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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01555, antiderivative size = 36, 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, 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 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 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 8

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

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*}

Mathematica [A]  time = 0.0252097, 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])

________________________________________________________________________________________

Maple [A]  time = 0.163, size = 24, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( x \right ) \sin \left ( x \right ) -x}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a}{ \left ( \sin \left ( x \right ) \right ) ^{4}}}}}}} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.49717, size = 34, normalized size = 0.94 \begin{align*} \frac{x}{2 \, \sqrt{a}} - \frac{\tan \left (x\right )}{2 \,{\left (\sqrt{a} \tan \left (x\right )^{2} + \sqrt{a}\right )}} \end{align*}

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))

________________________________________________________________________________________

Fricas [A]  time = 0.488808, size = 122, normalized size = 3.39 \begin{align*} -\frac{{\left (x \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sin \left (x\right ) - x\right )} \sqrt{\frac{a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{2 \, a} \end{align*}

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \csc ^{4}{\left (x \right )}}}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError