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

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

[Out]

-154/585*a^2*cot(x)*(a*csc(x)^3)^(1/2)-22/117*a^2*cot(x)*csc(x)^2*(a*csc(x)^3)^(1/2)-2/13*a^2*cot(x)*csc(x)^4*
(a*csc(x)^3)^(1/2)-154/195*a^2*cos(x)*sin(x)*(a*csc(x)^3)^(1/2)+154/195*a^2*(sin(1/4*Pi+1/2*x)^2)^(1/2)/sin(1/
4*Pi+1/2*x)*EllipticE(cos(1/4*Pi+1/2*x),2^(1/2))*sin(x)^(3/2)*(a*csc(x)^3)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 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]

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 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 \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*}

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Mathematica [A]  time = 0.18, 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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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \csc \relax (x)^{3}} a^{2} \csc \relax (x)^{6}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \csc \relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]

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)

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maple [C]  time = 0.77, size = 1313, normalized size = 10.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*csc(x)^3)^(5/2),x)

[Out]

-1/9360*(462*cos(x)^7*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin
(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))-231*cos(x)^7*2^(1/2)*(-I*(-1+co
s(x))/sin(x))^(1/2)*((I*cos(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticF(((I*cos(x
)-I+sin(x))/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)-I+sin(x))
/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))-2
31*cos(x)^6*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin
(x))^(1/2)*EllipticF(((I*cos(x)-I+sin(x))/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)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)-I+sin(x
))/sin(x))^(1/2),1/2*2^(1/2))+693*cos(x)^5*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)-I+sin(x))/sin(x))^
(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticF(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))-1386*cos(x
)^4*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/
2)*EllipticE(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))+693*cos(x)^4*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2
)*((I*cos(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticF(((I*cos(x)-I+sin(x))/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)-I+sin(x))/sin(x))^(1/2)*((
-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))-693*cos(x)^3*2^(1/
2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*Ellipt
icF(((I*cos(x)-I+sin(x))/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*co
s(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)-I+sin(x))/sin(x))^(1/2),
1/2*2^(1/2))-693*cos(x)^2*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)
+sin(x)+I)/sin(x))^(1/2)*EllipticF(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))-462*cos(x)^6-462*cos(x)*2^(
1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*Elli
pticE(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))+231*cos(x)*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos
(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticF(((I*cos(x)-I+sin(x))/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)-I+sin(x))/sin(x))^(1/2)*((-I*cos
(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))+231*2^(1/2)*(-I*(-1+cos(
x))/sin(x))^(1/2)*((I*cos(x)-I+sin(x))/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticF(((I*cos(x)-
I+sin(x))/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/sin(x
)/(-1+cos(x)^2)*a)^(5/2)*8^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \csc \relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {a}{{\sin \relax (x)}^3}\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/sin(x)^3)^(5/2),x)

[Out]

int((a/sin(x)^3)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \csc ^{3}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*csc(x)**3)**(5/2), x)

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