3.57 \(\int \sqrt {a \csc ^3(x)} \, dx\)

Optimal. Leaf size=48 \[ 2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt {a \csc ^3(x)} \]

[Out]

-2*cos(x)*sin(x)*(a*csc(x)^3)^(1/2)+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)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3768, 3771, 2639} \[ 2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt {a \csc ^3(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Csc[x]^3],x]

[Out]

-2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x] + 2*Sqrt[a*Csc[x]^3]*EllipticE[Pi/4 - x/2, 2]*Sin[x]^(3/2)

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 \sqrt {a \csc ^3(x)} \, dx &=\frac {\sqrt {a \csc ^3(x)} \int (-\csc (x))^{3/2} \, dx}{(-\csc (x))^{3/2}}\\ &=-2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\frac {\sqrt {a \csc ^3(x)} \int \frac {1}{\sqrt {-\csc (x)}} \, dx}{(-\csc (x))^{3/2}}\\ &=-2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\left (\sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sin (x)} \, dx\\ &=-2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+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 = 0.04, size = 46, normalized size = 0.96 \[ 2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sqrt {a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt {a \csc ^3(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Csc[x]^3],x]

[Out]

-2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x] + 2*Sqrt[a*Csc[x]^3]*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(3/2)

________________________________________________________________________________________

fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \csc \relax (x)^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^3)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*csc(x)^3), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^3)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*csc(x)^3), x)

________________________________________________________________________________________

maple [C]  time = 0.98, size = 343, normalized size = 7.15 \[ \frac {\left (2 \cos \relax (x ) \sqrt {2}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-i-\sin \relax (x )}{\sin \relax (x )}}\, \EllipticE \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )-\cos \relax (x ) \sqrt {2}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-i-\sin \relax (x )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {2}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-i-\sin \relax (x )}{\sin \relax (x )}}\, \EllipticE \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-i-\sin \relax (x )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )-2\right ) \sin \relax (x ) \sqrt {-\frac {2 a}{\sin \relax (x ) \left (-1+\cos ^{2}\relax (x )\right )}}\, \sqrt {8}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*csc(x)^3)^(1/2),x)

[Out]

1/4*(2*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)-I-sin(x))/s
in(x))^(1/2)*EllipticE(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))-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)-I-sin(x))/sin(x))^(1/2)*EllipticF(((I*cos(x)-I+sin(x))/si
n(x))^(1/2),1/2*2^(1/2))+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
)-I-sin(x))/sin(x))^(1/2)*EllipticE(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))-2^(1/2)*(-I*(-1+cos(x))/si
n(x))^(1/2)*((I*cos(x)-I+sin(x))/sin(x))^(1/2)*(-(I*cos(x)-I-sin(x))/sin(x))^(1/2)*EllipticF(((I*cos(x)-I+sin(
x))/sin(x))^(1/2),1/2*2^(1/2))-2)*sin(x)*(-2/sin(x)/(-1+cos(x)^2)*a)^(1/2)*8^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*csc(x)^3), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\frac {a}{{\sin \relax (x)}^3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/sin(x)^3)^(1/2),x)

[Out]

int((a/sin(x)^3)^(1/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)**3)**(1/2),x)

[Out]

Integral(sqrt(a*csc(x)**3), x)

________________________________________________________________________________________