3.580 \(\int \text{sech}^5(x) \, dx\)

Optimal. Leaf size=26 \[ \frac{3}{8} \tan ^{-1}(\sinh (x))+\frac{1}{4} \tanh (x) \text{sech}^3(x)+\frac{3}{8} \tanh (x) \text{sech}(x) \]

[Out]

(3*ArcTan[Sinh[x]])/8 + (3*Sech[x]*Tanh[x])/8 + (Sech[x]^3*Tanh[x])/4

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Rubi [A]  time = 0.0288209, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{3}{8} \tan ^{-1}(\sinh (x))+\frac{1}{4} \tanh (x) \text{sech}^3(x)+\frac{3}{8} \tanh (x) \text{sech}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sech[x]^5,x]

[Out]

(3*ArcTan[Sinh[x]])/8 + (3*Sech[x]*Tanh[x])/8 + (Sech[x]^3*Tanh[x])/4

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\cosh ^{5}{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/cosh(x)**5,x)

[Out]

Integral(cosh(x)**(-5), x)

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Mathematica [A]  time = 0.0110161, size = 30, normalized size = 1.15 \[ \frac{3}{4} \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+\frac{1}{4} \tanh (x) \text{sech}^3(x)+\frac{3}{8} \tanh (x) \text{sech}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sech[x]^5,x]

[Out]

(3*ArcTan[Tanh[x/2]])/4 + (3*Sech[x]*Tanh[x])/8 + (Sech[x]^3*Tanh[x])/4

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Maple [A]  time = 0.05, size = 21, normalized size = 0.8 \[ \left ({\frac{ \left ({\rm sech} \left (x\right ) \right ) ^{3}}{4}}+{\frac{3\,{\rm sech} \left (x\right )}{8}} \right ) \tanh \left ( x \right ) +{\frac{3\,\arctan \left ({{\rm e}^{x}} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/cosh(x)^5,x)

[Out]

(1/4*sech(x)^3+3/8*sech(x))*tanh(x)+3/4*arctan(exp(x))

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Maxima [A]  time = 1.49697, size = 82, normalized size = 3.15 \[ \frac{3 \, e^{\left (-x\right )} + 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{4 \,{\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} - \frac{3}{4} \, \arctan \left (e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(x)^(-5),x, algorithm="maxima")

[Out]

1/4*(3*e^(-x) + 11*e^(-3*x) - 11*e^(-5*x) - 3*e^(-7*x))/(4*e^(-2*x) + 6*e^(-4*x)
+ 4*e^(-6*x) + e^(-8*x) + 1) - 3/4*arctan(e^(-x))

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Fricas [A]  time = 0.213171, size = 622, normalized size = 23.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(x)^(-5),x, algorithm="fricas")

[Out]

1/4*(3*cosh(x)^7 + 21*cosh(x)*sinh(x)^6 + 3*sinh(x)^7 + (63*cosh(x)^2 + 11)*sinh
(x)^5 + 11*cosh(x)^5 + 5*(21*cosh(x)^3 + 11*cosh(x))*sinh(x)^4 + (105*cosh(x)^4
+ 110*cosh(x)^2 - 11)*sinh(x)^3 - 11*cosh(x)^3 + (63*cosh(x)^5 + 110*cosh(x)^3 -
33*cosh(x))*sinh(x)^2 + 3*(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*c
osh(x)^2 + 1)*sinh(x)^6 + 4*cosh(x)^6 + 8*(7*cosh(x)^3 + 3*cosh(x))*sinh(x)^5 +
2*(35*cosh(x)^4 + 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 + 1
0*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 + 15*cosh(x)^4 + 9*cosh(x)^2
+ 1)*sinh(x)^2 + 4*cosh(x)^2 + 8*(cosh(x)^7 + 3*cosh(x)^5 + 3*cosh(x)^3 + cosh(
x))*sinh(x) + 1)*arctan(cosh(x) + sinh(x)) + (21*cosh(x)^6 + 55*cosh(x)^4 - 33*c
osh(x)^2 - 3)*sinh(x) - 3*cosh(x))/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8
+ 4*(7*cosh(x)^2 + 1)*sinh(x)^6 + 4*cosh(x)^6 + 8*(7*cosh(x)^3 + 3*cosh(x))*sinh
(x)^5 + 2*(35*cosh(x)^4 + 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(
x)^5 + 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 + 15*cosh(x)^4 + 9*c
osh(x)^2 + 1)*sinh(x)^2 + 4*cosh(x)^2 + 8*(cosh(x)^7 + 3*cosh(x)^5 + 3*cosh(x)^3
+ cosh(x))*sinh(x) + 1)

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Sympy [A]  time = 8.57482, size = 422, normalized size = 16.23 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/cosh(x)**5,x)

[Out]

3*tanh(x/2)**8*atan(tanh(x/2))/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)*
*4 + 16*tanh(x/2)**2 + 4) - 5*tanh(x/2)**7/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 2
4*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 12*tanh(x/2)**6*atan(tanh(x/2))/(4*tanh(
x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 3*tanh(x/2)
**5/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) +
18*tanh(x/2)**4*atan(tanh(x/2))/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2
)**4 + 16*tanh(x/2)**2 + 4) - 3*tanh(x/2)**3/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 +
24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 12*tanh(x/2)**2*atan(tanh(x/2))/(4*tan
h(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 5*tanh(x/
2)/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) +
3*atan(tanh(x/2))/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(
x/2)**2 + 4)

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GIAC/XCAS [A]  time = 0.229546, size = 81, normalized size = 3.12 \[ \frac{3}{16} \, \pi - \frac{3 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 20 \, e^{\left (-x\right )} - 20 \, e^{x}}{4 \,{\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} + \frac{3}{8} \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(x)^(-5),x, algorithm="giac")

[Out]

3/16*pi - 1/4*(3*(e^(-x) - e^x)^3 + 20*e^(-x) - 20*e^x)/((e^(-x) - e^x)^2 + 4)^2
+ 3/8*arctan(1/2*(e^(2*x) - 1)*e^(-x))