3.56 \(\int (a \csc ^3(x))^{3/2} \, dx\)

Optimal. Leaf size=71 \[ -\frac{10}{21} a \sin ^{\frac{3}{2}}(x) \text{EllipticF}\left (\frac{\pi }{4}-\frac{x}{2},2\right ) \sqrt{a \csc ^3(x)}-\frac{10}{21} a \cos (x) \sqrt{a \csc ^3(x)}-\frac{2}{7} a \cot (x) \csc (x) \sqrt{a \csc ^3(x)} \]

[Out]

(-10*a*Cos[x]*Sqrt[a*Csc[x]^3])/21 - (2*a*Cot[x]*Csc[x]*Sqrt[a*Csc[x]^3])/7 - (10*a*Sqrt[a*Csc[x]^3]*EllipticF
[Pi/4 - x/2, 2]*Sin[x]^(3/2))/21

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Rubi [A]  time = 0.0375206, antiderivative size = 71, 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, 3768, 3771, 2641} \[ -\frac{10}{21} a \cos (x) \sqrt{a \csc ^3(x)}-\frac{2}{7} a \cot (x) \csc (x) \sqrt{a \csc ^3(x)}-\frac{10}{21} a \sin ^{\frac{3}{2}}(x) F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \csc ^3(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*a*Cos[x]*Sqrt[a*Csc[x]^3])/21 - (2*a*Cot[x]*Csc[x]*Sqrt[a*Csc[x]^3])/7 - (10*a*Sqrt[a*Csc[x]^3]*EllipticF
[Pi/4 - x/2, 2]*Sin[x]^(3/2))/21

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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[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 2641

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

Rubi steps

\begin{align*} \int \left (a \csc ^3(x)\right )^{3/2} \, dx &=-\frac{\left (a \sqrt{a \csc ^3(x)}\right ) \int (-\csc (x))^{9/2} \, dx}{(-\csc (x))^{3/2}}\\ &=-\frac{2}{7} a \cot (x) \csc (x) \sqrt{a \csc ^3(x)}-\frac{\left (5 a \sqrt{a \csc ^3(x)}\right ) \int (-\csc (x))^{5/2} \, dx}{7 (-\csc (x))^{3/2}}\\ &=-\frac{10}{21} a \cos (x) \sqrt{a \csc ^3(x)}-\frac{2}{7} a \cot (x) \csc (x) \sqrt{a \csc ^3(x)}-\frac{\left (5 a \sqrt{a \csc ^3(x)}\right ) \int \sqrt{-\csc (x)} \, dx}{21 (-\csc (x))^{3/2}}\\ &=-\frac{10}{21} a \cos (x) \sqrt{a \csc ^3(x)}-\frac{2}{7} a \cot (x) \csc (x) \sqrt{a \csc ^3(x)}+\frac{1}{21} \left (5 a \sqrt{a \csc ^3(x)} \sin ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sqrt{\sin (x)}} \, dx\\ &=-\frac{10}{21} a \cos (x) \sqrt{a \csc ^3(x)}-\frac{2}{7} a \cot (x) \csc (x) \sqrt{a \csc ^3(x)}-\frac{10}{21} a \sqrt{a \csc ^3(x)} F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sin ^{\frac{3}{2}}(x)\\ \end{align*}

Mathematica [A]  time = 0.113459, size = 46, normalized size = 0.65 \[ -\frac{1}{84} \left (a \csc ^3(x)\right )^{3/2} \left (40 \sin ^{\frac{9}{2}}(x) \text{EllipticF}\left (\frac{1}{4} (\pi -2 x),2\right )+22 \sin (2 x)-5 \sin (4 x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Csc[x]^3)^(3/2)*(40*EllipticF[(Pi - 2*x)/4, 2]*Sin[x]^(9/2) + 22*Sin[2*x] - 5*Sin[4*x]))/84

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Maple [C]  time = 0.319, size = 372, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/168*8^(1/2)*(cos(x)+1)^2*(-1+cos(x))^2*(5*I*sin(x)*cos(x)^3*EllipticF(((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)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2
)*2^(1/2)+5*I*sin(x)*cos(x)^2*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))*((I*cos(x)+sin(x)-I)/s
in(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*2^(1/2)-5*I*sin(x)*cos(x)*Ellip
ticF(((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)*((-I*cos(x)+sin(x)+I)/
sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*2^(1/2)-5*I*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*2^(1/2)*((-I*cos(x)
+sin(x)+I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2
))*sin(x)-10*cos(x)^3+16*cos(x))*(-2*a/sin(x)/(cos(x)^2-1))^(3/2)/sin(x)^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*csc(x)^3)*a*csc(x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((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 \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((a*csc(x)^3)^(3/2),x, algorithm="giac")

[Out]

integrate((a*csc(x)^3)^(3/2), x)