A]  time = 0.09, size = 32, normalized size = 1.68 \begin {gather*} \frac {x^{4} \left (- 16 e^{2} + 4 + 16 e^{4}\right )}{e^{4}} + \frac {x^{2} \left (128 - 256 e^{2}\right )}{e^{2}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x**3-512*x+1)*exp(2)**2+(-64*x**3+256*x)*exp(2)+16*x**3)/exp(2)**2,x)

[Out]

x**4*(-16*exp(2) + 4 + 16*exp(4))*exp(-4) + x**2*(128 - 256*exp(2))*exp(-2) + x

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