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

Optimal. Leaf size=30 \[ \frac{\coth ^3(x)}{3 a}-\frac{\text{csch}^3(x)}{3 a}-\frac{\text{csch}(x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.0785009, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 2607, 30, 2606} \[ \frac{\coth ^3(x)}{3 a}-\frac{\text{csch}^3(x)}{3 a}-\frac{\text{csch}(x)}{a} \]

Antiderivative was successfully verified.

[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 2706

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]

Rule 2607

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 30

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

Rule 2606

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])

Rubi steps

\begin{align*} \int \frac{\coth ^2(x)}{a+a \cosh (x)} \, dx &=\frac{\int \coth ^3(x) \text{csch}(x) \, dx}{a}-\frac{\int \coth ^2(x) \text{csch}^2(x) \, dx}{a}\\ &=\frac{i \operatorname{Subst}\left (\int x^2 \, dx,x,i \coth (x)\right )}{a}+\frac{i \operatorname{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]  time = 0.0480718, size = 25, normalized size = 0.83 \[ \frac{(-4 \cosh (x)+\cosh (2 x)-3) \text{csch}(x)}{6 a (\cosh (x)+1)} \]

Antiderivative was successfully verified.

[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]))

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Maple [A]  time = 0.021, size = 29, normalized size = 1. \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(coth(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 = 1.01716, size = 163, normalized size = 5.43 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.76574, size = 292, normalized size = 9.73 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\coth ^{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(coth(x)**2/(a+a*cosh(x)),x)

[Out]

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

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Giac [A]  time = 1.18759, size = 47, normalized size = 1.57 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)