\(\int \frac {\coth ^2(x)}{a+a \cosh (x)} \, dx\) [194]

Optimal result

Integrand size = 13, antiderivative size = 30 \[ \int \frac {\coth ^2(x)}{a+a \cosh (x)} \, dx=\frac {\coth ^3(x)}{3 a}-\frac {\text {csch}(x)}{a}-\frac {\text {csch}^3(x)}{3 a} \]

[Out]

1/3*coth(x)^3/a-csch(x)/a-1/3*csch(x)^3/a

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2785, 2687, 30, 2686} \[ \int \frac {\coth ^2(x)}{a+a \cosh (x)} \, dx=\frac {\coth ^3(x)}{3 a}-\frac {\text {csch}^3(x)}{3 a}-\frac {\text {csch}(x)}{a} \]

[In]

Int[Coth[x]^2/(a + a*Cosh[x]),x]

[Out]

Coth[x]^3/(3*a) - Csch[x]/a - Csch[x]^3/(3*a)

Rule 30

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

Rule 2686

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

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2785

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \coth ^3(x) \text {csch}(x) \, dx}{a}-\frac {\int \coth ^2(x) \text {csch}^2(x) \, dx}{a} \\ & = \frac {i \text {Subst}\left (\int x^2 \, dx,x,i \coth (x)\right )}{a}+\frac {i \text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,-i \text {csch}(x)\right )}{a} \\ & = \frac {\coth ^3(x)}{3 a}-\frac {\text {csch}(x)}{a}-\frac {\text {csch}^3(x)}{3 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {\coth ^2(x)}{a+a \cosh (x)} \, dx=\frac {(-3-4 \cosh (x)+\cosh (2 x)) \text {csch}(x)}{6 a (1+\cosh (x))} \]

[In]

Integrate[Coth[x]^2/(a + a*Cosh[x]),x]

[Out]

((-3 - 4*Cosh[x] + Cosh[2*x])*Csch[x])/(6*a*(1 + Cosh[x]))

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{3}}{3}+2 \tanh \left (\frac {x}{2}\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )}}{4 a}\)	\(29\)
risch	\(-\frac {2 \left (3 \,{\mathrm e}^{3 x}+3 \,{\mathrm e}^{2 x}+{\mathrm e}^{x}-1\right )}{3 \left ({\mathrm e}^{x}+1\right )^{3} a \left ({\mathrm e}^{x}-1\right )}\)	\(34\)

[In]

int(coth(x)^2/(a+a*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

1/4/a*(1/3*tanh(1/2*x)^3+2*tanh(1/2*x)-1/tanh(1/2*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.03 \[ \int \frac {\coth ^2(x)}{a+a \cosh (x)} \, dx=-\frac {2 \, {\left (3 \, \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right ) + 3 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )}}{3 \, {\left (a \cosh \left (x\right )^{3} + a \sinh \left (x\right )^{3} + 2 \, a \cosh \left (x\right )^{2} + {\left (3 \, a \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right )^{2} - a \cosh \left (x\right ) + {\left (3 \, a \cosh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - 2 \, a\right )}} \]

[In]

integrate(coth(x)^2/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\coth ^2(x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\coth ^{2}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(coth(x)**2/(a+a*cosh(x)),x)

[Out]

Integral(coth(x)**2/(cosh(x) + 1), x)/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (26) = 52\).

Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.03 \[ \int \frac {\coth ^2(x)}{a+a \cosh (x)} \, dx=-\frac {2 \, e^{\left (-x\right )}}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} - \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a} - \frac {2 \, e^{\left (-3 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a} + \frac {2}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} \]

[In]

integrate(coth(x)^2/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

-2/3*e^(-x)/(2*a*e^(-x) - 2*a*e^(-3*x) - a*e^(-4*x) + a) - 2*e^(-2*x)/(2*a*e^(-x) - 2*a*e^(-3*x) - a*e^(-4*x)
+ a) - 2*e^(-3*x)/(2*a*e^(-x) - 2*a*e^(-3*x) - a*e^(-4*x) + a) + 2/3/(2*a*e^(-x) - 2*a*e^(-3*x) - a*e^(-4*x) +
a)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {\coth ^2(x)}{a+a \cosh (x)} \, dx=-\frac {1}{2 \, a {\left (e^{x} - 1\right )}} - \frac {9 \, e^{\left (2 \, x\right )} + 12 \, e^{x} + 7}{6 \, a {\left (e^{x} + 1\right )}^{3}} \]

[In]

integrate(coth(x)^2/(a+a*cosh(x)),x, algorithm="giac")

[Out]

-1/2/(a*(e^x - 1)) - 1/6*(9*e^(2*x) + 12*e^x + 7)/(a*(e^x + 1)^3)

Mupad [B] (verification not implemented)

Time = 1.64 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.07 \[ \int \frac {\coth ^2(x)}{a+a \cosh (x)} \, dx=-\frac {\frac {{\mathrm {e}}^{2\,x}}{2\,a}+\frac {1}{2\,a}+\frac {{\mathrm {e}}^x}{3\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {1}{6\,a}+\frac {{\mathrm {e}}^x}{2\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x+1\right )} \]

[In]

int(coth(x)^2/(a + a*cosh(x)),x)

[Out]

- (exp(2*x)/(2*a) + 1/(2*a) + exp(x)/(3*a))/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1) - (1/(6*a) + exp(x)/(2*a))/
(exp(2*x) + 2*exp(x) + 1) - 1/(2*a*(exp(x) - 1)) - 1/(2*a*(exp(x) + 1))