3.602 \(\int \frac{e^x}{\cosh (x)-\sinh (x)} \, dx\)

Optimal. Leaf size=9 \[ \frac{e^{2 x}}{2} \]

[Out]

E^(2*x)/2

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Rubi [A]  time = 0.027043, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{e^{2 x}}{2} \]

Antiderivative was successfully verified.

[In]  Int[E^x/(Cosh[x] - Sinh[x]),x]

[Out]

E^(2*x)/2

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Rubi in Sympy [A]  time = 2.5804, size = 5, normalized size = 0.56 \[ \frac{e^{2 x}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(cosh(x)-sinh(x)),x)

[Out]

exp(2*x)/2

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Mathematica [A]  time = 0.00199029, size = 9, normalized size = 1. \[ \frac{e^{2 x}}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(Cosh[x] - Sinh[x]),x]

[Out]

E^(2*x)/2

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Maple [B]  time = 0.001, size = 14, normalized size = 1.6 \[{\frac{{{\rm e}^{x}}}{2\,\cosh \left ( x \right ) -2\,\sinh \left ( x \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(cosh(x)-sinh(x)),x)

[Out]

1/2*exp(x)/(cosh(x)-sinh(x))

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Maxima [A]  time = 1.35773, size = 8, normalized size = 0.89 \[ \frac{1}{2} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(cosh(x) - sinh(x)),x, algorithm="maxima")

[Out]

1/2*e^(2*x)

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Fricas [A]  time = 0.204769, size = 22, normalized size = 2.44 \[ \frac{\cosh \left (x\right ) + \sinh \left (x\right )}{2 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(cosh(x) - sinh(x)),x, algorithm="fricas")

[Out]

1/2*(cosh(x) + sinh(x))/(cosh(x) - sinh(x))

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Sympy [A]  time = 0.719767, size = 12, normalized size = 1.33 \[ \frac{e^{x}}{- 2 \sinh{\left (x \right )} + 2 \cosh{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)/(cosh(x)-sinh(x)),x)

[Out]

exp(x)/(-2*sinh(x) + 2*cosh(x))

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GIAC/XCAS [A]  time = 0.195779, size = 8, normalized size = 0.89 \[ \frac{1}{2} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(cosh(x) - sinh(x)),x, algorithm="giac")

[Out]

1/2*e^(2*x)