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

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Rubi [A]  time = 0.0111177, 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, Rules used = {3473, 8} \[ x-\frac{1}{3} \tanh ^3(x)-\tanh (x) \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^4,x]

[Out]

x - Tanh[x] - Tanh[x]^3/3

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \tanh ^4(x) \, dx &=-\frac{1}{3} \tanh ^3(x)+\int \tanh ^2(x) \, dx\\ &=-\tanh (x)-\frac{\tanh ^3(x)}{3}+\int 1 \, dx\\ &=x-\tanh (x)-\frac{\tanh ^3(x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.0030748, 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

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Maple [B]  time = 0.002, size = 26, normalized size = 1.9 \begin{align*} -{\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}} \end{align*}

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))

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Maxima [B]  time = 0.948404, size = 51, normalized size = 3.64 \begin{align*} 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 )}} \end{align*}

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)

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Fricas [B]  time = 2.10164, size = 221, normalized size = 15.79 \begin{align*} \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 )}} \end{align*}

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)*co
sh(x))/(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + 3*cosh(x))

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Sympy [A]  time = 0.216398, size = 10, normalized size = 0.71 \begin{align*} x - \frac{\tanh ^{3}{\left (x \right )}}{3} - \tanh{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)**4,x)

[Out]

x - tanh(x)**3/3 - tanh(x)

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Giac [B]  time = 1.13156, size = 35, normalized size = 2.5 \begin{align*} 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}} \end{align*}

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