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]

-arctanh(cot(x)*a^(1/2)/(a*csc(x)^2)^(1/2))*a^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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])

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 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
]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.55, size = 64, normalized size = 2.46 \[ \left [\frac {1}{2} \, \sqrt {-\frac {a}{\cos \relax (x)^{2} - 1}} \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right ) \sin \relax (x), \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {-\frac {a}{\cos \relax (x)^{2} - 1}} \cos \relax (x) \sin \relax (x)}{a}\right )\right ] \]

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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giac [A]  time = 0.56, size = 13, normalized size = 0.50 \[ \sqrt {a} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \mathrm {sgn}\left (\sin \relax (x)\right ) \]

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))

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maple [A]  time = 0.68, size = 32, normalized size = 1.23 \[ \frac {\sin \relax (x ) \sqrt {-\frac {a}{-1+\cos ^{2}\relax (x )}}\, \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right ) \sqrt {4}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.79, size = 24, normalized size = 0.92 \[ -\sqrt {-a} {\left (\arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) - \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {\frac {a}{{\sin \relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/sin(x)^2)^(1/2),x)

[Out]

int((a/sin(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \csc ^{2}{\relax (x )}}\, dx \]

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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