3.29 \(\int \sqrt{-1-\coth ^2(x)} \, dx\)

Optimal. Leaf size=45 \[ \tan ^{-1}\left (\frac{\coth (x)}{\sqrt{-\coth ^2(x)-1}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{-\coth ^2(x)-1}}\right ) \]

[Out]

ArcTan[Coth[x]/Sqrt[-1 - Coth[x]^2]] - Sqrt[2]*ArcTan[(Sqrt[2]*Coth[x])/Sqrt[-1 - Coth[x]^2]]

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Rubi [A]  time = 0.0336847, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3661, 402, 217, 203, 377} \[ \tan ^{-1}\left (\frac{\coth (x)}{\sqrt{-\coth ^2(x)-1}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{-\coth ^2(x)-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - Coth[x]^2],x]

[Out]

ArcTan[Coth[x]/Sqrt[-1 - Coth[x]^2]] - Sqrt[2]*ArcTan[(Sqrt[2]*Coth[x])/Sqrt[-1 - Coth[x]^2]]

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rubi steps

\begin{align*} \int \sqrt{-1-\coth ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{-1-x^2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x^2} \left (1-x^2\right )} \, dx,x,\coth (x)\right )\right )+\operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x^2}} \, dx,x,\coth (x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{\coth (x)}{\sqrt{-1-\coth ^2(x)}}\right )\right )+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\coth (x)}{\sqrt{-1-\coth ^2(x)}}\right )\\ &=\tan ^{-1}\left (\frac{\coth (x)}{\sqrt{-1-\coth ^2(x)}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{-1-\coth ^2(x)}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0504301, size = 62, normalized size = 1.38 \[ \frac{\sinh (x) \sqrt{-\coth ^2(x)-1} \left (\sqrt{2} \log \left (\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right )-\tanh ^{-1}\left (\frac{\cosh (x)}{\sqrt{\cosh (2 x)}}\right )\right )}{\sqrt{\cosh (2 x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 - Coth[x]^2],x]

[Out]

(Sqrt[-1 - Coth[x]^2]*(-ArcTanh[Cosh[x]/Sqrt[Cosh[2*x]]] + Sqrt[2]*Log[Sqrt[2]*Cosh[x] + Sqrt[Cosh[2*x]]])*Sin
h[x])/Sqrt[Cosh[2*x]]

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Maple [B]  time = 0.05, size = 142, normalized size = 3.2 \begin{align*}{\frac{1}{2}\sqrt{- \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}+2\,{\rm coth} \left (x\right )}}+{\frac{1}{2}\arctan \left ({{\rm coth} \left (x\right ){\frac{1}{\sqrt{- \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}+2\,{\rm coth} \left (x\right )}}}} \right ) }-{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 2\,{\rm coth} \left (x\right )-2 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}+2\,{\rm coth} \left (x\right )}}}} \right ) }-{\frac{1}{2}\sqrt{- \left ({\rm coth} \left (x\right )-1 \right ) ^{2}-2\,{\rm coth} \left (x\right )}}+{\frac{1}{2}\arctan \left ({{\rm coth} \left (x\right ){\frac{1}{\sqrt{- \left ({\rm coth} \left (x\right )-1 \right ) ^{2}-2\,{\rm coth} \left (x\right )}}}} \right ) }+{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( -2-2\,{\rm coth} \left (x\right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ({\rm coth} \left (x\right )-1 \right ) ^{2}-2\,{\rm coth} \left (x\right )}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1-coth(x)^2)^(1/2),x)

[Out]

1/2*(-(1+coth(x))^2+2*coth(x))^(1/2)+1/2*arctan(coth(x)/(-(1+coth(x))^2+2*coth(x))^(1/2))-1/2*2^(1/2)*arctan(1
/4*(2*coth(x)-2)*2^(1/2)/(-(1+coth(x))^2+2*coth(x))^(1/2))-1/2*(-(coth(x)-1)^2-2*coth(x))^(1/2)+1/2*arctan(cot
h(x)/(-(coth(x)-1)^2-2*coth(x))^(1/2))+1/2*2^(1/2)*arctan(1/4*(-2-2*coth(x))*2^(1/2)/(-(coth(x)-1)^2-2*coth(x)
)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-\coth \left (x\right )^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-coth(x)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-coth(x)^2 - 1), x)

