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

Optimal. Leaf size=74 \[ \frac{6 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{5 d f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}} \]

[Out]

(-2*Cos[e + f*x])/(5*f*(d*Csc[e + f*x])^(3/2)) + (6*EllipticE[(e - Pi/2 + f*x)/2, 2])/(5*d*f*Sqrt[d*Csc[e + f*
x]]*Sqrt[Sin[e + f*x]])

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Rubi [A]  time = 0.0456729, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {16, 3769, 3771, 2639} \[ \frac{6 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{5 d f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cos[e + f*x])/(5*f*(d*Csc[e + f*x])^(3/2)) + (6*EllipticE[(e - Pi/2 + f*x)/2, 2])/(5*d*f*Sqrt[d*Csc[e + f*
x]]*Sqrt[Sin[e + f*x]])

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 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]

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]

Rubi steps

\begin{align*} \int \frac{\sin (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx &=d \int \frac{1}{(d \csc (e+f x))^{5/2}} \, dx\\ &=-\frac{2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac{3 \int \frac{1}{\sqrt{d \csc (e+f x)}} \, dx}{5 d}\\ &=-\frac{2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac{3 \int \sqrt{\sin (e+f x)} \, dx}{5 d \sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}}\\ &=-\frac{2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac{6 E\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right )}{5 d f \sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.0324978, size = 60, normalized size = 0.81 \[ \frac{-2 \sin (2 (e+f x))-\frac{12 E\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )}{\sqrt{\sin (e+f x)}}}{10 d f \sqrt{d \csc (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/Sqrt[Sin[e + f*x]] - 2*Sin[2*(e + f*x)])/(10*d*f*Sqrt[d*Csc[e + f*x
]])

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Maple [C]  time = 0.128, size = 546, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5/f*2^(1/2)*(-6*cos(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)*EllipticE(((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2),1
/2*2^(1/2))+3*cos(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))+2^(1/2)*cos(f*x+e)^3-6*(-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)*EllipticE(((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2)
,1/2*2^(1/2))+3*(-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))-4
*2^(1/2)*cos(f*x+e)+3*2^(1/2))/(d/sin(f*x+e))^(3/2)/sin(f*x+e)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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