3.1.56 \(\int (a \csc ^3(x))^{3/2} \, dx\) [56]

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

[Out]

-10/21*a*cos(x)*(a*csc(x)^3)^(1/2)-2/7*a*cot(x)*csc(x)*(a*csc(x)^3)^(1/2)-10/21*a*(sin(1/4*Pi+1/2*x)^2)^(1/2)/
sin(1/4*Pi+1/2*x)*EllipticF(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 = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2720} \begin {gather*} -\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)} \end {gather*}

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 2720

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

Rule 3853

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 3856

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 4208

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 )^{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*}

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Mathematica [A]
time = 0.13, size = 46, normalized size = 0.65 \begin {gather*} -\frac {1}{84} \left (a \csc ^3(x)\right )^{3/2} \left (40 F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {9}{2}}(x)+22 \sin (2 x)-5 \sin (4 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/84*((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]))

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Maple [C] Result contains complex when optimal does not.
time = 0.19, size = 372, normalized size = 5.24

method	result	size
default	\(-\frac {\left (\cos \left (x \right )+1\right )^{2} \left (\cos \left (x \right )-1\right )^{2} \left (5 i \sin \left (x \right ) \left (\cos ^{3}\left (x \right )\right ) \sqrt {2}\, \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (\cos \left (x \right )-1\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right )+5 i \sin \left (x \right ) \left (\cos ^{2}\left (x \right )\right ) \sqrt {2}\, \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (\cos \left (x \right )-1\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right )-5 i \sin \left (x \right ) \cos \left (x \right ) \sqrt {2}\, \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (\cos \left (x \right )-1\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right )-5 i \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {2}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (\cos \left (x \right )-1\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (x \right )-10 \left (\cos ^{3}\left (x \right )\right )+16 \cos \left (x \right )\right ) \left (-\frac {2 a}{\sin \left (x \right ) \left (\cos ^{2}\left (x \right )-1\right )}\right )^{\frac {3}{2}} \sqrt {8}}{168 \sin \left (x \right )^{3}}\)	\(372\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.10, size = 99, normalized size = 1.39 \begin {gather*} -\frac {5 \, {\left (i \, a \cos \left (x\right )^{2} - i \, a\right )} \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 5 \, {\left (-i \, a \cos \left (x\right )^{2} + i \, a\right )} \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (5 \, a \cos \left (x\right )^{3} - 8 \, a \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}}{21 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(5*(I*a*cos(x)^2 - I*a)*sqrt(2*I*a)*weierstrassPInverse(4, 0, cos(x) + I*sin(x)) + 5*(-I*a*cos(x)^2 + I*
a)*sqrt(-2*I*a)*weierstrassPInverse(4, 0, cos(x) - I*sin(x)) + 2*(5*a*cos(x)^3 - 8*a*cos(x))*sqrt(-a/((cos(x)^
2 - 1)*sin(x))))/(cos(x)^2 - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \csc ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (\frac {a}{{\sin \left (x\right )}^3}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a/sin(x)^3)^(3/2), x)

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