3.600 \(\int e^{2 x} \text{csch}^4(x) \, dx\)

Optimal. Leaf size=20 \[ \frac{8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]

[Out]

(8*E^(6*x))/(3*(1 - E^(2*x))^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0381935, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[E^(2*x)*Csch[x]^4,x]

[Out]

(8*E^(6*x))/(3*(1 - E^(2*x))^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.14405, size = 27, normalized size = 1.35 \[ - \frac{e^{2 x}}{3 \sinh ^{2}{\left (x \right )}} - \frac{e^{2 x} \cosh{\left (x \right )}}{3 \sinh ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(2*x)/sinh(x)**4,x)

[Out]

-exp(2*x)/(3*sinh(x)**2) - exp(2*x)*cosh(x)/(3*sinh(x)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0120733, size = 29, normalized size = 1.45 \[ -\frac{8 \left (-3 e^{2 x}+3 e^{4 x}+1\right )}{3 \left (e^{2 x}-1\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(2*x)*Csch[x]^4,x]

[Out]

(-8*(1 - 3*E^(2*x) + 3*E^(4*x)))/(3*(-1 + E^(2*x))^3)

_______________________________________________________________________________________

Maple [A]  time = 0.11, size = 20, normalized size = 1. \[ - \left ( \tanh \left ( x \right ) \right ) ^{-2}-{\frac{1}{3\, \left ( \tanh \left ( x \right ) \right ) ^{3}}}- \left ( \tanh \left ( x \right ) \right ) ^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(2*x)/sinh(x)^4,x)

[Out]

-1/tanh(x)^2-1/3/tanh(x)^3-1/tanh(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.3664, size = 30, normalized size = 1.5 \[ \frac{8}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/sinh(x)^4,x, algorithm="maxima")

[Out]

8/3/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.203387, size = 101, normalized size = 5.05 \[ -\frac{8 \,{\left (4 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sinh \left (x\right )^{2} - 3\right )}}{3 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 2\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/sinh(x)^4,x, algorithm="fricas")

[Out]

-8/3*(4*cosh(x)^2 + 4*cosh(x)*sinh(x) + 4*sinh(x)^2 - 3)/(cosh(x)^4 + 4*cosh(x)*
sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 2)*sinh(x)^2 - 4*cosh(x)^2 + 4*(cosh(x)
^3 - cosh(x))*sinh(x) + 3)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{2 x}}{\sinh ^{4}{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(2*x)/sinh(x)**4,x)

[Out]

Integral(exp(2*x)/sinh(x)**4, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.23012, size = 32, normalized size = 1.6 \[ -\frac{8 \,{\left (3 \, e^{\left (4 \, x\right )} - 3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/sinh(x)^4,x, algorithm="giac")

[Out]

-8/3*(3*e^(4*x) - 3*e^(2*x) + 1)/(e^(2*x) - 1)^3