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

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

[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 = 0.04, 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 = {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 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]

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 )^{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 = 0.11, size = 46, normalized size = 0.65 \[ -\frac {1}{84} \left (a \csc ^3(x)\right )^{3/2} \left (22 \sin (2 x)-5 \sin (4 x)+40 \sin ^{\frac {9}{2}}(x) F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )\right ) \]

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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fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \csc \relax (x)^{3}} a \csc \relax (x)^{3}, x\right ) \]

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

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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maple [C]  time = 0.90, size = 372, normalized size = 5.24 \[ -\frac {\left (\cos \relax (x )+1\right )^{2} \left (-1+\cos \relax (x )\right )^{2} \left (5 i \sin \relax (x ) \left (\cos ^{3}\relax (x )\right ) \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}+5 i \sin \relax (x ) \left (\cos ^{2}\relax (x )\right ) \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}-5 i \sin \relax (x ) \cos \relax (x ) \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}-5 i \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {2}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sin \relax (x )-10 \left (\cos ^{3}\relax (x )\right )+16 \cos \relax (x )\right ) \left (-\frac {2 a}{\sin \relax (x ) \left (-1+\cos ^{2}\relax (x )\right )}\right )^{\frac {3}{2}} \sqrt {8}}{168 \sin \relax (x )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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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