3.508 \(\int \sqrt{d \csc (e+f x)} \sin ^3(e+f x) \, dx\)

Optimal. Leaf size=75 \[ \frac{6 d E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 d^2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0542168, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 3769, 3771, 2639} \[ \frac{6 d E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 d^2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^3,x]

[Out]

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

Mathematica [A]  time = 0.115072, size = 62, normalized size = 0.83 \[ -\frac{2 \sqrt{d \csc (e+f x)} \left (\sin ^2(e+f x) \cos (e+f x)+3 \sqrt{\sin (e+f x)} E\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{5 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^3,x]

[Out]

(-2*Sqrt[d*Csc[e + f*x]]*(3*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + Cos[e + f*x]*Sin[e + f*x]
^2))/(5*f)

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Maple [C]  time = 0.198, size = 538, normalized size = 7.2 \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)^3*(d*csc(f*x+e))^(1/2),x)

[Out]

1/5/f*2^(1/2)*(d/sin(f*x+e))^(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)/si
n(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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^3*(d*csc(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt{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(sin(f*x+e)^3*(d*csc(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(-(cos(f*x + e)^2 - 1)*sqrt(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(sin(f*x+e)**3*(d*csc(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [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(sin(f*x+e)^3*(d*csc(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out