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

Optimal. Leaf size=67 \[ \frac{6 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b}-\frac{2 \cos (a+b x)}{5 b \csc ^{\frac{3}{2}}(a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0258078, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3769, 3771, 2639} \[ \frac{6 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b}-\frac{2 \cos (a+b x)}{5 b \csc ^{\frac{3}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^(-5/2),x]

[Out]

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

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

Mathematica [A]  time = 0.106665, size = 60, normalized size = 0.9 \[ -\frac{2 \sqrt{\csc (a+b x)} \left (\sin ^2(a+b x) \cos (a+b x)+3 \sqrt{\sin (a+b x)} E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^(-5/2),x]

[Out]

(-2*Sqrt[Csc[a + b*x]]*(3*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + Cos[a + b*x]*Sin[a + b*x]^2
))/(5*b)

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Maple [A]  time = 1.229, size = 142, normalized size = 2.1 \begin{align*}{\frac{1}{\cos \left ( bx+a \right ) b} \left ({\frac{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{5}}-{\frac{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{5}}-{\frac{6}{5}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticE} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }+{\frac{3}{5}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2/5*sin(b*x+a)^4-2/5*sin(b*x+a)^2-6/5*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*Ellipt
icE((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))+3/5*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*Ell
ipticF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2)))/cos(b*x+a)/sin(b*x+a)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\csc \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(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{1}{\csc \left (b x + a\right )^{\frac{5}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(csc(b*x + a)^(-5/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\csc ^{\frac{5}{2}}{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(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}{\csc \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(b*x + a)^(-5/2), x)