3.90 \(\int \frac{\sinh ^3(x)}{1+\coth (x)} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0474609, 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} \[ \frac{4 \cosh ^3(x)}{15}-\frac{4 \cosh (x)}{5}-\frac{\sinh ^3(x)}{5 (\coth (x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^3/(1 + Coth[x]),x]

[Out]

(-4*Cosh[x])/5 + (4*Cosh[x]^3)/15 - Sinh[x]^3/(5*(1 + Coth[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{\sinh ^3(x)}{1+\coth (x)} \, dx &=-\frac{\sinh ^3(x)}{5 (1+\coth (x))}+\frac{4}{5} \int \sinh ^3(x) \, dx\\ &=-\frac{\sinh ^3(x)}{5 (1+\coth (x))}-\frac{4}{5} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )\\ &=-\frac{4 \cosh (x)}{5}+\frac{4 \cosh ^3(x)}{15}-\frac{\sinh ^3(x)}{5 (1+\coth (x))}\\ \end{align*}

Mathematica [A]  time = 0.0763492, size = 36, normalized size = 1.24 \[ \frac{\text{csch}(x) (-40 \sinh (2 x)+4 \sinh (4 x)-20 \cosh (2 x)+\cosh (4 x)-45)}{120 (\coth (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^3/(1 + Coth[x]),x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.03, 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{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{3}{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{3}{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(sinh(x)^3/(1+coth(x)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.05927, 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(sinh(x)^3/(1+coth(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)

________________________________________________________________________________________

Fricas [B]  time = 2.40185, 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(sinh(x)^3/(1+coth(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))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{3}{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**3/(1+coth(x)),x)

[Out]

Integral(sinh(x)**3/(coth(x) + 1), x)

________________________________________________________________________________________

Giac [A]  time = 1.15439, 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(sinh(x)^3/(1+coth(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