3.603 \(\int \frac{e^{m x}}{\cosh (x)+\sinh (x)} \, dx\)

Optimal. Leaf size=13 \[ \frac{e^{(m-1) x}}{m-1} \]

[Out]

E^((-1 + m)*x)/(-1 + m)

_______________________________________________________________________________________

Rubi [A]  time = 0.047424, antiderivative size = 19, normalized size of antiderivative = 1.46, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{e^{-(1-m) x}}{1-m} \]

Antiderivative was successfully verified.

[In]  Int[E^(m*x)/(Cosh[x] + Sinh[x]),x]

[Out]

-(1/(E^((1 - m)*x)*(1 - m)))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.33957, size = 10, normalized size = 0.77 \[ - \frac{e^{x \left (m - 1\right )}}{- m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(m*x)/(cosh(x)+sinh(x)),x)

[Out]

-exp(x*(m - 1))/(-m + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0184985, size = 18, normalized size = 1.38 \[ \frac{e^{m x} (\cosh (x)-\sinh (x))}{m-1} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(m*x)/(Cosh[x] + Sinh[x]),x]

[Out]

(E^(m*x)*(Cosh[x] - Sinh[x]))/(-1 + m)

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 18, normalized size = 1.4 \[{\frac{{{\rm e}^{mx}}}{ \left ( -1+m \right ) \left ( \cosh \left ( x \right ) +\sinh \left ( x \right ) \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(m*x)/(cosh(x)+sinh(x)),x)

[Out]

1/(-1+m)*exp(m*x)/(cosh(x)+sinh(x))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.203085, size = 34, normalized size = 2.62 \[ \frac{\cosh \left (m x\right ) + \sinh \left (m x\right )}{{\left (m - 1\right )} \cosh \left (x\right ) +{\left (m - 1\right )} \sinh \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.964104, size = 32, normalized size = 2.46 \[ \begin{cases} \frac{e^{m x}}{m \sinh{\left (x \right )} + m \cosh{\left (x \right )} - \sinh{\left (x \right )} - \cosh{\left (x \right )}} & \text{for}\: m \neq 1 \\\frac{x e^{x}}{\sinh{\left (x \right )} + \cosh{\left (x \right )}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(m*x)/(cosh(x)+sinh(x)),x)

[Out]

Piecewise((exp(m*x)/(m*sinh(x) + m*cosh(x) - sinh(x) - cosh(x)), Ne(m, 1)), (x*e
xp(x)/(sinh(x) + cosh(x)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.203376, size = 22, normalized size = 1.69 \[ \frac{e^{\left (m x\right )}}{m e^{x} - e^{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(m*x)/(m*e^x - e^x)