3.513 \(\int \csc ^2(e+f x) \sqrt{d \csc (e+f x)} \, dx\)

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

[Out]

(-2*Cos[e + f*x]*(d*Csc[e + f*x])^(3/2))/(3*d*f) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sq
rt[Sin[e + f*x]])/(3*f)

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Rubi [A]  time = 0.0404922, antiderivative size = 74, 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, 3768, 3771, 2641} \[ \frac{2 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{3 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^2*Sqrt[d*Csc[e + f*x]],x]

[Out]

(-2*Cos[e + f*x]*(d*Csc[e + f*x])^(3/2))/(3*d*f) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sq
rt[Sin[e + f*x]])/(3*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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[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 \csc ^2(e+f x) \sqrt{d \csc (e+f x)} \, dx &=\frac{\int (d \csc (e+f x))^{5/2} \, dx}{d^2}\\ &=-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac{1}{3} \int \sqrt{d \csc (e+f x)} \, dx\\ &=-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac{1}{3} \left (\sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{1}{\sqrt{\sin (e+f x)}} \, dx\\ &=-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac{2 \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)}}{3 f}\\ \end{align*}

Mathematica [A]  time = 0.0896675, size = 55, normalized size = 0.74 \[ -\frac{2 (d \csc (e+f x))^{3/2} \left (\cos (e+f x)+\sin ^{\frac{3}{2}}(e+f x) F\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{3 d f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^2*Sqrt[d*Csc[e + f*x]],x]

[Out]

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

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Maple [C]  time = 0.127, size = 319, normalized size = 4.3 \begin{align*}{\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2}}{3\,f \left ( \sin \left ( fx+e \right ) \right ) ^{5}}\sqrt{{\frac{d}{\sin \left ( fx+e \right ) }}} \left ( i\cos \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 \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 ) }}}\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 ) +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 ) }}}\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{2}\cos \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*csc(f*x + e))*csc(f*x + e)^2, 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 )} \csc \left (f x + e\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**2*(d*csc(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(d*csc(e + f*x))*csc(e + f*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*csc(f*x + e))*csc(f*x + e)^2, x)