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+1/3*coth(x)^3/a

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Rubi [A]  time = 0.05, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [B]  time = 0.43, size = 100, normalized size = 5.26 \[ \frac {8 \, {\left (\cosh \relax (x) + 2 \, \sinh \relax (x)\right )}}{3 \, {\left (a \cosh \relax (x)^{5} + 5 \, a \cosh \relax (x) \sinh \relax (x)^{4} + a \sinh \relax (x)^{5} - 3 \, a \cosh \relax (x)^{3} + {\left (10 \, a \cosh \relax (x)^{2} - 3 \, a\right )} \sinh \relax (x)^{3} + {\left (10 \, a \cosh \relax (x)^{3} - 9 \, a \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + {\left (5 \, a \cosh \relax (x)^{4} - 9 \, a \cosh \relax (x)^{2} + 4 \, a\right )} \sinh \relax (x)\right )}} \]

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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giac [A]  time = 0.13, size = 21, normalized size = 1.11 \[ \frac {4 \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )}}{3 \, a {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \]

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)

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maple [B]  time = 0.11, size = 37, normalized size = 1.95 \[ \frac {\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-3 \tanh \left (\frac {x}{2}\right )+\frac {1}{3 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {3}{\tanh \left (\frac {x}{2}\right )}}{8 a} \]

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

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maxima [B]  time = 0.31, size = 61, normalized size = 3.21 \[ -\frac {4 \, e^{\left (-2 \, x\right )}}{3 \, a e^{\left (-2 \, x\right )} - 3 \, a e^{\left (-4 \, x\right )} + a e^{\left (-6 \, x\right )} - a} + \frac {4}{3 \, {\left (3 \, a e^{\left (-2 \, x\right )} - 3 \, a e^{\left (-4 \, x\right )} + a e^{\left (-6 \, x\right )} - a\right )}} \]

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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mupad [B]  time = 0.91, size = 21, normalized size = 1.11 \[ \frac {4\,\left (3\,{\mathrm {e}}^{2\,x}-1\right )}{3\,a\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2*(a - a*cosh(x)^2)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\operatorname {csch}^{2}{\relax (x )}}{\cosh ^{2}{\relax (x )} - 1}\, dx}{a} \]

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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