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

Optimal. Leaf size=44 \[ \frac{2 d \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{f} \]

[Out]

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

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Rubi [A]  time = 0.0280744, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {16, 3771, 2641} \[ \frac{2 d \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int (d \csc (e+f x))^{3/2} \sin (e+f x) \, dx &=d \int \sqrt{d \csc (e+f x)} \, dx\\ &=\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 \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)}}{f}\\ \end{align*}

Mathematica [A]  time = 0.0159633, size = 43, normalized size = 0.98 \[ -\frac{2 d \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right ) \sqrt{d \csc (e+f x)}}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.125, size = 165, normalized size = 3.8 \begin{align*}{\frac{-i\sqrt{2} \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{f\sin \left ( fx+e \right ) }\sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \left ({\frac{d}{\sin \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \csc \left (f x + e\right )\right )^{\frac{3}{2}} \sin \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{d \csc \left (f x + e\right )} d \csc \left (f x + e\right ) \sin \left (f x + e\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*csc(f*x + e))*d*csc(f*x + e)*sin(f*x + e), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \csc \left (f x + e\right )\right )^{\frac{3}{2}} \sin \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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