3.1.29 \(\int \sqrt {-1-\coth ^2(x)} \, dx\) [29]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 = {3742, 399, 223, 209, 385} \begin {gather*} \text {ArcTan}\left (\frac {\coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )-\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-\coth ^2(x)-1}}\right ) \end {gather*}

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 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 223

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 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 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 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 \sqrt {-1-\coth ^2(x)} \, dx &=\text {Subst}\left (\int \frac {\sqrt {-1-x^2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \left (1-x^2\right )} \, dx,x,\coth (x)\right )\right )+\text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2}} \, dx,x,\coth (x)\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )\right )+\text {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*}

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Mathematica [A]
time = 0.03, size = 62, normalized size = 1.38 \begin {gather*} \frac {\sqrt {-1-\coth ^2(x)} \left (-\tanh ^{-1}\left (\frac {\cosh (x)}{\sqrt {\cosh (2 x)}}\right )+\sqrt {2} \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)}\right )\right ) \sinh (x)}{\sqrt {\cosh (2 x)}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(141\) vs. \(2(37)=74\).
time = 0.95, size = 142, normalized size = 3.16

method	result	size
derivativedivides	\(\frac {\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}{2}+\frac {\arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{2}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{2}-\frac {\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}{2}+\frac {\arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{2}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{2}\)	\(142\)
default	\(\frac {\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}{2}+\frac {\arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{2}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{2}-\frac {\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}{2}+\frac {\arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{2}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{2}\)	\(142\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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+2*coth(x))*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(co
th(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 = 0.00, 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-coth(x)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-coth(x)^2 - 1), x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.35, size = 226, normalized size = 5.02 \begin {gather*} -\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 (-4 \, {\left (i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}\right ) - \frac {1}{2} i \, \log \left (-4 \, {\left (-i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}\right ) \end {gather*}

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

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

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] Result contains complex when optimal does not.
time = 0.41, size = 124, normalized size = 2.76 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 1.30, size = 43, normalized size = 0.96 \begin {gather*} -\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {coth}\left (x\right )}{\sqrt {-{\mathrm {coth}\left (x\right )}^2-1}}\right )-\ln \left (\mathrm {coth}\left (x\right )-\sqrt {-{\mathrm {coth}\left (x\right )}^2-1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- coth(x)^2 - 1)^(1/2),x)

[Out]

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

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