3.114 \(\int \frac{\frac{b B}{a}+B \cosh (x)}{a+b \cosh (x)} \, dx\)

Optimal. Leaf size=56 \[ \frac{B x}{b}-\frac{2 B \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a b} \]

[Out]

(B*x)/b - (2*Sqrt[a - b]*Sqrt[a + b]*B*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*b)

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Rubi [A]  time = 0.0784398, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2735, 2659, 208} \[ \frac{B x}{b}-\frac{2 B \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a b} \]

Antiderivative was successfully verified.

[In]

Int[((b*B)/a + B*Cosh[x])/(a + b*Cosh[x]),x]

[Out]

(B*x)/b - (2*Sqrt[a - b]*Sqrt[a + b]*B*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*b)

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0689199, size = 56, normalized size = 1. \[ \frac{B \left (\frac{2 \sqrt{b^2-a^2} \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{b}+\frac{a x}{b}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.021, size = 107, normalized size = 1.9 \begin{align*} -2\,{\frac{aB}{b\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tanh \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+2\,{\frac{Bb}{a\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tanh \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+{\frac{B}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{B}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*B/a+B*cosh(x))/(a+b*cosh(x)),x)

[Out]

-2/b/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))*a*B+2*B/a*b/((a+b)*(a-b))^(1/2)*arctan
h((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))+B/b*ln(tanh(1/2*x)+1)-B/b*ln(tanh(1/2*x)-1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.28327, size = 491, normalized size = 8.77 \begin{align*} \left [\frac{B a x + \sqrt{a^{2} - b^{2}} B \log \left (\frac{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right )}{a b}, \frac{B a x + 2 \, \sqrt{-a^{2} + b^{2}} B \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right )}{a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(B*a*x + sqrt(a^2 - b^2)*B*log((b^2*cosh(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*cosh(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)))/(a*b), (B*a*x + 2*sqrt(-a^2 + b^2)*B*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sin
h(x) + a)/(a^2 - b^2)))/(a*b)]

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Sympy [A]  time = 154.659, size = 170, normalized size = 3.04 \begin{align*} \begin{cases} \text{NaN} & \text{for}\: a = 0 \wedge b = 0 \\\frac{B x}{b} & \text{for}\: a = b \\\frac{B \sinh{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{B x}{b} & \text{for}\: a = - b \\\frac{B x}{b} + \frac{B \log{\left (- \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} - \frac{B \log{\left (\sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} + \frac{B \log{\left (- \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{a \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} - \frac{B \log{\left (\sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{a \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((nan, Eq(a, 0) & Eq(b, 0)), (B*x/b, Eq(a, b)), (B*sinh(x)/a, Eq(b, 0)), (B*x/b, Eq(a, -b)), (B*x/b +
B*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(b*sqrt(a/(a - b) + b/(a - b))) - B*log(sqrt(a/(a - b) + b/(a
- b)) + tanh(x/2))/(b*sqrt(a/(a - b) + b/(a - b))) + B*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*sqrt(
a/(a - b) + b/(a - b))) - B*log(sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*sqrt(a/(a - b) + b/(a - b))), True
))

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Giac [A]  time = 1.20697, size = 77, normalized size = 1.38 \begin{align*} \frac{B x}{b} - \frac{2 \,{\left (B a^{2} - B b^{2}\right )} \arctan \left (\frac{b e^{x} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}} a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*B/a+B*cosh(x))/(a+b*cosh(x)),x, algorithm="giac")

[Out]

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