3.1.49 \(\int \frac {1}{\sqrt {-1-\coth ^2(x)}} \, dx\) [49]

Optimal. Leaf size=27 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )}{\sqrt {2}} \]

[Out]

1/2*arctan(coth(x)*2^(1/2)/(-1-coth(x)^2)^(1/2))*2^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3742, 385, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 - Coth[x]^2],x]

[Out]

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

Rule 209

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

Rule 385

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]

Rule 3742

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])

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 46, normalized size = 1.70 \begin {gather*} \frac {\sqrt {\cosh (2 x)} \text {csch}(x) \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)}\right )}{\sqrt {2} \sqrt {-1-\coth ^2(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 - Coth[x]^2],x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(22)=44\).
time = 1.04, size = 66, normalized size = 2.44

method	result	size
derivativedivides	\(\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{4}\)	\(66\)
default	\(\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{4}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-1-coth(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*2^(1/2)*arctan(1/4*(-2+2*coth(x))*2^(1/2)/(-(1+coth(x))^2+2*coth(x))^(1/2))-1/4*2^(1/2)*arctan(1/4*(-2-2*c
oth(x))*2^(1/2)/(-(coth(x)-1)^2-2*coth(x))^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-coth(x)^2 - 1), x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.37, size = 175, normalized size = 6.48 \begin {gather*} \frac {1}{8} i \, \sqrt {2} \log \left (\frac {1}{2} \, {\left (i \, \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}{8} i \, \sqrt {2} \log \left (\frac {1}{2} \, {\left (-i \, \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}{8} i \, \sqrt {2} \log \left ({\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} + i \, \sqrt {2} e^{\left (4 \, x\right )} + i \, \sqrt {2} e^{\left (2 \, x\right )} + 2 i \, \sqrt {2}\right )} e^{\left (-4 \, x\right )}\right ) + \frac {1}{8} i \, \sqrt {2} \log \left ({\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} - i \, \sqrt {2} e^{\left (4 \, x\right )} - i \, \sqrt {2} e^{\left (2 \, x\right )} - 2 i \, \sqrt {2}\right )} e^{\left (-4 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*I*sqrt(2)*log(1/2*(I*sqrt(2)*sqrt(-2*e^(4*x) - 2) + 2*e^(2*x) - 2)*e^(-2*x)) - 1/8*I*sqrt(2)*log(1/2*(-I*s
qrt(2)*sqrt(-2*e^(4*x) - 2) + 2*e^(2*x) - 2)*e^(-2*x)) - 1/8*I*sqrt(2)*log((sqrt(-2*e^(4*x) - 2)*(e^(2*x) + 2)
+ I*sqrt(2)*e^(4*x) + I*sqrt(2)*e^(2*x) + 2*I*sqrt(2))*e^(-4*x)) + 1/8*I*sqrt(2)*log((sqrt(-2*e^(4*x) - 2)*(e
^(2*x) + 2) - I*sqrt(2)*e^(4*x) - I*sqrt(2)*e^(2*x) - 2*I*sqrt(2))*e^(-4*x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \coth ^{2}{\left (x \right )} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-coth(x)**2 - 1), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.39, size = 73, normalized size = 2.70 \begin {gather*} -\frac {\sqrt {2} {\left (-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 )}}{4 \, \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.25, size = 22, normalized size = 0.81 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {coth}\left (x\right )}{\sqrt {-{\mathrm {coth}\left (x\right )}^2-1}}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(- coth(x)^2 - 1)^(1/2),x)

[Out]

(2^(1/2)*atan((2^(1/2)*coth(x))/(- coth(x)^2 - 1)^(1/2)))/2

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