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

Optimal. Leaf size=67 \[ \frac {\text {csch}(x) (b-a \cosh (x))}{a^2-b^2}+\frac {2 b^2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}} \]

[Out]

2*b^2*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/(a-b)^(3/2)/(a+b)^(3/2)+(b-a*cosh(x))*csch(x)/(a^2-b^2)

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Rubi [A]  time = 0.09, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 = {2696, 12, 2659, 208} \[ \frac {\text {csch}(x) (b-a \cosh (x))}{a^2-b^2}+\frac {2 b^2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(a + b*Cosh[x]),x]

[Out]

(2*b^2*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)) + ((b - a*Cosh[x])*Csch[x])
/(a^2 - b^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 208

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

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 2696

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[((g*Co
s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b - a*Sin[e + f*x]))/(f*g*(a^2 - b^2)*(p + 1)), x] + Dist[1/
(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*(a^2*(p + 2) - b^2*(m + p + 2)
+ a*b*(m + p + 3)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] &&
IntegersQ[2*m, 2*p]

Rubi steps

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

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Mathematica [A]  time = 0.21, size = 77, normalized size = 1.15 \[ \frac {2 b^2 \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}}-\frac {\tanh \left (\frac {x}{2}\right )}{2 (a-b)}-\frac {\coth \left (\frac {x}{2}\right )}{2 (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^2*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - Coth[x/2]/(2*(a + b)) - Tanh[x/2]/(2
*(a - b))

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fricas [B]  time = 0.60, size = 470, normalized size = 7.01 \[ \left [\frac {2 \, a^{3} - 2 \, a b^{2} + {\left (b^{2} \cosh \relax (x)^{2} + 2 \, b^{2} \cosh \relax (x) \sinh \relax (x) + b^{2} \sinh \relax (x)^{2} - b^{2}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) + b}\right ) - 2 \, {\left (a^{2} b - b^{3}\right )} \cosh \relax (x) - 2 \, {\left (a^{2} b - b^{3}\right )} \sinh \relax (x)}{a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x)^{2} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x) \sinh \relax (x) - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \relax (x)^{2}}, \frac {2 \, {\left (a^{3} - a b^{2} + {\left (b^{2} \cosh \relax (x)^{2} + 2 \, b^{2} \cosh \relax (x) \sinh \relax (x) + b^{2} \sinh \relax (x)^{2} - b^{2}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{a^{2} - b^{2}}\right ) - {\left (a^{2} b - b^{3}\right )} \cosh \relax (x) - {\left (a^{2} b - b^{3}\right )} \sinh \relax (x)\right )}}{a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x)^{2} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x) \sinh \relax (x) - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \relax (x)^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 76, normalized size = 1.13 \[ \frac {2 \, b^{2} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (b e^{x} - a\right )}}{{\left (a^{2} - b^{2}\right )} {\left (e^{\left (2 \, x\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^2*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/((a^2 - b^2)*sqrt(-a^2 + b^2)) + 2*(b*e^x - a)/((a^2 - b^2)*(e^(2*x
) - 1))

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maple [A]  time = 0.08, size = 78, normalized size = 1.16 \[ -\frac {\tanh \left (\frac {x}{2}\right )}{2 \left (a -b \right )}-\frac {1}{2 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )}+\frac {2 b^{2} \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a-b)*tanh(1/2*x)-1/2/(a+b)/tanh(1/2*x)+2/(a+b)/(a-b)*b^2/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/(
(a+b)*(a-b))^(1/2))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B]  time = 1.48, size = 327, normalized size = 4.88 \[ -\frac {\frac {2\,a}{a^2-b^2}-\frac {2\,b\,{\mathrm {e}}^x}{a^2-b^2}}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2}{{\left (a^2-b^2\right )}^2\,\sqrt {b^4}}+\frac {2\,a\,\left (a^3\,\sqrt {b^4}-a\,b^2\,\sqrt {b^4}\right )}{b^4\,\left (a^2-b^2\right )\,\sqrt {-{\left (a^2-b^2\right )}^3}\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}\right )-\frac {2\,a\,\left (b^3\,\sqrt {b^4}-a^2\,b\,\sqrt {b^4}\right )}{b^4\,\left (a^2-b^2\right )\,\sqrt {-{\left (a^2-b^2\right )}^3}\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}\right )\,\left (\frac {b^3\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{2}-\frac {a^2\,b\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{2}\right )\right )\,\sqrt {b^4}}{\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2*(a + b*cosh(x))),x)

[Out]

- ((2*a)/(a^2 - b^2) - (2*b*exp(x))/(a^2 - b^2))/(exp(2*x) - 1) - (2*atan((exp(x)*(2/((a^2 - b^2)^2*(b^4)^(1/2
)) + (2*a*(a^3*(b^4)^(1/2) - a*b^2*(b^4)^(1/2)))/(b^4*(a^2 - b^2)*(-(a^2 - b^2)^3)^(1/2)*(b^6 - a^6 - 3*a^2*b^
4 + 3*a^4*b^2)^(1/2))) - (2*a*(b^3*(b^4)^(1/2) - a^2*b*(b^4)^(1/2)))/(b^4*(a^2 - b^2)*(-(a^2 - b^2)^3)^(1/2)*(
b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)))*((b^3*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/2 - (a^2*b*(b^6 -
a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/2))*(b^4)^(1/2))/(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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