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]

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

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Rubi [A]  time = 0.06, antiderivative size = 42, 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 = {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 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]

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 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 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]

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*}

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Mathematica [C]  time = 0.14, 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))

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fricas [B]  time = 0.64, size = 349, normalized size = 8.31 \[ \left [\frac {\sqrt {-\frac {b}{a}} \log \left (\frac {b \cosh \relax (x)^{4} + 4 \, b \cosh \relax (x) \sinh \relax (x)^{3} + b \sinh \relax (x)^{4} - 2 \, {\left (2 \, a - b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b \cosh \relax (x)^{2} - 2 \, a + b\right )} \sinh \relax (x)^{2} + 4 \, {\left (b \cosh \relax (x)^{3} - {\left (2 \, a - b\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 4 \, {\left (a \cosh \relax (x)^{3} + 3 \, a \cosh \relax (x) \sinh \relax (x)^{2} + a \sinh \relax (x)^{3} + a \cosh \relax (x) + {\left (3 \, a \cosh \relax (x)^{2} + a\right )} \sinh \relax (x)\right )} \sqrt {-\frac {b}{a}} + b}{b \cosh \relax (x)^{4} + 4 \, b \cosh \relax (x) \sinh \relax (x)^{3} + b \sinh \relax (x)^{4} + 2 \, {\left (2 \, a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b \cosh \relax (x)^{2} + 2 \, a + b\right )} \sinh \relax (x)^{2} + 4 \, {\left (b \cosh \relax (x)^{3} + {\left (2 \, a + b\right )} \cosh \relax (x)\right )} \sinh \relax (x) + b}\right ) - 2 \, \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right )}{2 \, {\left (a + b\right )}}, -\frac {\sqrt {\frac {b}{a}} \arctan \left (\frac {1}{2} \, \sqrt {\frac {b}{a}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )}\right ) - \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (b \cosh \relax (x)^{3} + 3 \, b \cosh \relax (x) \sinh \relax (x)^{2} + b \sinh \relax (x)^{3} + {\left (4 \, a + b\right )} \cosh \relax (x) + {\left (3 \, b \cosh \relax (x)^{2} + 4 \, a + b\right )} \sinh \relax (x)\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right ) + \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right )}{a + b}\right ] \]

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)]

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

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[-26,-97]Warning, need to choose a branch for the root of a polynomial with parameters. Th
is might be wrong.The choice was done assuming [a,b]=[-71,-13]Undef/Unsigned Inf encountered in limitLimit: Ma
x order reached or unable to make series expansion Error: Bad Argument Value

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maple [A]  time = 0.10, size = 52, normalized size = 1.24 \[ -\frac {b \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a +2 b}{4 \sqrt {a b}}\right )}{\left (a +b \right ) \sqrt {a b}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a +b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left (e^{x} + 1\right )}{a + b} + \frac {\log \left (e^{x} - 1\right )}{a + b} - 2 \, \int \frac {b e^{\left (3 \, x\right )} - b e^{x}}{a b + b^{2} + {\left (a b + b^{2}\right )} e^{\left (4 \, x\right )} + 2 \, {\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} e^{\left (2 \, x\right )}}\,{d x} \]

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)

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mupad [B]  time = 1.39, size = 462, normalized size = 11.00 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\left (16\,a^2\,\sqrt {-a^2-2\,a\,b-b^2}+b^2\,\sqrt {-a^2-2\,a\,b-b^2}+8\,a\,b\,\sqrt {-a^2-2\,a\,b-b^2}\right )}{16\,a^3+24\,a^2\,b+9\,a\,b^2+b^3}\right )}{\sqrt {-a^2-2\,a\,b-b^2}}-\frac {\sqrt {b}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {b}\,{\mathrm {e}}^x\,\sqrt {a\,{\left (a+b\right )}^2}}{2\,a\,\left (a+b\right )}\right )-2\,\mathrm {atan}\left (\frac {\left (a^3\,b^{5/2}\,\sqrt {a^3+2\,a^2\,b+a\,b^2}+a^2\,b^{7/2}\,\sqrt {a^3+2\,a^2\,b+a\,b^2}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {64\,\left (8\,a^3+10\,a^2\,b+2\,a\,b^2\right )}{a\,b^3\,\sqrt {a\,{\left (a+b\right )}^2}\,\left (a^2+b\,a\right )\,\sqrt {a^3+2\,a^2\,b+a\,b^2}}+\frac {32\,\left (b^{3/2}\,\sqrt {a^3+2\,a^2\,b+a\,b^2}+4\,a\,\sqrt {b}\,\sqrt {a^3+2\,a^2\,b+a\,b^2}\right )}{a^2\,b^{5/2}\,\left (a+b\right )\,\left (a^2+b\,a\right )\,\sqrt {a^3+2\,a^2\,b+a\,b^2}}\right )+\frac {32\,{\mathrm {e}}^{3\,x}\,\left (b^{3/2}\,\sqrt {a^3+2\,a^2\,b+a\,b^2}+4\,a\,\sqrt {b}\,\sqrt {a^3+2\,a^2\,b+a\,b^2}\right )}{a^2\,b^{5/2}\,\left (a+b\right )\,\left (a^2+b\,a\right )\,\sqrt {a^3+2\,a^2\,b+a\,b^2}}\right )}{256\,a+64\,b}\right )\right )}{2\,\sqrt {a^3+2\,a^2\,b+a\,b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)*(a + b*cosh(x)^2)),x)

[Out]

- (2*atan((exp(x)*(16*a^2*(- 2*a*b - a^2 - b^2)^(1/2) + b^2*(- 2*a*b - a^2 - b^2)^(1/2) + 8*a*b*(- 2*a*b - a^2
- b^2)^(1/2)))/(9*a*b^2 + 24*a^2*b + 16*a^3 + b^3)))/(- 2*a*b - a^2 - b^2)^(1/2) - (b^(1/2)*(2*atan((b^(1/2)*
exp(x)*(a*(a + b)^2)^(1/2))/(2*a*(a + b))) - 2*atan(((a^3*b^(5/2)*(a*b^2 + 2*a^2*b + a^3)^(1/2) + a^2*b^(7/2)*
(a*b^2 + 2*a^2*b + a^3)^(1/2))*(exp(x)*((64*(2*a*b^2 + 10*a^2*b + 8*a^3))/(a*b^3*(a*(a + b)^2)^(1/2)*(a*b + a^
2)*(a*b^2 + 2*a^2*b + a^3)^(1/2)) + (32*(b^(3/2)*(a*b^2 + 2*a^2*b + a^3)^(1/2) + 4*a*b^(1/2)*(a*b^2 + 2*a^2*b
+ a^3)^(1/2)))/(a^2*b^(5/2)*(a + b)*(a*b + a^2)*(a*b^2 + 2*a^2*b + a^3)^(1/2))) + (32*exp(3*x)*(b^(3/2)*(a*b^2
+ 2*a^2*b + a^3)^(1/2) + 4*a*b^(1/2)*(a*b^2 + 2*a^2*b + a^3)^(1/2)))/(a^2*b^(5/2)*(a + b)*(a*b + a^2)*(a*b^2
+ 2*a^2*b + a^3)^(1/2))))/(256*a + 64*b))))/(2*(a*b^2 + 2*a^2*b + a^3)^(1/2))

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

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)

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