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

Optimal. Leaf size=29 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{\sqrt{b} \sqrt{a+b}} \]

[Out]

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

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Rubi [A]  time = 0.032059, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3186, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{\sqrt{b} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(a + b*Cosh[x]^2),x]

[Out]

ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]]/(Sqrt[b]*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 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{\cosh (x)}{a+b \cosh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\sinh (x)\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{\sqrt{b} \sqrt{a+b}}\\ \end{align*}

Mathematica [A]  time = 0.0110267, size = 29, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{\sqrt{b} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.021, size = 66, normalized size = 2.3 \begin{align*}{\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}{\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}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.97649, size = 988, normalized size = 34.07 \begin{align*} \left [-\frac{\sqrt{-a b - b^{2}} \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 (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )\right )} \sqrt{-a b - b^{2}} + 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 \,{\left (a b + b^{2}\right )}}, \frac{\sqrt{a b + b^{2}} \arctan \left (\frac{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 )}{2 \, \sqrt{a b + b^{2}}}\right ) + \sqrt{a b + b^{2}} \arctan \left (\frac{\sqrt{a b + b^{2}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{2 \,{\left (a + b\right )}}\right )}{a b + b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

Exception raised: TypeError