3.52 \(\int \frac {1}{(a \csc ^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac {2 \cot (x)}{3 a \sqrt {a \csc ^2(x)}}-\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}} \]

[Out]

-1/3*cot(x)/(a*csc(x)^2)^(3/2)-2/3*cot(x)/a/(a*csc(x)^2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 36, 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, 192, 191} \[ -\frac {2 \cot (x)}{3 a \sqrt {a \csc ^2(x)}}-\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csc[x]^2)^(-3/2),x]

[Out]

-Cot[x]/(3*(a*Csc[x]^2)^(3/2)) - (2*Cot[x])/(3*a*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 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

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}{\left (a \csc ^2(x)\right )^{3/2}} \, dx &=-\left (a \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\right )\\ &=-\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}}-\frac {2 \cot (x)}{3 a \sqrt {a \csc ^2(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 0.75 \[ \frac {(\cos (3 x)-9 \cos (x)) \csc ^3(x)}{12 \left (a \csc ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csc[x]^2)^(-3/2),x]

[Out]

((-9*Cos[x] + Cos[3*x])*Csc[x]^3)/(12*(a*Csc[x]^2)^(3/2))

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fricas [A]  time = 0.53, size = 29, normalized size = 0.81 \[ \frac {{\left (\cos \relax (x)^{3} - 3 \, \cos \relax (x)\right )} \sqrt {-\frac {a}{\cos \relax (x)^{2} - 1}} \sin \relax (x)}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/3*(cos(x)^3 - 3*cos(x))*sqrt(-a/(cos(x)^2 - 1))*sin(x)/a^2

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giac [A]  time = 0.43, size = 43, normalized size = 1.19 \[ -\frac {4 \, {\left (\frac {3 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3}} - \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )\right )}}{3 \, a^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^2)^(3/2),x, algorithm="giac")

[Out]

-4/3*((3*sgn(tan(1/2*x))*tan(1/2*x)^2 + sgn(tan(1/2*x)))/(tan(1/2*x)^2 + 1)^3 - sgn(tan(1/2*x)))/a^(3/2)

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maple [A]  time = 0.41, size = 31, normalized size = 0.86 \[ \frac {\sin \relax (x ) \left (\cos \relax (x )-2\right ) \sqrt {4}}{6 \left (-1+\cos \relax (x )\right )^{2} \left (-\frac {a}{-1+\cos ^{2}\relax (x )}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csc(x)^2)^(3/2),x)

[Out]

1/6*sin(x)*(cos(x)-2)/(-1+cos(x))^2/(-1/(-1+cos(x)^2)*a)^(3/2)*4^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \csc \relax (x)^{2}\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*csc(x)^2)^(-3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left (\frac {a}{{\sin \relax (x)}^2}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a/sin(x)^2)^(3/2),x)

[Out]

int(1/(a/sin(x)^2)^(3/2), x)

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sympy [A]  time = 1.04, size = 39, normalized size = 1.08 \[ - \frac {2 \cot ^{3}{\relax (x )}}{3 a^{\frac {3}{2}} \left (\csc ^{2}{\relax (x )}\right )^{\frac {3}{2}}} - \frac {\cot {\relax (x )}}{a^{\frac {3}{2}} \left (\csc ^{2}{\relax (x )}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)**2)**(3/2),x)

[Out]

-2*cot(x)**3/(3*a**(3/2)*(csc(x)**2)**(3/2)) - cot(x)/(a**(3/2)*(csc(x)**2)**(3/2))

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