3.100 \(\int \frac{\sinh (x)}{a+b \coth (x)} \, dx\)

Optimal. Leaf size=73 \[ -\frac{b \sinh (x)}{a^2-b^2}+\frac{a \cosh (x)}{a^2-b^2}-\frac{b^2 \tanh ^{-1}\left (\frac{\sinh (x) (a \coth (x)+b)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.106747, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {3511, 3486, 2638, 3509, 206} \[ -\frac{b \sinh (x)}{a^2-b^2}+\frac{a \cosh (x)}{a^2-b^2}-\frac{b^2 \tanh ^{-1}\left (\frac{\sinh (x) (a \coth (x)+b)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(a + b*Coth[x]),x]

[Out]

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

Rule 3511

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^
2), Int[(d*Sec[e + f*x])^m*(a - b*Tan[e + f*x]), x], x] + Dist[b^2/(d^2*(a^2 + b^2)), Int[(d*Sec[e + f*x])^(m
+ 2)/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[m, 0]

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 2638

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

Rule 3509

Int[sec[(e_.) + (f_.)*(x_)]/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Dist[f^(-1), Subst[Int[1/(a^
2 + b^2 - x^2), x], x, (b - a*Tan[e + f*x])/Sec[e + f*x]], x] /; FreeQ[{a, b, e, f}, 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{\sinh (x)}{a+b \coth (x)} \, dx &=\frac{\int (a-b \coth (x)) \sinh (x) \, dx}{a^2-b^2}+\frac{b^2 \int \frac{\text{csch}(x)}{a+b \coth (x)} \, dx}{a^2-b^2}\\ &=-\frac{b \sinh (x)}{a^2-b^2}+\frac{a \int \sinh (x) \, dx}{a^2-b^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a^2-b^2-x^2} \, dx,x,i (-i b-i a \coth (x)) \sinh (x)\right )}{a^2-b^2}\\ &=-\frac{b^2 \tanh ^{-1}\left (\frac{(b+a \coth (x)) \sinh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac{a \cosh (x)}{a^2-b^2}-\frac{b \sinh (x)}{a^2-b^2}\\ \end{align*}

Mathematica [A]  time = 0.435651, size = 80, normalized size = 1.1 \[ \frac{a \cosh (x)}{a^2-b^2}+b \left (\frac{\sinh (x)}{b^2-a^2}-\frac{2 b \tan ^{-1}\left (\frac{a+b \tanh \left (\frac{x}{2}\right )}{\sqrt{b-a} \sqrt{a+b}}\right )}{(b-a)^{3/2} (a+b)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(a + b*Coth[x]),x]

[Out]

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

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Maple [A]  time = 0.032, size = 93, normalized size = 1.3 \begin{align*} -8\,{\frac{1}{ \left ( 8\,a+8\,b \right ) \left ( \tanh \left ( x/2 \right ) -1 \right ) }}+8\,{\frac{1}{ \left ( 8\,a-8\,b \right ) \left ( \tanh \left ( x/2 \right ) +1 \right ) }}+2\,{\frac{{b}^{2}}{ \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(sinh(x)/(a+b*coth(x)),x)

[Out]

-8/(8*a+8*b)/(tanh(1/2*x)-1)+8/(8*a-8*b)/(tanh(1/2*x)+1)+2*b^2/(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(sinh(x)/(a+b*coth(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.66047, size = 1103, normalized size = 15.11 \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 (b^{2} \cosh \left (x\right ) + b^{2} \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 (b^{2} \cosh \left (x\right ) + b^{2} \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(sinh(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*(b^2*cosh(x) + b^2*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*(b^2*cosh(x) + b^2*sin
h(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{\sinh{\left (x \right )}}{a + b \coth{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(x)/(a + b*coth(x)), x)

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Giac [A]  time = 1.15365, size = 97, normalized size = 1.33 \begin{align*} \frac{2 \, b^{2} \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(sinh(x)/(a+b*coth(x)),x, algorithm="giac")

[Out]

2*b^2*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)