3.117 \(\int \frac{\cosh (x)}{a+b \coth (x)} \, dx\)

Optimal. Leaf size=72 \[ \frac{a \sinh (x)}{a^2-b^2}-\frac{b \cosh (x)}{a^2-b^2}+\frac{a b \tanh ^{-1}\left (\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.112425, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {3518, 3109, 2637, 2638, 3074, 206} \[ \frac{a \sinh (x)}{a^2-b^2}-\frac{b \cosh (x)}{a^2-b^2}+\frac{a b \tanh ^{-1}\left (\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(a + b*Coth[x]),x]

[Out]

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

Rule 3518

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[(Sin[e + f*x]
^m*(a*Cos[e + f*x] + b*Sin[e + f*x])^n)/Cos[e + f*x]^n, x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
ILtQ[n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))

Rule 3109

Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(
c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Dist[
a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Dist[(a*b)/(a^2 + b^2), Int[(Cos[c + d*x]^(m
- 1)*Sin[c + d*x]^(n - 1))/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b
^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3074

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

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])

Rubi steps

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

Mathematica [A]  time = 0.301377, size = 79, normalized size = 1.1 \[ \frac{a \sinh (x)}{a^2-b^2}+\frac{b \cosh (x)}{b^2-a^2}+\frac{2 a b \tan ^{-1}\left (\frac{a+b \tanh \left (\frac{x}{2}\right )}{\sqrt{b-a} \sqrt{a+b}}\right )}{(b-a)^{3/2} (a+b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/(a + b*Coth[x]),x]

[Out]

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

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Maple [A]  time = 0.033, size = 92, normalized size = 1.3 \begin{align*} -4\,{\frac{1}{ \left ( 4\,a+4\,b \right ) \left ( \tanh \left ( x/2 \right ) -1 \right ) }}-4\,{\frac{1}{ \left ( 4\,a-4\,b \right ) \left ( \tanh \left ( x/2 \right ) +1 \right ) }}-2\,{\frac{ab}{ \left ( a+b \right ) \left ( a-b \right ) \sqrt{-{a}^{2}+{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b+2\,a}{\sqrt{-{a}^{2}+{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/(4*a+4*b)/(tanh(1/2*x)-1)-4/(4*a-4*b)/(tanh(1/2*x)+1)-2*a*b/(a+b)/(a-b)/(-a^2+b^2)^(1/2)*arctan(1/2*(2*tanh
(1/2*x)*b+2*a)/(-a^2+b^2)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.87074, size = 1106, normalized size = 15.36 \begin{align*} \left [-\frac{a^{3} + a^{2} b - a b^{2} - b^{3} -{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2} + 2 \,{\left (a b \cosh \left (x\right ) + a b \sinh \left (x\right )\right )} \sqrt{a^{2} - b^{2}} \log \left (\frac{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt{a^{2} - b^{2}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + a - b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right )^{2} - a + b}\right )}{2 \,{\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) +{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}, -\frac{a^{3} + a^{2} b - a b^{2} - b^{3} -{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2} + 4 \,{\left (a b \cosh \left (x\right ) + a b \sinh \left (x\right )\right )} \sqrt{-a^{2} + b^{2}} \arctan \left (\frac{\sqrt{-a^{2} + b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right )}\right )}{2 \,{\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) +{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)/(a + b*coth(x)), x)

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Giac [A]  time = 1.13028, size = 96, normalized size = 1.33 \begin{align*} -\frac{2 \, a b \arctan \left (\frac{a e^{x} + b e^{x}}{\sqrt{-a^{2} + b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )} \sqrt{-a^{2} + b^{2}}} - \frac{e^{\left (-x\right )}}{2 \,{\left (a - b\right )}} + \frac{e^{x}}{2 \,{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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