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

Optimal. Leaf size=29 \[ -\frac {2 \cot (x)}{3 \sqrt {\csc ^2(x)}}-\frac {\cot (x)}{3 \csc ^2(x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4122, 192, 191} \[ -\frac {2 \cot (x)}{3 \sqrt {\csc ^2(x)}}-\frac {\cot (x)}{3 \csc ^2(x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.79 \[ \frac {(\cos (3 x)-9 \cos (x)) \csc (x)}{12 \sqrt {\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.57, size = 11, normalized size = 0.38 \[ \frac {1}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*cos(x)^3 - cos(x)

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giac [B]  time = 0.42, size = 44, normalized size = 1.52 \[ -\frac {4 \, {\left (\frac {3 \, {\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{\cos \relax (x) + 1} - \mathrm {sgn}\left (\sin \relax (x)\right )\right )}}{3 \, {\left (\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 1\right )}^{3}} + \frac {4}{3} \, \mathrm {sgn}\left (\sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(3*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) - sgn(sin(x)))/((cos(x) - 1)/(cos(x) + 1) - 1)^3 + 4/3*sgn(sin(x
))

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maple [A]  time = 0.44, size = 30, normalized size = 1.03 \[ \frac {\sin \relax (x ) \left (\cos \relax (x )-2\right ) \sqrt {4}}{6 \left (-1+\cos \relax (x )\right )^{2} \left (-\frac {1}{-1+\cos ^{2}\relax (x )}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.52, size = 11, normalized size = 0.38 \[ \frac {1}{12} \, \cos \left (3 \, x\right ) - \frac {3}{4} \, \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*cos(3*x) - 3/4*cos(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(1/sin(x)^2)^(3/2), x)

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sympy [A]  time = 0.94, size = 29, normalized size = 1.00 \[ - \frac {2 \cot ^{3}{\relax (x )}}{3 \left (\csc ^{2}{\relax (x )}\right )^{\frac {3}{2}}} - \frac {\cot {\relax (x )}}{\left (\csc ^{2}{\relax (x )}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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