3.10 \(\int \frac{\text{csch}(x)}{a+b \cosh ^2(x)} \, dx\)

Optimal. Leaf size=42 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \cosh (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)}-\frac{\tanh ^{-1}(\cosh (x))}{a+b} \]

[Out]

-((Sqrt[b]*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*(a + b))) - ArcTanh[Cosh[x]]/(a + b)

________________________________________________________________________________________

Rubi [A]  time = 0.0622791, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3190, 391, 206, 205} \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \cosh (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)}-\frac{\tanh ^{-1}(\cosh (x))}{a+b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]/(a + b*Cosh[x]^2),x]

[Out]

-((Sqrt[b]*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*(a + b))) - ArcTanh[Cosh[x]]/(a + b)

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 391

Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^n),
x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 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])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [C]  time = 0.135437, size = 99, normalized size = 2.36 \[ \frac{-\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b}-i \sqrt{a+b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a}}\right )-\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b}+i \sqrt{a+b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a}}\right )+\sqrt{a} \log \left (\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{a} (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[b]*ArcTan[(Sqrt[b] - I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]]) - Sqrt[b]*ArcTan[(Sqrt[b] + I*Sqrt[a + b]*Tanh
[x/2])/Sqrt[a]] + Sqrt[a]*Log[Tanh[x/2]])/(Sqrt[a]*(a + b))

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 52, normalized size = 1.2 \begin{align*}{\frac{1}{a+b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{b}{a+b}\arctan \left ({\frac{1}{4} \left ( 2\, \left ( a+b \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,a+2\,b \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\log \left (e^{x} + 1\right )}{a + b} + \frac{\log \left (e^{x} - 1\right )}{a + b} - 2 \, \int \frac{b e^{\left (3 \, x\right )} - b e^{x}}{a b + b^{2} +{\left (a b + b^{2}\right )} e^{\left (4 \, x\right )} + 2 \,{\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} e^{\left (2 \, x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(a+b*cosh(x)^2),x, algorithm="maxima")

[Out]

-log(e^x + 1)/(a + b) + log(e^x - 1)/(a + b) - 2*integrate((b*e^(3*x) - b*e^x)/(a*b + b^2 + (a*b + b^2)*e^(4*x
) + 2*(2*a^2 + 3*a*b + b^2)*e^(2*x)), x)

________________________________________________________________________________________

Fricas [B]  time = 2.08679, size = 1085, normalized size = 25.83 \begin{align*} \left [\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} - 2 \,{\left (2 \, a - b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, b \cosh \left (x\right )^{2} - 2 \, a + b\right )} \sinh \left (x\right )^{2} + 4 \,{\left (b \cosh \left (x\right )^{3} -{\left (2 \, a - b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4 \,{\left (a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right ) \sinh \left (x\right )^{2} + a \sinh \left (x\right )^{3} + a \cosh \left (x\right ) +{\left (3 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )\right )} \sqrt{-\frac{b}{a}} + b}{b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \,{\left (2 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, b \cosh \left (x\right )^{2} + 2 \, a + b\right )} \sinh \left (x\right )^{2} + 4 \,{\left (b \cosh \left (x\right )^{3} +{\left (2 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}\right ) - 2 \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{2 \,{\left (a + b\right )}}, -\frac{\sqrt{\frac{b}{a}} \arctan \left (\frac{1}{2} \, \sqrt{\frac{b}{a}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}\right ) - \sqrt{\frac{b}{a}} \arctan \left (\frac{{\left (b \cosh \left (x\right )^{3} + 3 \, b \cosh \left (x\right ) \sinh \left (x\right )^{2} + b \sinh \left (x\right )^{3} +{\left (4 \, a + b\right )} \cosh \left (x\right ) +{\left (3 \, b \cosh \left (x\right )^{2} + 4 \, a + b\right )} \sinh \left (x\right )\right )} \sqrt{\frac{b}{a}}}{2 \, b}\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a + b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(-b/a)*log((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*(2*a - b)*cosh(x)^2 + 2*(3*b*cosh(
x)^2 - 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 - (2*a - b)*cosh(x))*sinh(x) - 4*(a*cosh(x)^3 + 3*a*cosh(x)*sinh(x)
^2 + a*sinh(x)^3 + a*cosh(x) + (3*a*cosh(x)^2 + a)*sinh(x))*sqrt(-b/a) + b)/(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)
^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*
cosh(x))*sinh(x) + b)) - 2*log(cosh(x) + sinh(x) + 1) + 2*log(cosh(x) + sinh(x) - 1))/(a + b), -(sqrt(b/a)*arc
tan(1/2*sqrt(b/a)*(cosh(x) + sinh(x))) - sqrt(b/a)*arctan(1/2*(b*cosh(x)^3 + 3*b*cosh(x)*sinh(x)^2 + b*sinh(x)
^3 + (4*a + b)*cosh(x) + (3*b*cosh(x)^2 + 4*a + b)*sinh(x))*sqrt(b/a)/b) + log(cosh(x) + sinh(x) + 1) - log(co
sh(x) + sinh(x) - 1))/(a + b)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}{\left (x \right )}}{a + b \cosh ^{2}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(a+b*cosh(x)**2),x)

[Out]

Integral(csch(x)/(a + b*cosh(x)**2), x)

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError