3.22 \(\int \frac{1}{(c \csc (a+b x))^{3/2}} \, dx\)

Optimal. Leaf size=77 \[ \frac{2 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right ) \sqrt{c \csc (a+b x)}}{3 b c^2}-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}} \]

[Out]

(-2*Cos[a + b*x])/(3*b*c*Sqrt[c*Csc[a + b*x]]) + (2*Sqrt[c*Csc[a + b*x]]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt
[Sin[a + b*x]])/(3*b*c^2)

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Rubi [A]  time = 0.032662, antiderivative size = 77, 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 = {3769, 3771, 2641} \[ \frac{2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \csc (a+b x)}}{3 b c^2}-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Csc[a + b*x])^(-3/2),x]

[Out]

(-2*Cos[a + b*x])/(3*b*c*Sqrt[c*Csc[a + b*x]]) + (2*Sqrt[c*Csc[a + b*x]]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt
[Sin[a + b*x]])/(3*b*c^2)

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 \frac{1}{(c \csc (a+b x))^{3/2}} \, dx &=-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}}+\frac{\int \sqrt{c \csc (a+b x)} \, dx}{3 c^2}\\ &=-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}}+\frac{\left (\sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{3 c^2}\\ &=-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}}+\frac{2 \sqrt{c \csc (a+b x)} F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{3 b c^2}\\ \end{align*}

Mathematica [A]  time = 0.0823655, size = 63, normalized size = 0.82 \[ -\frac{\csc ^2(a+b x) \left (2 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+\sin (2 (a+b x))\right )}{3 b (c \csc (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Csc[a + b*x])^(-3/2),x]

[Out]

-(Csc[a + b*x]^2*(2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + Sin[2*(a + b*x)]))/(3*b*(c*Csc[a
+ b*x])^(3/2))

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Maple [C]  time = 0.215, size = 189, normalized size = 2.5 \begin{align*} -{\frac{\sqrt{2}}{3\,b \left ( -1+\cos \left ( bx+a \right ) \right ) \sin \left ( bx+a \right ) } \left ( i\sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{-{\frac{i\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}\sin \left ( bx+a \right ){\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +\sqrt{2} \left ( \cos \left ( bx+a \right ) \right ) ^{2}-\sqrt{2}\cos \left ( bx+a \right ) \right ) \left ({\frac{c}{\sin \left ( bx+a \right ) }} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*csc(b*x+a))^(3/2),x)

[Out]

-1/3/b*2^(1/2)*(I*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2)*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*(-(I*co
s(b*x+a)-sin(b*x+a)-I)/sin(b*x+a))^(1/2)*sin(b*x+a)*EllipticF(((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2),1
/2*2^(1/2))+2^(1/2)*cos(b*x+a)^2-2^(1/2)*cos(b*x+a))/(-1+cos(b*x+a))/(c/sin(b*x+a))^(3/2)/sin(b*x+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \csc \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate((c*csc(b*x + a))^(-3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c \csc \left (b x + a\right )}}{c^{2} \csc \left (b x + a\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*csc(b*x + a))/(c^2*csc(b*x + a)^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \csc{\left (a + b x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))**(3/2),x)

[Out]

Integral((c*csc(a + b*x))**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \csc \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((c*csc(b*x + a))^(-3/2), x)