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

Optimal. Leaf size=57 \[ \frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a b}+\frac{x}{a}-\frac{\tanh ^{-1}(\cosh (x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.17816, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {3894, 4051, 3770, 3919, 3831, 2660, 618, 206} \[ \frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a b}+\frac{x}{a}-\frac{\tanh ^{-1}(\cosh (x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2/(a + b*Csch[x]),x]

[Out]

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

Rule 3894

Int[cot[(c_.) + (d_.)*(x_)]^2*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[(-1 + Csc[c + d*x]
^2)*(a + b*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0]

Rule 4051

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[C/b, I
nt[Csc[e + f*x], x], x] + Dist[1/b, Int[(A*b - a*C*Csc[e + f*x])/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b,
e, f, A, C}, x]

Rule 3770

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

Rule 3919

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

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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{\coth ^2(x)}{a+b \text{csch}(x)} \, dx &=-\int \frac{-1-\text{csch}^2(x)}{a+b \text{csch}(x)} \, dx\\ &=\frac{i \int \frac{-i b+i a \text{csch}(x)}{a+b \text{csch}(x)} \, dx}{b}+\frac{\int \text{csch}(x) \, dx}{b}\\ &=\frac{x}{a}-\frac{\tanh ^{-1}(\cosh (x))}{b}-\frac{\left (a^2+b^2\right ) \int \frac{\text{csch}(x)}{a+b \text{csch}(x)} \, dx}{a b}\\ &=\frac{x}{a}-\frac{\tanh ^{-1}(\cosh (x))}{b}-\left (\frac{1}{a}+\frac{a}{b^2}\right ) \int \frac{1}{1+\frac{a \sinh (x)}{b}} \, dx\\ &=\frac{x}{a}-\frac{\tanh ^{-1}(\cosh (x))}{b}-\left (2 \left (\frac{1}{a}+\frac{a}{b^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{x}{a}-\frac{\tanh ^{-1}(\cosh (x))}{b}+\left (4 \left (\frac{1}{a}+\frac{a}{b^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (1+\frac{a^2}{b^2}\right )-x^2} \, dx,x,\frac{2 a}{b}-2 \tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{x}{a}-\frac{\tanh ^{-1}(\cosh (x))}{b}+\frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b \left (\frac{a}{b}-\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{a^2+b^2}}\right )}{a b}\\ \end{align*}

Mathematica [A]  time = 0.0813885, size = 65, normalized size = 1.14 \[ \frac{2 \sqrt{-a^2-b^2} \tan ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )+a \log \left (\tanh \left (\frac{x}{2}\right )\right )+b x}{a b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.027, size = 110, normalized size = 1.9 \begin{align*}{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-2\,{\frac{a}{b\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b-2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }-2\,{\frac{b}{a\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b-2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }+{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*ln(tanh(1/2*x)+1)-2*a/b/(a^2+b^2)^(1/2)*arctanh(1/2*(2*tanh(1/2*x)*b-2*a)/(a^2+b^2)^(1/2))-2/a*b/(a^2+b^2)
^(1/2)*arctanh(1/2*(2*tanh(1/2*x)*b-2*a)/(a^2+b^2)^(1/2))+1/b*ln(tanh(1/2*x))-1/a*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(coth(x)^2/(a+b*csch(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.81417, size = 424, normalized size = 7.44 \begin{align*} \frac{b x - a \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + a \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + \sqrt{a^{2} + b^{2}} \log \left (\frac{a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \,{\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} + b^{2}}{\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \,{\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x - a*log(cosh(x) + sinh(x) + 1) + a*log(cosh(x) + sinh(x) - 1) + sqrt(a^2 + b^2)*log((a^2*cosh(x)^2 + a^2*
sinh(x)^2 + 2*a*b*cosh(x) + a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(a^2 + b^2)*(a*cosh(x) + a*sin
h(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*cosh(x) + b)*sinh(x) - a)))/(a*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16172, size = 120, normalized size = 2.11 \begin{align*} \frac{x}{a} - \frac{\log \left (e^{x} + 1\right )}{b} + \frac{\log \left ({\left | e^{x} - 1 \right |}\right )}{b} - \frac{\sqrt{a^{2} + b^{2}} \log \left (\frac{{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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