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]

-2/3*coth(x)/a+1/3*csch(x)/(a+a*cosh(x))

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 8

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

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*}

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Mathematica [A]  time = 0.05, 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]

-1/12*((2*Cosh[x] + Cosh[2*x])*Csch[x/2]*Sech[x/2]^3)/a

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fricas [B]  time = 1.62, size = 94, normalized size = 3.92 \[ -\frac {4 \, {\left (2 \, \cosh \relax (x) + 2 \, \sinh \relax (x) + 1\right )}}{3 \, {\left (a \cosh \relax (x)^{4} + a \sinh \relax (x)^{4} + 2 \, a \cosh \relax (x)^{3} + 2 \, {\left (2 \, a \cosh \relax (x) + a\right )} \sinh \relax (x)^{3} + 6 \, {\left (a \cosh \relax (x)^{2} + a \cosh \relax (x)\right )} \sinh \relax (x)^{2} - 2 \, a \cosh \relax (x) + 2 \, {\left (2 \, a \cosh \relax (x)^{3} + 3 \, a \cosh \relax (x)^{2} - a\right )} \sinh \relax (x) - a\right )}} \]

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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giac [A]  time = 0.15, size = 35, normalized size = 1.46 \[ -\frac {1}{2 \, a {\left (e^{x} - 1\right )}} + \frac {3 \, e^{\left (2 \, x\right )} + 12 \, e^{x} + 5}{6 \, a {\left (e^{x} + 1\right )}^{3}} \]

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)

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maple [A]  time = 0.08, size = 29, normalized size = 1.21 \[ \frac {\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-2 \tanh \left (\frac {x}{2}\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )}}{4 a} \]

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 = 0.30, size = 59, normalized size = 2.46 \[ -\frac {8 \, e^{\left (-x\right )}}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} - \frac {4}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} \]

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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mupad [B]  time = 0.92, size = 89, normalized size = 3.71 \[ \frac {\frac {{\mathrm {e}}^{2\,x}}{6\,a}+\frac {1}{6\,a}+\frac {{\mathrm {e}}^x}{a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}+\frac {\frac {1}{2\,a}+\frac {{\mathrm {e}}^x}{6\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {1}{6\,a\,\left ({\mathrm {e}}^x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2*(a + a*cosh(x))),x)

[Out]

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

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

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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