3.23 \(\int \frac{1}{(c \csc (a+b x))^{5/2}} \, dx\)

Optimal. Leaf size=77 \[ \frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b c^2 \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}} \]

[Out]

(-2*Cos[a + b*x])/(5*b*c*(c*Csc[a + b*x])^(3/2)) + (6*EllipticE[(a - Pi/2 + b*x)/2, 2])/(5*b*c^2*Sqrt[c*Csc[a
+ b*x]]*Sqrt[Sin[a + b*x]])

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Rubi [A]  time = 0.0319249, 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, 2639} \[ \frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b c^2 \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Csc[a + b*x])^(-5/2),x]

[Out]

(-2*Cos[a + b*x])/(5*b*c*(c*Csc[a + b*x])^(3/2)) + (6*EllipticE[(a - Pi/2 + b*x)/2, 2])/(5*b*c^2*Sqrt[c*Csc[a
+ b*x]]*Sqrt[Sin[a + b*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]

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 \frac{1}{(c \csc (a+b x))^{5/2}} \, dx &=-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac{3 \int \frac{1}{\sqrt{c \csc (a+b x)}} \, dx}{5 c^2}\\ &=-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac{3 \int \sqrt{\sin (a+b x)} \, dx}{5 c^2 \sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}}\\ &=-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac{6 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{5 b c^2 \sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.143341, size = 60, normalized size = 0.78 \[ \frac{-2 \sin (2 (a+b x))-\frac{12 E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )}{\sqrt{\sin (a+b x)}}}{10 b c^2 \sqrt{c \csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Csc[a + b*x])^(-5/2),x]

[Out]

((-12*EllipticE[(-2*a + Pi - 2*b*x)/4, 2])/Sqrt[Sin[a + b*x]] - 2*Sin[2*(a + b*x)])/(10*b*c^2*Sqrt[c*Csc[a + b
*x]])

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Maple [C]  time = 0.252, size = 547, normalized size = 7.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*csc(b*x+a))^(5/2),x)

[Out]

-1/5/b*2^(1/2)*(6*((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)
*cos(b*x+a)-3*((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)*E
llipticF(((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)+6*(-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))-3*(-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)*cos(b*x+a)^3+4
*2^(1/2)*cos(b*x+a)-3*2^(1/2))/(c/sin(b*x+a))^(5/2)/sin(b*x+a)^3

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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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate((c*csc(b*x + a))^(-5/2), 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^{3} \csc \left (b x + a\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*csc(b*x + a))/(c^3*csc(b*x + a)^3), 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{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))**(5/2),x)

[Out]

Integral((c*csc(a + b*x))**(-5/2), 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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))^(5/2),x, algorithm="giac")

[Out]

integrate((c*csc(b*x + a))^(-5/2), x)