3.578 \(\int \tanh ^4(x) \, dx\)

Optimal. Leaf size=14 \[ x-\frac{1}{3} \tanh ^3(x)-\tanh (x) \]

[Out]

x - Tanh[x] - Tanh[x]^3/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0179731, antiderivative size = 14, 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 \[ x-\frac{1}{3} \tanh ^3(x)-\tanh (x) \]

Antiderivative was successfully verified.

[In]  Int[Tanh[x]^4,x]

[Out]

x - Tanh[x] - Tanh[x]^3/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 40.0089, size = 14, normalized size = 1. \[ - \frac{\tanh ^{3}{\left (x \right )}}{3} - \tanh{\left (x \right )} + \operatorname{atanh}{\left (\tanh{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(tanh(x)**4,x)

[Out]

-tanh(x)**3/3 - tanh(x) + atanh(tanh(x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.00551939, size = 18, normalized size = 1.29 \[ x-\frac{4 \tanh (x)}{3}+\frac{1}{3} \tanh (x) \text{sech}^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Tanh[x]^4,x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 26, normalized size = 1.9 \[ -{\frac{ \left ( \tanh \left ( x \right ) \right ) ^{3}}{3}}-\tanh \left ( x \right ) -{\frac{\ln \left ( -1+\tanh \left ( x \right ) \right ) }{2}}+{\frac{\ln \left ( 1+\tanh \left ( x \right ) \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(tanh(x)^4,x)

[Out]

-1/3*tanh(x)^3-tanh(x)-1/2*ln(-1+tanh(x))+1/2*ln(1+tanh(x))

_______________________________________________________________________________________

Maxima [A]  time = 1.3713, size = 51, normalized size = 3.64 \[ x - \frac{4 \,{\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 2\right )}}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.2035, size = 92, normalized size = 6.57 \[ \frac{{\left (3 \, x + 4\right )} \cosh \left (x\right )^{3} + 3 \,{\left (3 \, x + 4\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} - 12 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) - 4 \, \sinh \left (x\right )^{3} + 3 \,{\left (3 \, x + 4\right )} \cosh \left (x\right )}{3 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((3*x + 4)*cosh(x)^3 + 3*(3*x + 4)*cosh(x)*sinh(x)^2 - 12*cosh(x)^2*sinh(x)
- 4*sinh(x)^3 + 3*(3*x + 4)*cosh(x))/(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + 3*cosh(x
))

_______________________________________________________________________________________

Sympy [A]  time = 0.271181, size = 10, normalized size = 0.71 \[ x - \frac{\tanh ^{3}{\left (x \right )}}{3} - \tanh{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tanh(x)**4,x)

[Out]

x - tanh(x)**3/3 - tanh(x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.20209, size = 35, normalized size = 2.5 \[ x + \frac{4 \,{\left (3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 2\right )}}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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