3.51 \(\int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx\)

Optimal. Leaf size=14 \[ -\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4122, 191} \[ -\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Csc[x]^2],x]

[Out]

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 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 \frac {1}{\sqrt {a \csc ^2(x)}} \, dx &=-\left (a \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\right )\\ &=-\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \[ -\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Csc[x]^2],x]

[Out]

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

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fricas [A]  time = 0.83, size = 22, normalized size = 1.57 \[ -\frac {\sqrt {-\frac {a}{\cos \relax (x)^{2} - 1}} \cos \relax (x) \sin \relax (x)}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-a/(cos(x)^2 - 1))*cos(x)*sin(x)/a

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giac [B]  time = 0.36, size = 34, normalized size = 2.43 \[ \frac {2 \, \mathrm {sgn}\left (\sin \relax (x)\right )}{\sqrt {a}} + \frac {2}{\sqrt {a} {\left (\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sgn(sin(x))/sqrt(a) + 2/(sqrt(a)*((cos(x) - 1)/(cos(x) + 1) - 1)*sgn(sin(x)))

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maple [B]  time = 0.62, size = 27, normalized size = 1.93 \[ \frac {\sin \relax (x ) \sqrt {4}}{2 \sqrt {-\frac {a}{-1+\cos ^{2}\relax (x )}}\, \left (-1+\cos \relax (x )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 13, normalized size = 0.93 \[ -\frac {1}{\sqrt {\tan \relax (x)^{2} + 1} \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(tan(x)^2 + 1)*sqrt(a))

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mupad [B]  time = 0.22, size = 15, normalized size = 1.07 \[ -\frac {\sin \left (2\,x\right )}{2\,\sqrt {a}\,\sqrt {{\sin \relax (x)}^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin(2*x)/(2*a^(1/2)*(sin(x)^2)^(1/2))

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sympy [A]  time = 0.45, size = 17, normalized size = 1.21 \[ - \frac {\cot {\relax (x )}}{\sqrt {a} \sqrt {\csc ^{2}{\relax (x )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cot(x)/(sqrt(a)*sqrt(csc(x)**2))

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