3.55 \(\int (a \csc ^3(x))^{5/2} \, dx\)

Optimal. Leaf size=123 \[ -\frac{2}{13} a^2 \cot (x) \csc ^4(x) \sqrt{a \csc ^3(x)}-\frac{22}{117} a^2 \cot (x) \csc ^2(x) \sqrt{a \csc ^3(x)}-\frac{154}{585} a^2 \cot (x) \sqrt{a \csc ^3(x)}-\frac{154}{195} a^2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)}+\frac{154}{195} a^2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \csc ^3(x)} \]

[Out]

(-154*a^2*Cot[x]*Sqrt[a*Csc[x]^3])/585 - (22*a^2*Cot[x]*Csc[x]^2*Sqrt[a*Csc[x]^3])/117 - (2*a^2*Cot[x]*Csc[x]^
4*Sqrt[a*Csc[x]^3])/13 - (154*a^2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x])/195 + (154*a^2*Sqrt[a*Csc[x]^3]*EllipticE[Pi
/4 - x/2, 2]*Sin[x]^(3/2))/195

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Rubi [A]  time = 0.0595877, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3768, 3771, 2639} \[ -\frac{2}{13} a^2 \cot (x) \csc ^4(x) \sqrt{a \csc ^3(x)}-\frac{22}{117} a^2 \cot (x) \csc ^2(x) \sqrt{a \csc ^3(x)}-\frac{154}{585} a^2 \cot (x) \sqrt{a \csc ^3(x)}-\frac{154}{195} a^2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)}+\frac{154}{195} a^2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \csc ^3(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csc[x]^3)^(5/2),x]

[Out]

(-154*a^2*Cot[x]*Sqrt[a*Csc[x]^3])/585 - (22*a^2*Cot[x]*Csc[x]^2*Sqrt[a*Csc[x]^3])/117 - (2*a^2*Cot[x]*Csc[x]^
4*Sqrt[a*Csc[x]^3])/13 - (154*a^2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x])/195 + (154*a^2*Sqrt[a*Csc[x]^3]*EllipticE[Pi
/4 - x/2, 2]*Sin[x]^(3/2))/195

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 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 \left (a \csc ^3(x)\right )^{5/2} \, dx &=\frac{\left (a^2 \sqrt{a \csc ^3(x)}\right ) \int (-\csc (x))^{15/2} \, dx}{(-\csc (x))^{3/2}}\\ &=-\frac{2}{13} a^2 \cot (x) \csc ^4(x) \sqrt{a \csc ^3(x)}+\frac{\left (11 a^2 \sqrt{a \csc ^3(x)}\right ) \int (-\csc (x))^{11/2} \, dx}{13 (-\csc (x))^{3/2}}\\ &=-\frac{22}{117} a^2 \cot (x) \csc ^2(x) \sqrt{a \csc ^3(x)}-\frac{2}{13} a^2 \cot (x) \csc ^4(x) \sqrt{a \csc ^3(x)}+\frac{\left (77 a^2 \sqrt{a \csc ^3(x)}\right ) \int (-\csc (x))^{7/2} \, dx}{117 (-\csc (x))^{3/2}}\\ &=-\frac{154}{585} a^2 \cot (x) \sqrt{a \csc ^3(x)}-\frac{22}{117} a^2 \cot (x) \csc ^2(x) \sqrt{a \csc ^3(x)}-\frac{2}{13} a^2 \cot (x) \csc ^4(x) \sqrt{a \csc ^3(x)}+\frac{\left (77 a^2 \sqrt{a \csc ^3(x)}\right ) \int (-\csc (x))^{3/2} \, dx}{195 (-\csc (x))^{3/2}}\\ &=-\frac{154}{585} a^2 \cot (x) \sqrt{a \csc ^3(x)}-\frac{22}{117} a^2 \cot (x) \csc ^2(x) \sqrt{a \csc ^3(x)}-\frac{2}{13} a^2 \cot (x) \csc ^4(x) \sqrt{a \csc ^3(x)}-\frac{154}{195} a^2 \cos (x) \sqrt{a \csc ^3(x)} \sin (x)-\frac{\left (77 a^2 \sqrt{a \csc ^3(x)}\right ) \int \frac{1}{\sqrt{-\csc (x)}} \, dx}{195 (-\csc (x))^{3/2}}\\ &=-\frac{154}{585} a^2 \cot (x) \sqrt{a \csc ^3(x)}-\frac{22}{117} a^2 \cot (x) \csc ^2(x) \sqrt{a \csc ^3(x)}-\frac{2}{13} a^2 \cot (x) \csc ^4(x) \sqrt{a \csc ^3(x)}-\frac{154}{195} a^2 \cos (x) \sqrt{a \csc ^3(x)} \sin (x)-\frac{1}{195} \left (77 a^2 \sqrt{a \csc ^3(x)} \sin ^{\frac{3}{2}}(x)\right ) \int \sqrt{\sin (x)} \, dx\\ &=-\frac{154}{585} a^2 \cot (x) \sqrt{a \csc ^3(x)}-\frac{22}{117} a^2 \cot (x) \csc ^2(x) \sqrt{a \csc ^3(x)}-\frac{2}{13} a^2 \cot (x) \csc ^4(x) \sqrt{a \csc ^3(x)}-\frac{154}{195} a^2 \cos (x) \sqrt{a \csc ^3(x)} \sin (x)+\frac{154}{195} a^2 \sqrt{a \csc ^3(x)} E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sin ^{\frac{3}{2}}(x)\\ \end{align*}

Mathematica [A]  time = 0.189496, size = 58, normalized size = 0.47 \[ \frac{\left (a \csc ^3(x)\right )^{5/2} \left (-9414 \sin (2 x)+5346 \sin (4 x)-1694 \sin (6 x)+231 \sin (8 x)+29568 \sin ^{\frac{15}{2}}(x) E\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right )\right )}{37440} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csc[x]^3)^(5/2),x]

[Out]

((a*Csc[x]^3)^(5/2)*(29568*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(15/2) - 9414*Sin[2*x] + 5346*Sin[4*x] - 1694*Sin
[6*x] + 231*Sin[8*x]))/37440

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Maple [C]  time = 0.222, size = 1313, normalized size = 10.7 \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)^(5/2),x)

[Out]

-1/9360*8^(1/2)*(462*cos(x)^7*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*Ellipt
icE(((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)-231*cos(x)^7*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))+462*cos(x)^6*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)-231*cos(x)^6*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))-1386*cos(x)^5*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)+693*cos(x)^5*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))-13
86*cos(x)^4*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)+693*cos(x)^4*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))+1386*cos(x)^3*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)-693*cos(x)
^3*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))+1386*cos(x)^2*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)-693*cos(x)^2*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))-462*cos(x)^6-462*co
s(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)+231*cos(x)*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))+154*cos(x)^5-462*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)+231*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))+1386*cos(x)^4-418*cos(x)^3-1386*cos(x)^2+354*cos(x)+462)*sin(x)*(
-2*a/sin(x)/(cos(x)^2-1))^(5/2)

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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{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*csc(x)^3)^(5/2), 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^{2} \csc \left (x\right )^{6}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^3)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)**3)**(5/2),x)

[Out]

Timed out

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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{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*csc(x)^3)^(5/2), x)