3.59 \(\int \frac{1}{(a \csc ^3(x))^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{14 \cos (x)}{45 a \sqrt{a \csc ^3(x)}}-\frac{2 \sin ^2(x) \cos (x)}{9 a \sqrt{a \csc ^3(x)}}-\frac{14 E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{15 a \sin ^{\frac{3}{2}}(x) \sqrt{a \csc ^3(x)}} \]

[Out]

(-14*Cos[x])/(45*a*Sqrt[a*Csc[x]^3]) - (14*EllipticE[Pi/4 - x/2, 2])/(15*a*Sqrt[a*Csc[x]^3]*Sin[x]^(3/2)) - (2
*Cos[x]*Sin[x]^2)/(9*a*Sqrt[a*Csc[x]^3])

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Rubi [A]  time = 0.0383605, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2639} \[ -\frac{14 \cos (x)}{45 a \sqrt{a \csc ^3(x)}}-\frac{2 \sin ^2(x) \cos (x)}{9 a \sqrt{a \csc ^3(x)}}-\frac{14 E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{15 a \sin ^{\frac{3}{2}}(x) \sqrt{a \csc ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csc[x]^3)^(-3/2),x]

[Out]

(-14*Cos[x])/(45*a*Sqrt[a*Csc[x]^3]) - (14*EllipticE[Pi/4 - x/2, 2])/(15*a*Sqrt[a*Csc[x]^3]*Sin[x]^(3/2)) - (2
*Cos[x]*Sin[x]^2)/(9*a*Sqrt[a*Csc[x]^3])

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 3769

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx &=-\frac{(-\csc (x))^{3/2} \int \frac{1}{(-\csc (x))^{9/2}} \, dx}{a \sqrt{a \csc ^3(x)}}\\ &=-\frac{2 \cos (x) \sin ^2(x)}{9 a \sqrt{a \csc ^3(x)}}-\frac{\left (7 (-\csc (x))^{3/2}\right ) \int \frac{1}{(-\csc (x))^{5/2}} \, dx}{9 a \sqrt{a \csc ^3(x)}}\\ &=-\frac{14 \cos (x)}{45 a \sqrt{a \csc ^3(x)}}-\frac{2 \cos (x) \sin ^2(x)}{9 a \sqrt{a \csc ^3(x)}}-\frac{\left (7 (-\csc (x))^{3/2}\right ) \int \frac{1}{\sqrt{-\csc (x)}} \, dx}{15 a \sqrt{a \csc ^3(x)}}\\ &=-\frac{14 \cos (x)}{45 a \sqrt{a \csc ^3(x)}}-\frac{2 \cos (x) \sin ^2(x)}{9 a \sqrt{a \csc ^3(x)}}+\frac{7 \int \sqrt{\sin (x)} \, dx}{15 a \sqrt{a \csc ^3(x)} \sin ^{\frac{3}{2}}(x)}\\ &=-\frac{14 \cos (x)}{45 a \sqrt{a \csc ^3(x)}}-\frac{14 E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{15 a \sqrt{a \csc ^3(x)} \sin ^{\frac{3}{2}}(x)}-\frac{2 \cos (x) \sin ^2(x)}{9 a \sqrt{a \csc ^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0954014, size = 52, normalized size = 0.66 \[ \frac{\sin ^{\frac{3}{2}}(x) (5 \cos (3 x)-33 \cos (x))-84 E\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right )}{90 \sin ^{\frac{9}{2}}(x) \left (a \csc ^3(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csc[x]^3)^(-3/2),x]

[Out]

(-84*EllipticE[(Pi - 2*x)/4, 2] + (-33*Cos[x] + 5*Cos[3*x])*Sin[x]^(3/2))/(90*(a*Csc[x]^3)^(3/2)*Sin[x]^(9/2))

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Maple [C]  time = 0.235, size = 349, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csc(x)^3)^(3/2),x)

[Out]

-1/45*8^(1/2)*(42*cos(x)*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE((
(I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)-21*cos(x)*2^(1/2)*(-I*(-1+co
s(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*EllipticF(((I*cos(x
)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))+10*cos(x)^5+42*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)
+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)
-21*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/
2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-34*cos(x)^3+66*cos(x)-42)/(-2*a/sin(x)/(cos(x)^2-
1))^(3/2)/sin(x)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \csc \left (x\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^3)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*csc(x)^3)^(-3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \csc \left (x\right )^{3}}}{a^{2} \csc \left (x\right )^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^3)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*csc(x)^3)/(a^2*csc(x)^6), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \csc ^{3}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)**3)**(3/2),x)

[Out]

Integral((a*csc(x)**3)**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \csc \left (x\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^3)^(3/2),x, algorithm="giac")

[Out]

integrate((a*csc(x)^3)^(-3/2), x)