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

Optimal. Leaf size=102 \[ -\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {10 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{21 d f} \]

[Out]

-2/7*d^2*cos(f*x+e)/f/(d*csc(f*x+e))^(5/2)-10/21*cos(f*x+e)/f/(d*csc(f*x+e))^(1/2)-10/21*(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)/d/f

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Rubi [A]  time = 0.07, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3769, 3771, 2641} \[ -\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {10 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{21 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.11, size = 70, normalized size = 0.69 \[ -\frac {\sqrt {d \csc (e+f x)} \left (26 \sin (2 (e+f x))-3 \sin (4 (e+f x))+40 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{84 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/84*(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)]))/(d*f)

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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right )}{d \csc \left (f x + e\right )}, x\right ) \]

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)/(d*csc(f*x + e)), x)

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

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]

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

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maple [C]  time = 0.20, size = 208, normalized size = 2.04 \[ -\frac {\left (5 i \sin \left (f x +e \right ) \sqrt {-\frac {i \cos \left (f x +e \right )-\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}-3 \left (\cos ^{4}\left (f x +e \right )\right ) \sqrt {2}+3 \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}+8 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-8 \cos \left (f x +e \right ) \sqrt {2}\right ) \sqrt {2}}{21 f \left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {d}{\sin \left (f x +e \right )}}} \]

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]

-1/21/f*(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)*(
-(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))-3*cos(f*x+e)^4*2^(1/2)+3*cos(f*x+e)^3*2^(1/2)+8*cos(f*x+e)^2*2^(1/2)-8*cos(f*x+e)*2^(1/2))/(-1+cos(f*x+
e))/(d/sin(f*x+e))^(1/2)*2^(1/2)

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(e + f*x)^3/(d/sin(e + f*x))^(1/2),x)

[Out]

int(sin(e + f*x)^3/(d/sin(e + f*x))^(1/2), 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(sin(f*x+e)**3/(d*csc(f*x+e))**(1/2),x)

[Out]

Timed out

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