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

Optimal. Leaf size=33 \[ \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right )+\tanh ^{-1}\left (\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right ) \]

[Out]

ArcTan[Cot[x]/Sqrt[-2 - Cot[x]^2]] + ArcTanh[Cot[x]/Sqrt[-2 - Cot[x]^2]]

________________________________________________________________________________________

Rubi [A]  time = 0.0248605, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4128, 402, 217, 203, 377, 206} \[ \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right )+\tanh ^{-1}\left (\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - Csc[x]^2],x]

[Out]

ArcTan[Cot[x]/Sqrt[-2 - Cot[x]^2]] + 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 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]

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 \sqrt{-1-\csc ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{-2-x^2}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{-2-x^2}} \, dx,x,\cot (x)\right )+\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 )+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\cot (x)}{\sqrt{-2-\cot ^2(x)}}\right )\\ &=\tan ^{-1}\left (\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 = 0.0383694, size = 70, normalized size = 2.12 \[ \frac{\sqrt{2} \sin (x) \sqrt{-\csc ^2(x)-1} \left (\log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right )+\tan ^{-1}\left (\frac{\sqrt{2} \cos (x)}{\sqrt{\cos (2 x)-3}}\right )\right )}{\sqrt{\cos (2 x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 - Csc[x]^2],x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.187, size = 139, normalized size = 4.2 \begin{align*}{\frac{\sqrt{4} \left ( -1+\cos \left ( x \right ) \right ) \left ( \cos \left ( x \right ) +1 \right ) ^{2}}{ \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}-8 \right ) \sin \left ( x \right ) }\sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \sin \left ( x \right ) \right ) ^{2}}}} \left ( \arcsin \left ({\frac{\sqrt{2} \left ( \cos \left ( x \right ) +2 \right ) }{2\,\cos \left ( x \right ) +2}} \right ) +2\,{\it Artanh} \left ( 1/2\,{\frac{\cos \left ( x \right ) \sqrt{4} \left ( -1+\cos \left ( x \right ) \right ) }{ \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 ) -\arctan \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-3\,\cos \left ( x \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 ) \right ) \sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*4^(1/2)*((cos(x)^2-2)/sin(x)^2)^(1/2)*(-1+cos(x))*(arcsin(1/2*2^(1/2)*(cos(x)+2)/(cos(x)+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))-arctan((cos(x)^2-3*cos(x)+2)/((cos(x)^
2-2)/(cos(x)+1)^2)^(1/2)/sin(x)^2))*((cos(x)^2-2)/(cos(x)+1)^2)^(1/2)*(cos(x)+1)^2/(cos(x)^2-2)/sin(x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-csc(x)^2 - 1), x)

________________________________________________________________________________________

Fricas [C]  time = 0.475077, size = 387, normalized size = 11.73 \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 ) - i \, \log \left (\sqrt{e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1} - e^{\left (2 i \, x\right )} + 2 i + 1\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 ) + i \, \log \left (\sqrt{e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1} - e^{\left (2 i \, x\right )} - 2 i + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-csc(x)**2 - 1), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-csc(x)^2 - 1), x)