3.64 \(\int \sqrt{a \csc ^4(x)} \, dx\)

Optimal. Leaf size=16 \[ \sin (x) (-\cos (x)) \sqrt{a \csc ^4(x)} \]

[Out]

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

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Rubi [A]  time = 0.0147763, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4123, 3767, 8} \[ \sin (x) (-\cos (x)) \sqrt{a \csc ^4(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4123

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]

Rule 3767

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 8

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

Rubi steps

\begin{align*} \int \sqrt{a \csc ^4(x)} \, dx &=\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 ) \operatorname{Subst}(\int 1 \, dx,x,\cot (x))\right )\\ &=-\cos (x) \sqrt{a \csc ^4(x)} \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0060438, size = 16, normalized size = 1. \[ \sin (x) (-\cos (x)) \sqrt{a \csc ^4(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.138, size = 18, normalized size = 1.1 \begin{align*} -{\frac{\sqrt{16}\sin \left ( x \right ) \cos \left ( x \right ) }{4}\sqrt{{\frac{a}{ \left ( \sin \left ( x \right ) \right ) ^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.52784, size = 12, normalized size = 0.75 \begin{align*} -\frac{\sqrt{a}}{\tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a)/tan(x)

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Fricas [A]  time = 0.46573, size = 70, normalized size = 4.38 \begin{align*} -\sqrt{\frac{a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}} \cos \left (x\right ) \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \csc ^{4}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*csc(x)**4), x)

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Giac [A]  time = 1.35062, size = 12, normalized size = 0.75 \begin{align*} -\frac{\sqrt{a}}{\tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a)/tan(x)