\(\int \sqrt {a \csc ^4(x)} \, dx\) [64]

Optimal result

Integrand size = 10, antiderivative size = 16 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\cos (x) \sqrt {a \csc ^4(x)} \sin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 3852, 8} \[ \int \sqrt {a \csc ^4(x)} \, dx=\sin (x) (-\cos (x)) \sqrt {a \csc ^4(x)} \]

[In]

Int[Sqrt[a*Csc[x]^4],x]

[Out]

-(Cos[x]*Sqrt[a*Csc[x]^4]*Sin[x])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \int \csc ^2(x) \, dx \\ & = -\left (\left (\sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \text {Subst}(\int 1 \, dx,x,\cot (x))\right ) \\ & = -\cos (x) \sqrt {a \csc ^4(x)} \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\cos (x) \sqrt {a \csc ^4(x)} \sin (x) \]

[In]

Integrate[Sqrt[a*Csc[x]^4],x]

[Out]

-(Cos[x]*Sqrt[a*Csc[x]^4]*Sin[x])

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12

method	result	size
default	\(-\frac {\cos \left (x \right ) \sin \left (x \right ) \sqrt {a \csc \left (x \right )^{4}}\, \sqrt {16}}{4}\)	\(18\)
risch	\(2 i \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left (1-{\mathrm e}^{-2 i x}\right )\)	\(31\)

[In]

int((a*csc(x)^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*cos(x)*sin(x)*(a*csc(x)^4)^(1/2)*16^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}} \cos \left (x\right ) \sin \left (x\right ) \]

[In]

integrate((a*csc(x)^4)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \sqrt {a \csc ^4(x)} \, dx=\int \sqrt {a \csc ^{4}{\left (x \right )}}\, dx \]

[In]

integrate((a*csc(x)**4)**(1/2),x)

[Out]

Integral(sqrt(a*csc(x)**4), x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\frac {\sqrt {a}}{\tan \left (x\right )} \]

[In]

integrate((a*csc(x)^4)^(1/2),x, algorithm="maxima")

[Out]

-sqrt(a)/tan(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\frac {\sqrt {a}}{\tan \left (x\right )} \]

[In]

integrate((a*csc(x)^4)^(1/2),x, algorithm="giac")

[Out]

-sqrt(a)/tan(x)

Mupad [B] (verification not implemented)

Time = 20.74 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.44 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\sqrt {a}\,\mathrm {cot}\left (x\right ) \]

[In]

int((a/sin(x)^4)^(1/2),x)

[Out]

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