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

Optimal. Leaf size=71 \[ -\frac{2 d^2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 d \cos (e+f x) \sqrt{d \csc (e+f x)}}{f} \]

[Out]

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

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Rubi [A]  time = 0.0334052, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3771, 2639} \[ -\frac{2 d^2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 d \cos (e+f x) \sqrt{d \csc (e+f x)}}{f} \]

Antiderivative was successfully verified.

[In]

Int[(d*Csc[e + f*x])^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.078176, size = 54, normalized size = 0.76 \[ \frac{(d \csc (e+f x))^{3/2} \left (2 \sin ^{\frac{3}{2}}(e+f x) E\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )-\sin (2 (e+f x))\right )}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[e + f*x])^(3/2),x]

[Out]

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

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Maple [C]  time = 0.105, size = 520, normalized size = 7.3 \begin{align*}{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{f} \left ( 2\,\cos \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 ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}},1/2\,\sqrt{2} \right ) -\cos \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 ) }}}\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 ) +2\,\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 ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}},1/2\,\sqrt{2} \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 ) }}}\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} \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),x)

[Out]

1/f*2^(1/2)*(2*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))-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*(-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))-(-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))*(d/sin(f*x+e))^(3/2)
*sin(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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((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 (\sqrt{d \csc \left (f x + e\right )} d \csc \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),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((d*csc(f*x+e))**(3/2),x)

[Out]

Integral((d*csc(e + f*x))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((d*csc(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((d*csc(f*x + e))^(3/2), x)