3.517 \(\int (d \csc (e+f x))^{3/2} \sin ^3(e+f x) \, dx\)

Optimal. Leaf size=75 \[ \frac {2 d \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{3 f}-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}} \]

[Out]

-2/3*d^2*cos(f*x+e)/f/(d*csc(f*x+e))^(1/2)-2/3*d*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)
*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2))*(d*csc(f*x+e))^(1/2)*sin(f*x+e)^(1/2)/f

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Rubi [A]  time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3769, 3771, 2641} \[ \frac {2 d \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{3 f}-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^3,x]

[Out]

(-2*d^2*Cos[e + f*x])/(3*f*Sqrt[d*Csc[e + f*x]]) + (2*d*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*
Sqrt[Sin[e + f*x]])/(3*f)

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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]

Rubi steps

\begin {align*} \int (d \csc (e+f x))^{3/2} \sin ^3(e+f x) \, dx &=d^3 \int \frac {1}{(d \csc (e+f x))^{3/2}} \, dx\\ &=-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\frac {1}{3} d \int \sqrt {d \csc (e+f x)} \, dx\\ &=-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\frac {1}{3} \left (d \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx\\ &=-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\frac {2 d \sqrt {d \csc (e+f x)} F\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {\sin (e+f x)}}{3 f}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 56, normalized size = 0.75 \[ -\frac {d \sqrt {d \csc (e+f x)} \left (\sin (2 (e+f x))+2 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{3 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^3,x]

[Out]

-1/3*(d*Sqrt[d*Csc[e + f*x]]*(2*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + Sin[2*(e + f*x)]))/f

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fricas [F]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (d \cos \left (f x + e\right )^{2} - d\right )} \sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right ) \sin \left (f x + e\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(f*x+e))^(3/2)*sin(f*x+e)^3,x, algorithm="fricas")

[Out]

integral(-(d*cos(f*x + e)^2 - d)*sqrt(d*csc(f*x + e))*csc(f*x + e)*sin(f*x + e), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(f*x+e))^(3/2)*sin(f*x+e)^3,x, algorithm="giac")

[Out]

integrate((d*csc(f*x + e))^(3/2)*sin(f*x + e)^3, x)

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maple [C]  time = 0.15, size = 189, normalized size = 2.52 \[ -\frac {\left (i \sin \left (f x +e \right ) \sqrt {-\frac {i \cos \left (f x +e \right )-\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}+\left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-\cos \left (f x +e \right ) \sqrt {2}\right ) \left (\frac {d}{\sin \left (f x +e \right )}\right )^{\frac {3}{2}} \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {2}}{3 f \left (-1+\cos \left (f x +e \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*csc(f*x+e))^(3/2)*sin(f*x+e)^3,x)

[Out]

-1/3/f*(I*sin(f*x+e)*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2)*(-(I
*cos(f*x+e)-sin(f*x+e)-I)/sin(f*x+e))^(1/2)*EllipticF(((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2),1/2*2^(1/
2))+cos(f*x+e)^2*2^(1/2)-cos(f*x+e)*2^(1/2))*(d/sin(f*x+e))^(3/2)*sin(f*x+e)^2/(-1+cos(f*x+e))*2^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(f*x+e))^(3/2)*sin(f*x+e)^3,x, algorithm="maxima")

[Out]

integrate((d*csc(f*x + e))^(3/2)*sin(f*x + e)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(e + f*x)^3*(d/sin(e + f*x))^(3/2),x)

[Out]

int(sin(e + f*x)^3*(d/sin(e + f*x))^(3/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(f*x+e))**(3/2)*sin(f*x+e)**3,x)

[Out]

Timed out

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