3.524 \(\int \frac{\coth (x)}{\sqrt{a+b \sinh ^n(x)}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

[Out]

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

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Rubi [A]  time = 0.0865435, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3230, 266, 63, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/Sqrt[a + b*Sinh[x]^n],x]

[Out]

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

Rule 3230

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + b*(c*ff*x)^n)^p)/(1 - ff^2*x^2)^(
(m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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

Mathematica [A]  time = 0.0237434, size = 29, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/Sqrt[a + b*Sinh[x]^n],x]

[Out]

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

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Maple [A]  time = 0.033, size = 24, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{n\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( \sinh \left ( x \right ) \right ) ^{n}}}{\sqrt{a}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(a+b*sinh(x)^n)^(1/2),x)

[Out]

-2*arctanh((a+b*sinh(x)^n)^(1/2)/a^(1/2))/n/a^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \left (x\right )}{\sqrt{b \sinh \left (x\right )^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*sinh(x)^n)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.83506, size = 390, normalized size = 13.45 \begin{align*} \left [\frac{\log \left (\frac{b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) - 2 \, \sqrt{b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) + a} \sqrt{a} + 2 \, a}{\cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + \sinh \left (n \log \left (\sinh \left (x\right )\right )\right )}\right )}{\sqrt{a} n}, \frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) + a} \sqrt{-a}}{a}\right )}{a n}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*sinh(x)^n)^(1/2),x, algorithm="fricas")

[Out]

[log((b*cosh(n*log(sinh(x))) + b*sinh(n*log(sinh(x))) - 2*sqrt(b*cosh(n*log(sinh(x))) + b*sinh(n*log(sinh(x)))
+ a)*sqrt(a) + 2*a)/(cosh(n*log(sinh(x))) + sinh(n*log(sinh(x)))))/(sqrt(a)*n), 2*sqrt(-a)*arctan(sqrt(b*cosh
(n*log(sinh(x))) + b*sinh(n*log(sinh(x))) + a)*sqrt(-a)/a)/(a*n)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*sinh(x)**n)**(1/2),x)

[Out]

Integral(coth(x)/sqrt(a + b*sinh(x)**n), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \left (x\right )}{\sqrt{b \sinh \left (x\right )^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*sinh(x)^n)^(1/2),x, algorithm="giac")

[Out]

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