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

Optimal. Leaf size=19 \[ \frac{\coth ^3(x)}{3 a}-\frac{\coth (x)}{a} \]

[Out]

-(Coth[x]/a) + Coth[x]^3/(3*a)

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Rubi [A]  time = 0.0478624, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 3767} \[ \frac{\coth ^3(x)}{3 a}-\frac{\coth (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(a - a*Cosh[x]^2),x]

[Out]

-(Coth[x]/a) + Coth[x]^3/(3*a)

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

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]

Rubi steps

\begin{align*} \int \frac{\text{csch}^2(x)}{a-a \cosh ^2(x)} \, dx &=-\frac{\int \text{csch}^4(x) \, dx}{a}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (x)\right )}{a}\\ &=-\frac{\coth (x)}{a}+\frac{\coth ^3(x)}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0039976, size = 22, normalized size = 1.16 \[ -\frac{\frac{2 \coth (x)}{3}-\frac{1}{3} \coth (x) \text{csch}^2(x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.025, size = 37, normalized size = 2. \begin{align*}{\frac{1}{8\,a} \left ({\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-3\,\tanh \left ( x/2 \right ) -3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{-1}+{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/a*(1/3*tanh(1/2*x)^3-3*tanh(1/2*x)-3/tanh(1/2*x)+1/3/tanh(1/2*x)^3)

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Maxima [B]  time = 1.07968, size = 82, normalized size = 4.32 \begin{align*} -\frac{4 \, e^{\left (-2 \, x\right )}}{3 \, a e^{\left (-2 \, x\right )} - 3 \, a e^{\left (-4 \, x\right )} + a e^{\left (-6 \, x\right )} - a} + \frac{4}{3 \,{\left (3 \, a e^{\left (-2 \, x\right )} - 3 \, a e^{\left (-4 \, x\right )} + a e^{\left (-6 \, 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)^2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.78691, size = 317, normalized size = 16.68 \begin{align*} \frac{8 \,{\left (\cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right )}}{3 \,{\left (a \cosh \left (x\right )^{5} + 5 \, a \cosh \left (x\right ) \sinh \left (x\right )^{4} + a \sinh \left (x\right )^{5} - 3 \, a \cosh \left (x\right )^{3} +{\left (10 \, a \cosh \left (x\right )^{2} - 3 \, a\right )} \sinh \left (x\right )^{3} +{\left (10 \, a \cosh \left (x\right )^{3} - 9 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) +{\left (5 \, a \cosh \left (x\right )^{4} - 9 \, a \cosh \left (x\right )^{2} + 4 \, a\right )} \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3*(cosh(x) + 2*sinh(x))/(a*cosh(x)^5 + 5*a*cosh(x)*sinh(x)^4 + a*sinh(x)^5 - 3*a*cosh(x)^3 + (10*a*cosh(x)^2
- 3*a)*sinh(x)^3 + (10*a*cosh(x)^3 - 9*a*cosh(x))*sinh(x)^2 + 2*a*cosh(x) + (5*a*cosh(x)^4 - 9*a*cosh(x)^2 +
4*a)*sinh(x))

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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 ^{2}{\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)**2),x)

[Out]

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

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Giac [A]  time = 1.27094, size = 28, normalized size = 1.47 \begin{align*} \frac{4 \,{\left (3 \, e^{\left (2 \, x\right )} - 1\right )}}{3 \, a{\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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