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

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

[Out]

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

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Rubi [A]  time = 0.0319975, 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} \[ -\frac{2 c^2 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}-\frac{2 c \cos (a+b x) \sqrt{c \csc (a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c*Csc[a + b*x])^(3/2),x]

[Out]

(-2*c*Cos[a + b*x]*Sqrt[c*Csc[a + b*x]])/b - (2*c^2*EllipticE[(a - Pi/2 + b*x)/2, 2])/(b*Sqrt[c*Csc[a + b*x]]*
Sqrt[Sin[a + b*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 (c \csc (a+b x))^{3/2} \, dx &=-\frac{2 c \cos (a+b x) \sqrt{c \csc (a+b x)}}{b}-c^2 \int \frac{1}{\sqrt{c \csc (a+b x)}} \, dx\\ &=-\frac{2 c \cos (a+b x) \sqrt{c \csc (a+b x)}}{b}-\frac{c^2 \int \sqrt{\sin (a+b x)} \, dx}{\sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}}\\ &=-\frac{2 c \cos (a+b x) \sqrt{c \csc (a+b x)}}{b}-\frac{2 c^2 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{b \sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.126562, size = 54, normalized size = 0.76 \[ \frac{(c \csc (a+b x))^{3/2} \left (2 \sin ^{\frac{3}{2}}(a+b x) E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )-\sin (2 (a+b x))\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Csc[a + b*x])^(3/2),x]

[Out]

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

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Maple [C]  time = 0.219, size = 520, normalized size = 7.3 \begin{align*}{\frac{\sqrt{2}\sin \left ( bx+a \right ) }{b} \left ( 2\,\sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}\sqrt{-{\frac{i\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }}}\cos \left ( bx+a \right ) -\sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}\sqrt{-{\frac{i\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) -i}{\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{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }}}\cos \left ( bx+a \right ) +2\,\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 ) }}}\sqrt{-{\frac{i\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}},1/2\,\sqrt{2} \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 ) }}}\sqrt{-{\frac{i\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) -i}{\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} \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((c*csc(b*x+a))^(3/2),x)

[Out]

1/b*2^(1/2)*(2*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2)*(-(I*cos(b*x+a)-sin(b*x+a)-I)/sin(b*x+a))^(1/2)*
EllipticE(((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2),1/2*2^(1/2))*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*co
s(b*x+a)-((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2)*(-(I*cos(b*x+a)-sin(b*x+a)-I)/sin(b*x+a))^(1/2)*Ellipt
icF(((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2),1/2*2^(1/2))*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*cos(b*x+
a)+2*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2)*(-(I*cos(b*x+a)-sin(
b*x+a)-I)/sin(b*x+a))^(1/2)*EllipticE(((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2),1/2*2^(1/2))-(-I*(-1+cos(
b*x+a))/sin(b*x+a))^(1/2)*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2)*(-(I*cos(b*x+a)-sin(b*x+a)-I)/sin(b*x
+a))^(1/2)*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))*(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 \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((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 (\sqrt{c \csc \left (b x + a\right )} c \csc \left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((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 \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((c*csc(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((c*csc(b*x + a))^(3/2), x)