3.27 \(\int \frac{1}{\sqrt{-1-\csc ^2(x)}} \, dx\)

Optimal. Leaf size=18 \[ -\tanh ^{-1}\left (\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right ) \]

[Out]

-ArcTanh[Cot[x]/Sqrt[-2 - Cot[x]^2]]

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Rubi [A]  time = 0.0197279, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4128, 377, 206} \[ -\tanh ^{-1}\left (\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 - Csc[x]^2],x]

[Out]

-ArcTanh[Cot[x]/Sqrt[-2 - Cot[x]^2]]

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

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]

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

Mathematica [B]  time = 0.0310491, size = 51, normalized size = 2.83 \[ -\frac{\sqrt{\cos (2 x)-3} \csc (x) \log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right )}{\sqrt{2} \sqrt{-\csc ^2(x)-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 - Csc[x]^2],x]

[Out]

-((Sqrt[-3 + Cos[2*x]]*Csc[x]*Log[Sqrt[2]*Cos[x] + Sqrt[-3 + Cos[2*x]]])/(Sqrt[2]*Sqrt[-1 - Csc[x]^2]))

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Maple [B]  time = 0.157, size = 75, normalized size = 4.2 \begin{align*} -{\frac{\sin \left ( x \right ) }{-1+\cos \left ( x \right ) }\sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}}{\it Artanh} \left ({\frac{\cos \left ( x \right ) \sqrt{4} \left ( -1+\cos \left ( x \right ) \right ) }{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}}}}} \right ){\frac{1}{\sqrt{-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin(x)*((cos(x)^2-2)/(cos(x)+1)^2)^(1/2)*arctanh(1/2*cos(x)*4^(1/2)*(-1+cos(x))/sin(x)^2/((cos(x)^2-2)/(cos(x
)+1)^2)^(1/2))/(-(cos(x)^2-2)/(cos(x)^2-1))^(1/2)/(-1+cos(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-csc(x)^2 - 1), x)

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Fricas [C]  time = 0.466702, size = 211, normalized size = 11.72 \begin{align*} \frac{1}{2} \, \log \left (-2 \, \sqrt{e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1}{\left (e^{\left (2 i \, x\right )} - 1\right )} + 2 \, e^{\left (4 i \, x\right )} - 8 \, e^{\left (2 i \, x\right )} - 2\right ) - \frac{1}{2} \, \log \left (\sqrt{e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1} - e^{\left (2 i \, x\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(-2*sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1)*(e^(2*I*x) - 1) + 2*e^(4*I*x) - 8*e^(2*I*x) - 2) - 1/2*log(sqrt(e
^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-csc(x)**2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-csc(x)^2 - 1), x)