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

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

[Out]

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

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Rubi [A]  time = 0.0260246, 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 = {3768, 3771, 2641} \[ \frac{2 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b}-\frac{2 \cos (a+b x) \csc ^{\frac{3}{2}}(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^(5/2),x]

[Out]

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

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

Mathematica [A]  time = 0.08119, size = 50, normalized size = 0.75 \[ -\frac{2 \csc ^{\frac{3}{2}}(a+b x) \left (\sin ^{\frac{3}{2}}(a+b x) \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+\cos (a+b x)\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^(5/2),x]

[Out]

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

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Maple [A]  time = 1.2, size = 88, normalized size = 1.3 \begin{align*}{\frac{1}{3\,\cos \left ( bx+a \right ) b} \left ( \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 ) \sin \left ( bx+a \right ) -2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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(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 (\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(csc(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

integral(csc(b*x + a)^(5/2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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(csc(b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^(5/2), x)