3.50 \(\int \sqrt{a \csc ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ -\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right ) \]

[Out]

-(Sqrt[a]*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]])

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Rubi [A]  time = 0.0156433, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 217, 206} \[ -\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Csc[x]^2],x]

[Out]

-(Sqrt[a]*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]])

Rule 4122

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

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

Mathematica [A]  time = 0.0160915, size = 30, normalized size = 1.15 \[ \sin (x) \sqrt{a \csc ^2(x)} \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Csc[x]^2],x]

[Out]

Sqrt[a*Csc[x]^2]*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x]

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Maple [A]  time = 0.126, size = 32, normalized size = 1.2 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{2}\sqrt{-{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}}}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*csc(x)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.6668, size = 32, normalized size = 1.23 \begin{align*} -\sqrt{-a}{\left (\arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-a)*(arctan2(sin(x), cos(x) + 1) - arctan2(sin(x), cos(x) - 1))

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Fricas [A]  time = 0.494227, size = 188, normalized size = 7.23 \begin{align*} \left [\frac{1}{2} \, \sqrt{-\frac{a}{\cos \left (x\right )^{2} - 1}} \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \sin \left (x\right ), \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{-\frac{a}{\cos \left (x\right )^{2} - 1}} \cos \left (x\right ) \sin \left (x\right )}{a}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(-a/(cos(x)^2 - 1))*log(-(cos(x) - 1)/(cos(x) + 1))*sin(x), sqrt(-a)*arctan(sqrt(-a)*sqrt(-a/(cos(x)^
2 - 1))*cos(x)*sin(x)/a)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*csc(x)**2), x)

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Giac [A]  time = 1.23949, size = 18, normalized size = 0.69 \begin{align*} \sqrt{a} \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a)*log(abs(tan(1/2*x)))*sgn(sin(x))