3.160 \(\int \frac{\text{csch}(x)}{a+a \cosh (x)} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{2 (a \cosh (x)+a)}-\frac{\tanh ^{-1}(\cosh (x))}{2 a} \]

[Out]

-ArcTanh[Cosh[x]]/(2*a) + 1/(2*(a + a*Cosh[x]))

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Rubi [A]  time = 0.052373, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2667, 44, 206} \[ \frac{1}{2 (a \cosh (x)+a)}-\frac{\tanh ^{-1}(\cosh (x))}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]/(a + a*Cosh[x]),x]

[Out]

-ArcTanh[Cosh[x]]/(2*a) + 1/(2*(a + a*Cosh[x]))

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0274464, size = 42, normalized size = 1.83 \[ \frac{1-2 \cosh ^2\left (\frac{x}{2}\right ) \left (\log \left (\cosh \left (\frac{x}{2}\right )\right )-\log \left (\sinh \left (\frac{x}{2}\right )\right )\right )}{2 a (\cosh (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]/(a + a*Cosh[x]),x]

[Out]

(1 - 2*Cosh[x/2]^2*(Log[Cosh[x/2]] - Log[Sinh[x/2]]))/(2*a*(1 + Cosh[x]))

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Maple [A]  time = 0.016, size = 23, normalized size = 1. \begin{align*} -{\frac{1}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{1}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01682, size = 63, normalized size = 2.74 \begin{align*} \frac{e^{\left (-x\right )}}{2 \, a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a} - \frac{\log \left (e^{\left (-x\right )} + 1\right )}{2 \, a} + \frac{\log \left (e^{\left (-x\right )} - 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x)/(2*a*e^(-x) + a*e^(-2*x) + a) - 1/2*log(e^(-x) + 1)/a + 1/2*log(e^(-x) - 1)/a

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Fricas [B]  time = 1.83715, size = 397, normalized size = 17.26 \begin{align*} -\frac{{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, \cosh \left (x\right ) - 2 \, \sinh \left (x\right )}{2 \,{\left (a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((cosh(x)^2 + 2*(cosh(x) + 1)*sinh(x) + sinh(x)^2 + 2*cosh(x) + 1)*log(cosh(x) + sinh(x) + 1) - (cosh(x)^
2 + 2*(cosh(x) + 1)*sinh(x) + sinh(x)^2 + 2*cosh(x) + 1)*log(cosh(x) + sinh(x) - 1) - 2*cosh(x) - 2*sinh(x))/(
a*cosh(x)^2 + a*sinh(x)^2 + 2*a*cosh(x) + 2*(a*cosh(x) + a)*sinh(x) + a)

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

[Out]

Integral(csch(x)/(cosh(x) + 1), x)/a

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Giac [B]  time = 1.18903, size = 70, normalized size = 3.04 \begin{align*} -\frac{\log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, a} + \frac{\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, a} + \frac{e^{\left (-x\right )} + e^{x} + 6}{4 \, a{\left (e^{\left (-x\right )} + e^{x} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*log(e^(-x) + e^x + 2)/a + 1/4*log(e^(-x) + e^x - 2)/a + 1/4*(e^(-x) + e^x + 6)/(a*(e^(-x) + e^x + 2))