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

Optimal. Leaf size=75 \[ \frac{2 c^2 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right ) \sqrt{c \csc (a+b x)}}{3 b}-\frac{2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0310997, antiderivative size = 75, 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, 2641} \[ \frac{2 c^2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \csc (a+b x)}}{3 b}-\frac{2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*Csc[a + b*x])^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.187472, size = 55, normalized size = 0.73 \[ -\frac{(c \csc (a+b x))^{5/2} \left (2 \sin ^{\frac{5}{2}}(a+b x) \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+\sin (2 (a+b x))\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Csc[a + b*x])^(5/2),x]

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.241, size = 319, normalized size = 4.3 \begin{align*}{\frac{\sqrt{2} \left ( -1+\cos \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}{3\,b \left ( \sin \left ( bx+a \right ) \right ) ^{3}} \left ( i\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 \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 ) +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}\cos \left ( bx+a \right ) \right ) \left ({\frac{c}{\sin \left ( bx+a \right ) }} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/b*2^(1/2)*(-1+cos(b*x+a))^2*(I*cos(b*x+a)*((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*cos(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))+I*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2)*(-I*(-1+cos(b*x+a))/s
in(b*x+a))^(1/2)*(-(I*cos(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))*(cos(b*x+a)+1)^2*(c/sin(b*x+a))^(5/2)/sin(b*x+a)^3

________________________________________________________________________________________

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

[Out]

integrate((c*csc(b*x + a))^(5/2), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

integrate((c*csc(b*x + a))^(5/2), x)