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

Optimal. Leaf size=41 \[ \frac{\tan ^{-1}(\sinh (x))}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{a \sqrt{a+b}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0535018, antiderivative size = 41, 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 = {3186, 391, 203, 205} \[ \frac{\tan ^{-1}(\sinh (x))}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{a \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]/(a + b*Cosh[x]^2),x]

[Out]

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

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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{sech}(x)}{a+b \cosh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\sinh (x)\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (x)\right )}{a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\sinh (x)\right )}{a}\\ &=\frac{\tan ^{-1}(\sinh (x))}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{a \sqrt{a+b}}\\ \end{align*}

Mathematica [A]  time = 0.0954587, size = 45, normalized size = 1.1 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a+b} \text{csch}(x)}{\sqrt{b}}\right )}{a \sqrt{a+b}}+\frac{2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*ArcTan[(Sqrt[a + b]*Csch[x])/Sqrt[b]])/(a*Sqrt[a + b]) + (2*ArcTan[Tanh[x/2]])/a

________________________________________________________________________________________

Maple [B]  time = 0.039, size = 83, normalized size = 2. \begin{align*} -{\frac{1}{a}\sqrt{b}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( x/2 \right ) \sqrt{a+b}+2\,\sqrt{a} \right ){\frac{1}{\sqrt{b}}}} \right ){\frac{1}{\sqrt{a+b}}}}+{\frac{1}{a}\sqrt{b}\arctan \left ({\frac{1}{2} \left ( -2\,\tanh \left ( x/2 \right ) \sqrt{a+b}+2\,\sqrt{a} \right ){\frac{1}{\sqrt{b}}}} \right ){\frac{1}{\sqrt{a+b}}}}+2\,{\frac{\arctan \left ( \tanh \left ( x/2 \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a*b^(1/2)/(a+b)^(1/2)*arctan(1/2*(2*tanh(1/2*x)*(a+b)^(1/2)+2*a^(1/2))/b^(1/2))+1/a*b^(1/2)/(a+b)^(1/2)*arc
tan(1/2*(-2*tanh(1/2*x)*(a+b)^(1/2)+2*a^(1/2))/b^(1/2))+2/a*arctan(tanh(1/2*x))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.29607, size = 1096, normalized size = 26.73 \begin{align*} \left [\frac{\sqrt{-\frac{b}{a + b}} \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 + 3 \, b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, b \cosh \left (x\right )^{2} - 2 \, a - 3 \, b\right )} \sinh \left (x\right )^{2} + 4 \,{\left (b \cosh \left (x\right )^{3} -{\left (2 \, a + 3 \, b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4 \,{\left ({\left (a + b\right )} \cosh \left (x\right )^{3} + 3 \,{\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} +{\left (a + b\right )} \sinh \left (x\right )^{3} -{\left (a + b\right )} \cosh \left (x\right ) +{\left (3 \,{\left (a + b\right )} \cosh \left (x\right )^{2} - a - b\right )} \sinh \left (x\right )\right )} \sqrt{-\frac{b}{a + b}} + 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 ) + 4 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{2 \, a}, -\frac{\sqrt{\frac{b}{a + b}} \arctan \left (\frac{1}{2} \, \sqrt{\frac{b}{a + b}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}\right ) + \sqrt{\frac{b}{a + b}} \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 + 3 \, b\right )} \cosh \left (x\right ) +{\left (3 \, b \cosh \left (x\right )^{2} + 4 \, a + 3 \, b\right )} \sinh \left (x\right )\right )} \sqrt{\frac{b}{a + b}}}{2 \, b}\right ) - 2 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}{\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(sech(x)/(a+b*cosh(x)**2),x)

[Out]

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

[Out]

Exception raised: TypeError