3.150 \(\int \frac{x \text{csch}^2(x)}{(a+b \coth (x))^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac{a x}{b \left (a^2-b^2\right )}+\frac{\log (a \sinh (x)+b \cosh (x))}{a^2-b^2}+\frac{x}{b (a+b \coth (x))} \]

[Out]

-((a*x)/(b*(a^2 - b^2))) + x/(b*(a + b*Coth[x])) + Log[b*Cosh[x] + a*Sinh[x]]/(a^2 - b^2)

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Rubi [A]  time = 0.0850082, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5467, 3484, 3530} \[ -\frac{a x}{b \left (a^2-b^2\right )}+\frac{\log (a \sinh (x)+b \cosh (x))}{a^2-b^2}+\frac{x}{b (a+b \coth (x))} \]

Antiderivative was successfully verified.

[In]

Int[(x*Csch[x]^2)/(a + b*Coth[x])^2,x]

[Out]

-((a*x)/(b*(a^2 - b^2))) + x/(b*(a + b*Coth[x])) + Log[b*Cosh[x] + a*Sinh[x]]/(a^2 - b^2)

Rule 5467

Int[Csch[(c_.) + (d_.)*(x_)]^2*(Coth[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Sym
bol] :> -Simp[((e + f*x)^m*(a + b*Coth[c + d*x])^(n + 1))/(b*d*(n + 1)), x] + Dist[(f*m)/(b*d*(n + 1)), Int[(e
+ f*x)^(m - 1)*(a + b*Coth[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[
n, -1]

Rule 3484

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

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

\begin{align*} \int \frac{x \text{csch}^2(x)}{(a+b \coth (x))^2} \, dx &=\frac{x}{b (a+b \coth (x))}-\frac{\int \frac{1}{a+b \coth (x)} \, dx}{b}\\ &=-\frac{a x}{b \left (a^2-b^2\right )}+\frac{x}{b (a+b \coth (x))}+\frac{i \int \frac{-i b-i a \coth (x)}{a+b \coth (x)} \, dx}{a^2-b^2}\\ &=-\frac{a x}{b \left (a^2-b^2\right )}+\frac{x}{b (a+b \coth (x))}+\frac{\log (b \cosh (x)+a \sinh (x))}{a^2-b^2}\\ \end{align*}

Mathematica [A]  time = 0.158962, size = 49, normalized size = 0.91 \[ \frac{a x-b \log (a \sinh (x)+b \cosh (x))}{b^3-a^2 b}+\frac{x \sinh (x)}{a b \sinh (x)+b^2 \cosh (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.112, size = 73, normalized size = 1.4 \begin{align*} -2\,{\frac{x}{{a}^{2}-{b}^{2}}}-2\,{\frac{x}{ \left ( a{{\rm e}^{2\,x}}+b{{\rm e}^{2\,x}}-a+b \right ) \left ( a+b \right ) }}+{\frac{1}{{a}^{2}-{b}^{2}}\ln \left ({{\rm e}^{2\,x}}-{\frac{a-b}{a+b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a^2-b^2)*x-2*x/(a*exp(2*x)+b*exp(2*x)-a+b)/(a+b)+1/(a^2-b^2)*ln(exp(2*x)-(a-b)/(a+b))

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Maxima [A]  time = 1.952, size = 92, normalized size = 1.7 \begin{align*} \frac{2 \, x e^{\left (2 \, x\right )}}{a^{2} - 2 \, a b + b^{2} -{\left (a^{2} - b^{2}\right )} e^{\left (2 \, x\right )}} + \frac{\log \left (\frac{{\left (a + b\right )} e^{\left (2 \, x\right )} - a + b}{a + b}\right )}{a^{2} - b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*e^(2*x)/(a^2 - 2*a*b + b^2 - (a^2 - b^2)*e^(2*x)) + log(((a + b)*e^(2*x) - a + b)/(a + b))/(a^2 - b^2)

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Fricas [B]  time = 2.68442, size = 491, normalized size = 9.09 \begin{align*} \frac{2 \,{\left (a + b\right )} x \cosh \left (x\right )^{2} + 4 \,{\left (a + b\right )} x \cosh \left (x\right ) \sinh \left (x\right ) + 2 \,{\left (a + b\right )} x \sinh \left (x\right )^{2} -{\left ({\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 )} \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a + b)*x*cosh(x)^2 + 4*(a + b)*x*cosh(x)*sinh(x) + 2*(a + b)*x*sinh(x)^2 - ((a + b)*cosh(x)^2 + 2*(a + b)*
cosh(x)*sinh(x) + (a + b)*sinh(x)^2 - a + b)*log(2*(b*cosh(x) + a*sinh(x))/(cosh(x) - sinh(x))))/(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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{csch}^{2}{\left (x \right )}}{\left (a + b \coth{\left (x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*csch(x)**2/(a+b*coth(x))**2,x)

[Out]

Integral(x*csch(x)**2/(a + b*coth(x))**2, x)

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Giac [B]  time = 1.17361, size = 228, normalized size = 4.22 \begin{align*} -\frac{2 \, a x e^{\left (2 \, x\right )} + 2 \, b x e^{\left (2 \, x\right )} - a e^{\left (2 \, x\right )} \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b\right ) - b e^{\left (2 \, x\right )} \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b\right ) + a \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b\right ) - b \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b\right )}{a^{3} e^{\left (2 \, x\right )} + a^{2} b e^{\left (2 \, x\right )} - a b^{2} e^{\left (2 \, x\right )} - b^{3} e^{\left (2 \, x\right )} - a^{3} + a^{2} b + a b^{2} - b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*csch(x)^2/(a+b*coth(x))^2,x, algorithm="giac")

[Out]

-(2*a*x*e^(2*x) + 2*b*x*e^(2*x) - a*e^(2*x)*log(a*e^(2*x) + b*e^(2*x) - a + b) - b*e^(2*x)*log(a*e^(2*x) + b*e
^(2*x) - a + b) + a*log(a*e^(2*x) + b*e^(2*x) - a + b) - b*log(a*e^(2*x) + b*e^(2*x) - a + b))/(a^3*e^(2*x) +
a^2*b*e^(2*x) - a*b^2*e^(2*x) - b^3*e^(2*x) - a^3 + a^2*b + a*b^2 - b^3)