3.161 \(\int \frac{\text{csch}^2(x)}{a+a \cosh (x)} \, dx\)

Optimal. Leaf size=24 \[ \frac{\text{csch}(x)}{3 (a \cosh (x)+a)}-\frac{2 \coth (x)}{3 a} \]

[Out]

(-2*Coth[x])/(3*a) + Csch[x]/(3*(a + a*Cosh[x]))

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Rubi [A]  time = 0.0483311, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2672, 3767, 8} \[ \frac{\text{csch}(x)}{3 (a \cosh (x)+a)}-\frac{2 \coth (x)}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(a + a*Cosh[x]),x]

[Out]

(-2*Coth[x])/(3*a) + Csch[x]/(3*(a + a*Cosh[x]))

Rule 2672

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0460004, size = 30, normalized size = 1.25 \[ -\frac{(2 \cosh (x)+\cosh (2 x)) \text{csch}\left (\frac{x}{2}\right ) \text{sech}^3\left (\frac{x}{2}\right )}{12 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*Cosh[x] + Cosh[2*x])*Csch[x/2]*Sech[x/2]^3)/(12*a)

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Maple [A]  time = 0.016, size = 29, normalized size = 1.2 \begin{align*}{\frac{1}{4\,a} \left ({\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-2\,\tanh \left ( x/2 \right ) - \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^2/(a+a*cosh(x)),x)

[Out]

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

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Maxima [B]  time = 1.00997, size = 80, normalized size = 3.33 \begin{align*} -\frac{8 \, 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{4}{3 \,{\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/3*e^(-x)/(2*a*e^(-x) - 2*a*e^(-3*x) - a*e^(-4*x) + a) - 4/3/(2*a*e^(-x) - 2*a*e^(-3*x) - a*e^(-4*x) + a)

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Fricas [B]  time = 1.85497, size = 292, normalized size = 12.17 \begin{align*} -\frac{4 \,{\left (2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right ) + 1\right )}}{3 \,{\left (a \cosh \left (x\right )^{4} + a \sinh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{3} + 2 \,{\left (2 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{3} + 6 \,{\left (a \cosh \left (x\right )^{2} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 2 \, a \cosh \left (x\right ) + 2 \,{\left (2 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} - a\right )} \sinh \left (x\right ) - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{csch}^{2}{\left (x \right )}}{\cosh{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17293, size = 47, normalized size = 1.96 \begin{align*} -\frac{1}{2 \, a{\left (e^{x} - 1\right )}} + \frac{3 \, e^{\left (2 \, x\right )} + 12 \, e^{x} + 5}{6 \, a{\left (e^{x} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a*(e^x - 1)) + 1/6*(3*e^(2*x) + 12*e^x + 5)/(a*(e^x + 1)^3)