\(\int \frac {\sinh (x)}{1+\coth (x)} \, dx\) [92]

Optimal result

Integrand size = 9, antiderivative size = 19 \[ \int \frac {\sinh (x)}{1+\coth (x)} \, dx=\frac {2 \cosh (x)}{3}-\frac {\sinh (x)}{3 (1+\coth (x))} \]

[Out]

2/3*cosh(x)-1/3*sinh(x)/(1+coth(x))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3583, 2718} \[ \int \frac {\sinh (x)}{1+\coth (x)} \, dx=\frac {2 \cosh (x)}{3}-\frac {\sinh (x)}{3 (\coth (x)+1)} \]

[In]

Int[Sinh[x]/(1 + Coth[x]),x]

[Out]

(2*Cosh[x])/3 - Sinh[x]/(3*(1 + Coth[x]))

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 (x)}{3 (1+\coth (x))}+\frac {2}{3} \int \sinh (x) \, dx \\ & = \frac {2 \cosh (x)}{3}-\frac {\sinh (x)}{3 (1+\coth (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\sinh (x)}{1+\coth (x)} \, dx=\frac {1}{12} \left (9 \cosh (x)-\cosh (3 x)+4 \sinh ^3(x)\right ) \]

[In]

Integrate[Sinh[x]/(1 + Coth[x]),x]

[Out]

(9*Cosh[x] - Cosh[3*x] + 4*Sinh[x]^3)/12

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
risch	\(\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{-x}}{2}-\frac {{\mathrm e}^{-3 x}}{12}\)	\(18\)
parallelrisch	\(-\frac {\cosh \left (3 x \right )}{12}+\frac {3 \cosh \left (x \right )}{4}+\frac {\sinh \left (3 x \right )}{12}-\frac {\sinh \left (x \right )}{4}+\frac {1}{3}\)	\(23\)
default	\(-\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}\)	\(40\)

[In]

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

[Out]

1/4*exp(x)+1/2*exp(-x)-1/12*exp(-3*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int \frac {\sinh (x)}{1+\coth (x)} \, dx=\frac {\cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 3}{6 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]

[In]

integrate(sinh(x)/(1+coth(x)),x, algorithm="fricas")

[Out]

1/6*(cosh(x)^2 + 4*cosh(x)*sinh(x) + sinh(x)^2 + 3)/(cosh(x) + sinh(x))

Sympy [F]

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

[In]

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

[Out]

Integral(sinh(x)/(coth(x) + 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {\sinh (x)}{1+\coth (x)} \, dx=\frac {1}{2} \, e^{\left (-x\right )} - \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{4} \, e^{x} \]

[In]

integrate(sinh(x)/(1+coth(x)),x, algorithm="maxima")

[Out]

1/2*e^(-x) - 1/12*e^(-3*x) + 1/4*e^x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh (x)}{1+\coth (x)} \, dx=\frac {1}{12} \, {\left (6 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )} + \frac {1}{4} \, e^{x} \]

[In]

integrate(sinh(x)/(1+coth(x)),x, algorithm="giac")

[Out]

1/12*(6*e^(2*x) - 1)*e^(-3*x) + 1/4*e^x

Mupad [B] (verification not implemented)

Time = 1.88 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {\sinh (x)}{1+\coth (x)} \, dx=\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^{-3\,x}}{12}+\frac {{\mathrm {e}}^x}{4} \]

[In]

int(sinh(x)/(coth(x) + 1),x)

[Out]

exp(-x)/2 - exp(-3*x)/12 + exp(x)/4