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Fricas [C]  time = 2.52965, size = 710, normalized size = 15.78 \begin{align*} -\frac{1}{4} \, \sqrt{-2} \log \left (-{\left (\sqrt{-2} \sqrt{-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) + \frac{1}{4} \, \sqrt{-2} \log \left ({\left (\sqrt{-2} \sqrt{-2 \, e^{\left (4 \, x\right )} - 2} - 2 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-2 \, x\right )}\right ) + \frac{1}{4} \, \sqrt{-2} \log \left (-2 \,{\left (\sqrt{-2 \, e^{\left (4 \, x\right )} - 2}{\left (e^{\left (2 \, x\right )} + 2\right )} + \sqrt{-2} e^{\left (4 \, x\right )} + \sqrt{-2} e^{\left (2 \, x\right )} + 2 \, \sqrt{-2}\right )} e^{\left (-4 \, x\right )}\right ) - \frac{1}{4} \, \sqrt{-2} \log \left (-2 \,{\left (\sqrt{-2 \, e^{\left (4 \, x\right )} - 2}{\left (e^{\left (2 \, x\right )} + 2\right )} - \sqrt{-2} e^{\left (4 \, x\right )} - \sqrt{-2} e^{\left (2 \, x\right )} - 2 \, \sqrt{-2}\right )} e^{\left (-4 \, x\right )}\right ) - \frac{1}{2} i \, \log \left ({\left (4 i \, \sqrt{-2 \, e^{\left (4 \, x\right )} - 2} - 4 \, e^{\left (2 \, x\right )} - 4\right )} e^{\left (-2 \, x\right )}\right ) + \frac{1}{2} i \, \log \left ({\left (-4 i \, \sqrt{-2 \, e^{\left (4 \, x\right )} - 2} - 4 \, e^{\left (2 \, x\right )} - 4\right )} e^{\left (-2 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-coth(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-1/4*sqrt(-2)*log(-(sqrt(-2)*sqrt(-2*e^(4*x) - 2) + 2*e^(2*x) - 2)*e^(-2*x)) + 1/4*sqrt(-2)*log((sqrt(-2)*sqrt
(-2*e^(4*x) - 2) - 2*e^(2*x) + 2)*e^(-2*x)) + 1/4*sqrt(-2)*log(-2*(sqrt(-2*e^(4*x) - 2)*(e^(2*x) + 2) + sqrt(-
2)*e^(4*x) + sqrt(-2)*e^(2*x) + 2*sqrt(-2))*e^(-4*x)) - 1/4*sqrt(-2)*log(-2*(sqrt(-2*e^(4*x) - 2)*(e^(2*x) + 2
) - sqrt(-2)*e^(4*x) - sqrt(-2)*e^(2*x) - 2*sqrt(-2))*e^(-4*x)) - 1/2*I*log((4*I*sqrt(-2*e^(4*x) - 2) - 4*e^(2
*x) - 4)*e^(-2*x)) + 1/2*I*log((-4*I*sqrt(-2*e^(4*x) - 2) - 4*e^(2*x) - 4)*e^(-2*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- \coth ^{2}{\left (x \right )} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-coth(x)**2)**(1/2),x)

[Out]

Integral(sqrt(-coth(x)**2 - 1), x)

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Giac [C]  time = 1.23223, size = 167, normalized size = 3.71 \begin{align*} -\frac{1}{2} \, \sqrt{2}{\left (i \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 2 \right |}}{2 \,{\left (\sqrt{2} + \sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}\right ) + i \, \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) - i \, \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - i \, \log \left (-\sqrt{e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-coth(x)^2)^(1/2),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*(I*sqrt(2)*log(1/2*abs(-2*sqrt(2) + 2*sqrt(e^(4*x) + 1) - 2*e^(2*x) + 2)/(sqrt(2) + sqrt(e^(4*x)
+ 1) - e^(2*x) + 1)) + I*log(sqrt(e^(4*x) + 1) - e^(2*x) + 1) - I*log(sqrt(e^(4*x) + 1) - e^(2*x)) - I*log(-sq
rt(e^(4*x) + 1) + e^(2*x) + 1))*sgn(-e^(2*x) + 1)