\(\int \frac {\sinh ^3(x)}{1+\coth (x)} \, dx\) [90]

Optimal result

Integrand size = 11, antiderivative size = 29 \[ \int \frac {\sinh ^3(x)}{1+\coth (x)} \, dx=-\frac {4 \cosh (x)}{5}+\frac {4 \cosh ^3(x)}{15}-\frac {\sinh ^3(x)}{5 (1+\coth (x))} \]

[Out]

-4/5*cosh(x)+4/15*cosh(x)^3-1/5*sinh(x)^3/(1+coth(x))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3583, 2713} \[ \int \frac {\sinh ^3(x)}{1+\coth (x)} \, dx=\frac {4 \cosh ^3(x)}{15}-\frac {4 \cosh (x)}{5}-\frac {\sinh ^3(x)}{5 (\coth (x)+1)} \]

[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 2713

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]

Rule 3583

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]

Rubi steps \begin{align*} \text {integral}& = -\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} \text {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] (verified)

Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {\sinh ^3(x)}{1+\coth (x)} \, dx=\frac {\text {csch}(x) (-45-20 \cosh (2 x)+\cosh (4 x)-40 \sinh (2 x)+4 \sinh (4 x))}{120 (1+\coth (x))} \]

[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 [A] (verified)

Time = 0.56 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03

method	result	size
risch	\(\frac {{\mathrm e}^{3 x}}{48}-\frac {{\mathrm e}^{x}}{4}-\frac {3 \,{\mathrm e}^{-x}}{8}+\frac {{\mathrm e}^{-3 x}}{12}-\frac {{\mathrm e}^{-5 x}}{80}\)	\(30\)
parallelrisch	\(-\frac {8}{15}-\frac {5 \cosh \left (x \right )}{8}+\frac {\sinh \left (x \right )}{8}+\frac {\sinh \left (5 x \right )}{80}+\frac {5 \cosh \left (3 x \right )}{48}-\frac {\cosh \left (5 x \right )}{80}-\frac {\sinh \left (3 x \right )}{16}\)	\(35\)
default	\(-\frac {2}{5 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}\)	\(80\)

[In]

int(sinh(x)^3/(1+coth(x)),x,method=_RETURNVERBOSE)

[Out]

1/48*exp(3*x)-1/4*exp(x)-3/8*exp(-x)+1/12*exp(-3*x)-1/80*exp(-5*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (23) = 46\).

Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {\sinh ^3(x)}{1+\coth (x)} \, dx=\frac {\cosh \left (x\right )^{4} + 16 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 10\right )} \sinh \left (x\right )^{2} - 20 \, \cosh \left (x\right )^{2} + 16 \, {\left (\cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 45}{120 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]

[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]

\[ \int \frac {\sinh ^3(x)}{1+\coth (x)} \, dx=\int \frac {\sinh ^{3}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \]

[In]

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

[Out]

Integral(sinh(x)**3/(coth(x) + 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {\sinh ^3(x)}{1+\coth (x)} \, dx=-\frac {1}{48} \, {\left (12 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )} - \frac {3}{8} \, e^{\left (-x\right )} + \frac {1}{12} \, e^{\left (-3 \, x\right )} - \frac {1}{80} \, e^{\left (-5 \, x\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\sinh ^3(x)}{1+\coth (x)} \, dx=-\frac {1}{240} \, {\left (90 \, e^{\left (4 \, x\right )} - 20 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{48} \, e^{\left (3 \, x\right )} - \frac {1}{4} \, e^{x} \]

[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

Mupad [B] (verification not implemented)

Time = 1.93 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh ^3(x)}{1+\coth (x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{12}-\frac {3\,{\mathrm {e}}^{-x}}{8}+\frac {{\mathrm {e}}^{3\,x}}{48}-\frac {{\mathrm {e}}^{-5\,x}}{80}-\frac {{\mathrm {e}}^x}{4} \]

[In]

int(sinh(x)^3/(coth(x) + 1),x)

[Out]

exp(-3*x)/12 - (3*exp(-x))/8 + exp(3*x)/48 - exp(-5*x)/80 - exp(x)/4