3.92 \(\int \frac{\cosh ^3(x)}{1+\tanh (x)} \, dx\)

Optimal. Leaf size=29 \[ \frac{4 \sinh ^3(x)}{15}+\frac{4 \sinh (x)}{5}-\frac{\cosh ^3(x)}{5 (\tanh (x)+1)} \]

[Out]

(4*Sinh[x])/5 + (4*Sinh[x]^3)/15 - Cosh[x]^3/(5*(1 + Tanh[x]))

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Rubi [A]  time = 0.0420266, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3502, 2633} \[ \frac{4 \sinh ^3(x)}{15}+\frac{4 \sinh (x)}{5}-\frac{\cosh ^3(x)}{5 (\tanh (x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^3/(1 + Tanh[x]),x]

[Out]

(4*Sinh[x])/5 + (4*Sinh[x]^3)/15 - Cosh[x]^3/(5*(1 + Tanh[x]))

Rule 3502

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(b*f*(m + 2*n)), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[
e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \frac{\cosh ^3(x)}{1+\tanh (x)} \, dx &=-\frac{\cosh ^3(x)}{5 (1+\tanh (x))}+\frac{4}{5} \int \cosh ^3(x) \, dx\\ &=-\frac{\cosh ^3(x)}{5 (1+\tanh (x))}+\frac{4}{5} i \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (x)\right )\\ &=\frac{4 \sinh (x)}{5}+\frac{4 \sinh ^3(x)}{15}-\frac{\cosh ^3(x)}{5 (1+\tanh (x))}\\ \end{align*}

Mathematica [A]  time = 0.0465145, size = 36, normalized size = 1.24 \[ \frac{\text{sech}(x) (40 \sinh (2 x)+4 \sinh (4 x)+20 \cosh (2 x)+\cosh (4 x)-45)}{120 (\tanh (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^3/(1 + Tanh[x]),x]

[Out]

(Sech[x]*(-45 + 20*Cosh[2*x] + Cosh[4*x] + 40*Sinh[2*x] + 4*Sinh[4*x]))/(120*(1 + Tanh[x]))

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Maple [B]  time = 0.032, size = 80, normalized size = 2.8 \begin{align*} -{\frac{2}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}+ \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}-{\frac{5}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{3}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{11}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{5}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^3/(1+tanh(x)),x)

[Out]

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

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Maxima [A]  time = 1.06333, size = 45, normalized size = 1.55 \begin{align*} \frac{1}{48} \,{\left (12 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (3 \, x\right )} - \frac{3}{8} \, e^{\left (-x\right )} - \frac{1}{12} \, e^{\left (-3 \, x\right )} - \frac{1}{80} \, e^{\left (-5 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3/(1+tanh(x)),x, algorithm="maxima")

[Out]

1/48*(12*e^(-2*x) + 1)*e^(3*x) - 3/8*e^(-x) - 1/12*e^(-3*x) - 1/80*e^(-5*x)

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Fricas [B]  time = 2.06148, size = 221, normalized size = 7.62 \begin{align*} \frac{\cosh \left (x\right )^{4} + 16 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 10\right )} \sinh \left (x\right )^{2} + 20 \, \cosh \left (x\right )^{2} + 16 \,{\left (\cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 45}{120 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3/(1+tanh(x)),x, algorithm="fricas")

[Out]

1/120*(cosh(x)^4 + 16*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 10)*sinh(x)^2 + 20*cosh(x)^2 + 16*(cosh
(x)^3 + 5*cosh(x))*sinh(x) - 45)/(cosh(x) + sinh(x))

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Sympy [B]  time = 1.59062, size = 134, normalized size = 4.62 \begin{align*} - \frac{8 \sinh ^{3}{\left (x \right )} \tanh{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} - \frac{2 \sinh ^{3}{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} - \frac{6 \sinh ^{2}{\left (x \right )} \cosh{\left (x \right )} \tanh{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} + \frac{6 \sinh ^{2}{\left (x \right )} \cosh{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} + \frac{6 \sinh{\left (x \right )} \cosh ^{2}{\left (x \right )} \tanh{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} + \frac{9 \sinh{\left (x \right )} \cosh ^{2}{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} + \frac{3 \cosh ^{3}{\left (x \right )} \tanh{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} - \frac{3 \cosh ^{3}{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**3/(1+tanh(x)),x)

[Out]

-8*sinh(x)**3*tanh(x)/(15*tanh(x) + 15) - 2*sinh(x)**3/(15*tanh(x) + 15) - 6*sinh(x)**2*cosh(x)*tanh(x)/(15*ta
nh(x) + 15) + 6*sinh(x)**2*cosh(x)/(15*tanh(x) + 15) + 6*sinh(x)*cosh(x)**2*tanh(x)/(15*tanh(x) + 15) + 9*sinh
(x)*cosh(x)**2/(15*tanh(x) + 15) + 3*cosh(x)**3*tanh(x)/(15*tanh(x) + 15) - 3*cosh(x)**3/(15*tanh(x) + 15)

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Giac [A]  time = 1.17814, size = 42, normalized size = 1.45 \begin{align*} -\frac{1}{240} \,{\left (90 \, e^{\left (4 \, x\right )} + 20 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac{1}{48} \, e^{\left (3 \, x\right )} + \frac{1}{4} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3/(1+tanh(x)),x, algorithm="giac")

[Out]

-1/240*(90*e^(4*x) + 20*e^(2*x) + 3)*e^(-5*x) + 1/48*e^(3*x) + 1/4*e^x