3.515 \(\int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx\)

Optimal. Leaf size=103 \[ -\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{10 d \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{21 f} \]

[Out]

(-2*d^4*Cos[e + f*x])/(7*f*(d*Csc[e + f*x])^(5/2)) - (10*d^2*Cos[e + f*x])/(21*f*Sqrt[d*Csc[e + f*x]]) + (10*d
*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/(21*f)

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Rubi [A]  time = 0.0758318, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 3769, 3771, 2641} \[ -\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{10 d \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{21 f} \]

Antiderivative was successfully verified.

[In]

Int[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^5,x]

[Out]

(-2*d^4*Cos[e + f*x])/(7*f*(d*Csc[e + f*x])^(5/2)) - (10*d^2*Cos[e + f*x])/(21*f*Sqrt[d*Csc[e + f*x]]) + (10*d
*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/(21*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 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 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 ^5(e+f x) \, dx &=d^5 \int \frac{1}{(d \csc (e+f x))^{7/2}} \, dx\\ &=-\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}+\frac{1}{7} \left (5 d^3\right ) \int \frac{1}{(d \csc (e+f x))^{3/2}} \, dx\\ &=-\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{1}{21} (5 d) \int \sqrt{d \csc (e+f x)} \, dx\\ &=-\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{1}{21} \left (5 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^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{10 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)}}{21 f}\\ \end{align*}

Mathematica [A]  time = 0.123485, size = 68, normalized size = 0.66 \[ -\frac{d \sqrt{d \csc (e+f x)} \left (26 \sin (2 (e+f x))-3 \sin (4 (e+f x))+40 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{84 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^5,x]

[Out]

-(d*Sqrt[d*Csc[e + f*x]]*(40*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + 26*Sin[2*(e + f*x)] - 3*
Sin[4*(e + f*x)]))/(84*f)

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Maple [C]  time = 0.113, size = 216, normalized size = 2.1 \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{21\,f \left ( -1+\cos \left ( fx+e \right ) \right ) } \left ( 5\,i\sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \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 ) }}}{\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 ) \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}-3\,\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{4}+3\,\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{3}+8\,\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{2}-8\,\sqrt{2}\cos \left ( fx+e \right ) \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)^5,x)

[Out]

-1/21/f*2^(1/2)*(5*I*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*sin(f*x+e)*(-(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))*((I*cos(f*x+e)+sin(f*x+e)-I)/sin
(f*x+e))^(1/2)-3*2^(1/2)*cos(f*x+e)^4+3*2^(1/2)*cos(f*x+e)^3+8*2^(1/2)*cos(f*x+e)^2-8*2^(1/2)*cos(f*x+e))*(d/s
in(f*x+e))^(3/2)*sin(f*x+e)^2/(-1+cos(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 )^{5}\,{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)^5,x, algorithm="maxima")

[Out]

integrate((d*csc(f*x + e))^(3/2)*sin(f*x + e)^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d \cos \left (f x + e\right )^{4} - 2 \, 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 ) \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)^5,x, algorithm="fricas")

[Out]

integral((d*cos(f*x + e)^4 - 2*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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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)**5,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((d*csc(f*x+e))^(3/2)*sin(f*x+e)^5,x, algorithm="giac")

[Out]

Timed out