3.108 \(\int \frac{\cosh (x)}{1+\coth (x)} \, dx\)

Optimal. Leaf size=17 \[ \frac{\cosh ^3(x)}{3}-\frac{\sinh ^3(x)}{3} \]

[Out]

Cosh[x]^3/3 - Sinh[x]^3/3

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Rubi [A]  time = 0.113401, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {3518, 3108, 3107, 2565, 30, 2564} \[ \frac{\cosh ^3(x)}{3}-\frac{\sinh ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(1 + Coth[x]),x]

[Out]

Cosh[x]^3/3 - Sinh[x]^3/3

Rule 3518

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[(Sin[e + f*x]
^m*(a*Cos[e + f*x] + b*Sin[e + f*x])^n)/Cos[e + f*x]^n, x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
ILtQ[n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))

Rule 3108

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_
.) + (d_.)*(x_)])^(p_), x_Symbol] :> Dist[a^p*b^p, Int[(Cos[c + d*x]^m*Sin[c + d*x]^n)/(b*Cos[c + d*x] + a*Sin
[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a^2 + b^2, 0] && ILtQ[p, 0]

Rule 3107

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_
.) + (d_.)*(x_)])^(p_.), x_Symbol] :> Int[ExpandTrig[cos[c + d*x]^m*sin[c + d*x]^n*(a*cos[c + d*x] + b*sin[c +
d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rubi steps

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

Mathematica [A]  time = 0.0166151, size = 19, normalized size = 1.12 \[ \frac{1}{12} \left (-4 \sinh ^3(x)+3 \cosh (x)+\cosh (3 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/(1 + Coth[x]),x]

[Out]

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

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Maple [B]  time = 0.024, size = 42, normalized size = 2.5 \begin{align*}{\frac{2}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}- \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2} \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)/(1+coth(x)),x)

[Out]

2/3/(tanh(1/2*x)+1)^3-1/(tanh(1/2*x)+1)^2+1/2/(tanh(1/2*x)+1)-1/2/(tanh(1/2*x)-1)

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Maxima [A]  time = 1.01463, size = 15, normalized size = 0.88 \begin{align*} \frac{1}{12} \, 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)/(1+coth(x)),x, algorithm="maxima")

[Out]

1/12*e^(-3*x) + 1/4*e^x

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Fricas [A]  time = 2.48556, size = 90, normalized size = 5.29 \begin{align*} \frac{\cosh \left (x\right )^{2} + \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}{3 \,{\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)/(1+coth(x)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)/(coth(x) + 1), x)

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Giac [A]  time = 1.13034, size = 15, normalized size = 0.88 \begin{align*} \frac{1}{12} \, 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)/(1+coth(x)),x, algorithm="giac")

[Out]

1/12*e^(-3*x) + 1/4*e^x