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

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 8

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

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \sin (x) (-\cos (x)) \sqrt {a \csc ^4(x)} \]

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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fricas [A]  time = 0.42, size = 24, normalized size = 1.50 \[ -\sqrt {\frac {a}{\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1}} \cos \relax (x) \sin \relax (x) \]

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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giac [A]  time = 0.25, size = 9, normalized size = 0.56 \[ -\frac {\sqrt {a}}{\tan \relax (x)} \]

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)

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maple [A]  time = 0.63, size = 18, normalized size = 1.12 \[ -\frac {\sin \relax (x ) \cos \relax (x ) \sqrt {\frac {a}{\sin \relax (x )^{4}}}\, \sqrt {16}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 9, normalized size = 0.56 \[ -\frac {\sqrt {a}}{\tan \relax (x)} \]

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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mupad [B]  time = 0.18, size = 7, normalized size = 0.44 \[ -\sqrt {a}\,\mathrm {cot}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \csc ^{4}{\relax (x )}}\, dx \]

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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