\(\int \frac {1}{\csc ^2(x)^{3/2}} \, dx\) [44]

Optimal result

Integrand size = 8, antiderivative size = 29 \[ \int \frac {1}{\csc ^2(x)^{3/2}} \, dx=-\frac {\cot (x)}{3 \csc ^2(x)^{3/2}}-\frac {2 \cot (x)}{3 \sqrt {\csc ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4207, 198, 197} \[ \int \frac {1}{\csc ^2(x)^{3/2}} \, dx=-\frac {2 \cot (x)}{3 \sqrt {\csc ^2(x)}}-\frac {\cot (x)}{3 \csc ^2(x)^{3/2}} \]

[In]

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

[Out]

-1/3*Cot[x]/(Csc[x]^2)^(3/2) - (2*Cot[x])/(3*Sqrt[Csc[x]^2])

Rule 197

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 198

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 4207

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*} \text {integral}& = -\text {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} \text {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*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\csc ^2(x)^{3/2}} \, dx=\frac {(-9 \cos (x)+\cos (3 x)) \csc (x)}{12 \sqrt {\csc ^2(x)}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {\sin \left (x \right )^{2} \operatorname {csgn}\left (\csc \left (x \right )\right ) \left (-2+\cos \left (x \right )^{2}-\cos \left (x \right )\right ) \sqrt {4}}{6 \left (\cos \left (x \right )-1\right )}\)	\(29\)
risch	\(\frac {i {\mathrm e}^{4 i x}}{24 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {3 i {\mathrm e}^{2 i x}}{8 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {3 i}{8 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {i {\mathrm e}^{-2 i x}}{24 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\)	\(137\)

[In]

int(1/(csc(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\csc ^2(x)^{3/2}} \, dx=\frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

[In]

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

[Out]

1/3*cos(x)^3 - cos(x)

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\csc ^2(x)^{3/2}} \, dx=- \frac {2 \cot ^{3}{\left (x \right )}}{3 \left (\csc ^{2}{\left (x \right )}\right )^{\frac {3}{2}}} - \frac {\cot {\left (x \right )}}{\left (\csc ^{2}{\left (x \right )}\right )^{\frac {3}{2}}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\csc ^2(x)^{3/2}} \, dx=\frac {1}{12} \, \cos \left (3 \, x\right ) - \frac {3}{4} \, \cos \left (x\right ) \]

[In]

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

[Out]

1/12*cos(3*x) - 3/4*cos(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (21) = 42\).

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\csc ^2(x)^{3/2}} \, dx=-\frac {4 \, {\left (\frac {3 \, {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \mathrm {sgn}\left (\sin \left (x\right )\right )\right )}}{3 \, {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{3}} + \frac {4}{3} \, \mathrm {sgn}\left (\sin \left (x\right )\right ) \]

[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
))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\csc ^2(x)^{3/2}} \, dx=\int \frac {1}{{\left (\frac {1}{{\sin \left (x\right )}^2}\right )}^{3/2}} \,d x \]

[In]

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

[Out]

int(1/(1/sin(x)^2)^(3/2), x)