\(\int \frac {1}{\sqrt {\csc ^2(x)}} \, dx\) [43]

Optimal result

Integrand size = 8, antiderivative size = 12 \[ \int \frac {1}{\sqrt {\csc ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4207, 197} \[ \int \frac {1}{\sqrt {\csc ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]

[In]

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

[Out]

-(Cot[x]/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 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 )^{3/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\csc ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.52 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.58

method	result	size
default	\(\frac {\sin \left (x \right )^{2} \operatorname {csgn}\left (\csc \left (x \right )\right ) \sqrt {4}}{-2+2 \cos \left (x \right )}\)	\(19\)
risch	\(-\frac {i {\mathrm e}^{2 i x}}{2 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i}{2 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\)	\(67\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\sqrt {\csc ^2(x)}} \, dx=-\cos \left (x\right ) \]

[In]

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

[Out]

-cos(x)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\csc ^2(x)}} \, dx=- \frac {\cot {\left (x \right )}}{\sqrt {\csc ^{2}{\left (x \right )}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {\csc ^2(x)}} \, dx=-\frac {1}{\sqrt {\tan \left (x\right )^{2} + 1}} \]

[In]

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

[Out]

-1/sqrt(tan(x)^2 + 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {\csc ^2(x)}} \, dx=-\cos \left (x\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) + \mathrm {sgn}\left (\sin \left (x\right )\right ) \]

[In]

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

[Out]

-cos(x)*sgn(sin(x)) + sgn(sin(x))

Mupad [B] (verification not implemented)

Time = 17.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\csc ^2(x)}} \, dx=-\frac {\sin \left (2\,x\right )}{2\,\sqrt {{\sin \left (x\right )}^2}} \]

[In]

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

[Out]

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