3.51.37 \(\int \frac {1-2 e^4+e^8+4096 x^3-16384 e^2 x^3+16384 e^4 x^3}{1-2 e^4+e^8} \, dx\)

Optimal. Leaf size=25 \[ x+\frac {64 \left (-4+8 e^2\right )^2 x^4}{\left (1-e^4\right )^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6, 12} \begin {gather*} \frac {1024 \left (1-2 e^2\right )^2 x^4}{\left (1-e^4\right )^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*E^4 + E^8 + 4096*x^3 - 16384*E^2*x^3 + 16384*E^4*x^3)/(1 - 2*E^4 + E^8),x]

[Out]

x + (1024*(1 - 2*E^2)^2*x^4)/(1 - E^4)^2

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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} &=\int \frac {1-2 e^4+e^8+16384 e^4 x^3+\left (4096-16384 e^2\right ) x^3}{1-2 e^4+e^8} \, dx\\ &=\int \frac {1-2 e^4+e^8+\left (4096-16384 e^2+16384 e^4\right ) x^3}{1-2 e^4+e^8} \, dx\\ &=\frac {\int \left (1-2 e^4+e^8+\left (4096-16384 e^2+16384 e^4\right ) x^3\right ) \, dx}{1-2 e^4+e^8}\\ &=x+\frac {1024 \left (1-2 e^2\right )^2 x^4}{\left (1-e^4\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 42, normalized size = 1.68 \begin {gather*} \frac {x-2 e^4 x+e^8 x+1024 x^4-4096 e^2 x^4+4096 e^4 x^4}{\left (-1+e^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*E^4 + E^8 + 4096*x^3 - 16384*E^2*x^3 + 16384*E^4*x^3)/(1 - 2*E^4 + E^8),x]

[Out]

(x - 2*E^4*x + E^8*x + 1024*x^4 - 4096*E^2*x^4 + 4096*E^4*x^4)/(-1 + E^4)^2

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fricas [B]  time = 0.50, size = 46, normalized size = 1.84 \begin {gather*} -\frac {4096 \, x^{4} e^{2} - 1024 \, x^{4} - x e^{8} - 2 \, {\left (2048 \, x^{4} - x\right )} e^{4} - x}{e^{8} - 2 \, e^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)^2-2*exp(4)+16384*x^3*exp(2)^2-16384*x^3*exp(2)+4096*x^3+1)/(exp(4)^2-2*exp(4)+1),x, algorith
m="fricas")

[Out]

-(4096*x^4*e^2 - 1024*x^4 - x*e^8 - 2*(2048*x^4 - x)*e^4 - x)/(e^8 - 2*e^4 + 1)

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giac [A]  time = 0.12, size = 41, normalized size = 1.64 \begin {gather*} \frac {4096 \, x^{4} e^{4} - 4096 \, x^{4} e^{2} + 1024 \, x^{4} + x e^{8} - 2 \, x e^{4} + x}{e^{8} - 2 \, e^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)^2-2*exp(4)+16384*x^3*exp(2)^2-16384*x^3*exp(2)+4096*x^3+1)/(exp(4)^2-2*exp(4)+1),x, algorith
m="giac")

[Out]

(4096*x^4*e^4 - 4096*x^4*e^2 + 1024*x^4 + x*e^8 - 2*x*e^4 + x)/(e^8 - 2*e^4 + 1)

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maple [A]  time = 0.05, size = 38, normalized size = 1.52

method	result	size
norman	\(\frac {\left ({\mathrm e}^{4}-1\right ) x +\frac {1024 \left (4 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{2}+1\right ) x^{4}}{{\mathrm e}^{4}-1}}{{\mathrm e}^{4}-1}\)	\(38\)
gosper	\(\frac {x \left (4096 x^{3} {\mathrm e}^{4}-4096 x^{3} {\mathrm e}^{2}+1024 x^{3}+{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+1\right )}{{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+1}\)	\(46\)
default	\(\frac {x \,{\mathrm e}^{8}-2 x \,{\mathrm e}^{4}+4096 x^{4} {\mathrm e}^{4}-4096 x^{4} {\mathrm e}^{2}+1024 x^{4}+x}{{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+1}\)	\(48\)
risch	\(\frac {4096 x^{4} {\mathrm e}^{4}}{{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+1}+\frac {x \,{\mathrm e}^{8}}{{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+1}-\frac {4096 x^{4} {\mathrm e}^{2}}{{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+1}+\frac {1024 x^{4}}{{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+1}-\frac {2 x \,{\mathrm e}^{4}}{{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+1}+\frac {x}{{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+1}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4)^2-2*exp(4)+16384*x^3*exp(2)^2-16384*x^3*exp(2)+4096*x^3+1)/(exp(4)^2-2*exp(4)+1),x,method=_RETURNV
ERBOSE)

[Out]

((exp(4)-1)*x+1024*(4*exp(2)^2-4*exp(2)+1)/(exp(4)-1)*x^4)/(exp(4)-1)

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maxima [A]  time = 0.34, size = 41, normalized size = 1.64 \begin {gather*} \frac {4096 \, x^{4} e^{4} - 4096 \, x^{4} e^{2} + 1024 \, x^{4} + x e^{8} - 2 \, x e^{4} + x}{e^{8} - 2 \, e^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)^2-2*exp(4)+16384*x^3*exp(2)^2-16384*x^3*exp(2)+4096*x^3+1)/(exp(4)^2-2*exp(4)+1),x, algorith
m="maxima")

[Out]

(4096*x^4*e^4 - 4096*x^4*e^2 + 1024*x^4 + x*e^8 - 2*x*e^4 + x)/(e^8 - 2*e^4 + 1)

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mupad [B]  time = 0.08, size = 21, normalized size = 0.84 \begin {gather*} \frac {1024\,{\left (2\,{\mathrm {e}}^2-1\right )}^2\,x^4}{{\left ({\mathrm {e}}^4-1\right )}^2}+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(8) - 2*exp(4) - 16384*x^3*exp(2) + 16384*x^3*exp(4) + 4096*x^3 + 1)/(exp(8) - 2*exp(4) + 1),x)

[Out]

x + (1024*x^4*(2*exp(2) - 1)^2)/(exp(4) - 1)^2

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sympy [A]  time = 0.06, size = 26, normalized size = 1.04 \begin {gather*} \frac {x^{4} \left (- 4096 e^{2} + 1024 + 4096 e^{4}\right )}{- 2 e^{4} + 1 + e^{8}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)**2-2*exp(4)+16384*x**3*exp(2)**2-16384*x**3*exp(2)+4096*x**3+1)/(exp(4)**2-2*exp(4)+1),x)

[Out]

x**4*(-4096*exp(2) + 1024 + 4096*exp(4))/(-2*exp(4) + 1 + exp(8)) + x

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