3.499 \(\int \left (-e^{-x}+e^x\right )^4 \, dx\)

Optimal. Leaf size=36 \[ 6 x-\frac{e^{-4 x}}{4}+2 e^{-2 x}-2 e^{2 x}+\frac{e^{4 x}}{4} \]

[Out]

-1/(4*E^(4*x)) + 2/E^(2*x) - 2*E^(2*x) + E^(4*x)/4 + 6*x

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Rubi [A]  time = 0.052147, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ 6 x-\frac{e^{-4 x}}{4}+2 e^{-2 x}-2 e^{2 x}+\frac{e^{4 x}}{4} \]

Antiderivative was successfully verified.

[In]  Int[(-E^(-x) + E^x)^4,x]

[Out]

-1/(4*E^(4*x)) + 2/E^(2*x) - 2*E^(2*x) + E^(4*x)/4 + 6*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 2 e^{2 x} + 3 \log{\left (e^{2 x} \right )} + \frac{\int ^{e^{2 x}} x\, dx}{2} + 2 e^{- 2 x} - \frac{e^{- 4 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1/exp(x)+exp(x))**4,x)

[Out]

-2*exp(2*x) + 3*log(exp(2*x)) + Integral(x, (x, exp(2*x)))/2 + 2*exp(-2*x) - exp
(-4*x)/4

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Mathematica [A]  time = 0.0234141, size = 34, normalized size = 0.94 \[ \frac{1}{4} \left (24 x-e^{-4 x}+8 e^{-2 x}-8 e^{2 x}+e^{4 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-E^(-x) + E^x)^4,x]

[Out]

(-E^(-4*x) + 8/E^(2*x) - 8*E^(2*x) + E^(4*x) + 24*x)/4

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Maple [A]  time = 0.01, size = 31, normalized size = 0.9 \[{\frac{ \left ({{\rm e}^{x}} \right ) ^{4}}{4}}-2\, \left ({{\rm e}^{x}} \right ) ^{2}-{\frac{1}{4\, \left ({{\rm e}^{x}} \right ) ^{4}}}+2\, \left ({{\rm e}^{x}} \right ) ^{-2}+6\,\ln \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1/exp(x)+exp(x))^4,x)

[Out]

1/4*exp(x)^4-2*exp(x)^2-1/4/exp(x)^4+2/exp(x)^2+6*ln(exp(x))

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Maxima [A]  time = 1.35233, size = 38, normalized size = 1.06 \[ 6 \, x + \frac{1}{4} \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 2 \, e^{\left (-2 \, x\right )} - \frac{1}{4} \, e^{\left (-4 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(-x) - e^x)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.203782, size = 42, normalized size = 1.17 \[ \frac{1}{4} \,{\left (24 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 8 \, e^{\left (6 \, x\right )} + 8 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-4 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(-x) - e^x)^4,x, algorithm="fricas")

[Out]

1/4*(24*x*e^(4*x) + e^(8*x) - 8*e^(6*x) + 8*e^(2*x) - 1)*e^(-4*x)

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Sympy [A]  time = 0.153951, size = 31, normalized size = 0.86 \[ 6 x + \frac{e^{4 x}}{4} - 2 e^{2 x} + 2 e^{- 2 x} - \frac{e^{- 4 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-1/exp(x)+exp(x))**4,x)

[Out]

6*x + exp(4*x)/4 - 2*exp(2*x) + 2*exp(-2*x) - exp(-4*x)/4

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GIAC/XCAS [A]  time = 0.198424, size = 49, normalized size = 1.36 \[ -\frac{1}{4} \,{\left (18 \, e^{\left (4 \, x\right )} - 8 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-4 \, x\right )} + 6 \, x + \frac{1}{4} \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(-x) - e^x)^4,x, algorithm="giac")

[Out]

-1/4*(18*e^(4*x) - 8*e^(2*x) + 1)*e^(-4*x) + 6*x + 1/4*e^(4*x) - 2*e^(2*x)