3.65 \(\int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx\)

Optimal. Leaf size=36 \[ \frac {x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}}-\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}} \]

[Out]

-1/2*cot(x)/(a*csc(x)^4)^(1/2)+1/2*x*csc(x)^2/(a*csc(x)^4)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 36, 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, 2635, 8} \[ \frac {x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}}-\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Csc[x]^4],x]

[Out]

-Cot[x]/(2*Sqrt[a*Csc[x]^4]) + (x*Csc[x]^2)/(2*Sqrt[a*Csc[x]^4])

Rule 8

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

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 \frac {1}{\sqrt {a \csc ^4(x)}} \, dx &=\frac {\csc ^2(x) \int \sin ^2(x) \, dx}{\sqrt {a \csc ^4(x)}}\\ &=-\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}}+\frac {\csc ^2(x) \int 1 \, dx}{2 \sqrt {a \csc ^4(x)}}\\ &=-\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}}+\frac {x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 25, normalized size = 0.69 \[ \frac {x \csc ^2(x)-\cot (x)}{2 \sqrt {a \csc ^4(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Csc[x]^4],x]

[Out]

(-Cot[x] + x*Csc[x]^2)/(2*Sqrt[a*Csc[x]^4])

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fricas [A]  time = 0.55, size = 46, normalized size = 1.28 \[ -\frac {{\left (x \cos \relax (x)^{2} - {\left (\cos \relax (x)^{3} - \cos \relax (x)\right )} \sin \relax (x) - x\right )} \sqrt {\frac {a}{\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1}}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x*cos(x)^2 - (cos(x)^3 - cos(x))*sin(x) - x)*sqrt(a/(cos(x)^4 - 2*cos(x)^2 + 1))/a

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giac [A]  time = 0.41, size = 22, normalized size = 0.61 \[ \frac {x}{2 \, \sqrt {a}} - \frac {\tan \relax (x)}{2 \, {\left (\tan \relax (x)^{2} + 1\right )} \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/sqrt(a) - 1/2*tan(x)/((tan(x)^2 + 1)*sqrt(a))

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maple [A]  time = 0.74, size = 24, normalized size = 0.67 \[ -\frac {\cos \relax (x ) \sin \relax (x )-x}{2 \sin \relax (x )^{2} \sqrt {\frac {a}{\sin \relax (x )^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.47, size = 25, normalized size = 0.69 \[ \frac {x}{2 \, \sqrt {a}} - \frac {\tan \relax (x)}{2 \, {\left (\sqrt {a} \tan \relax (x)^{2} + \sqrt {a}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/sqrt(a) - 1/2*tan(x)/(sqrt(a)*tan(x)^2 + sqrt(a))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {\frac {a}{{\sin \relax (x)}^4}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a/sin(x)^4)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \csc ^{4}{\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*csc(x)**4), x)

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