\(\int (c \csc (a+b x))^{5/2} \, dx\) [18]

Optimal result

Integrand size = 12, antiderivative size = 75 \[ \int (c \csc (a+b x))^{5/2} \, dx=-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b}+\frac {2 c^2 \sqrt {c \csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{3 b} \]

[Out]

-2/3*c*cos(b*x+a)*(c*csc(b*x+a))^(3/2)/b-2/3*c^2*(sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*x)
*EllipticF(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))*(c*csc(b*x+a))^(1/2)*sin(b*x+a)^(1/2)/b

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3856, 2720} \[ \int (c \csc (a+b x))^{5/2} \, dx=\frac {2 c^2 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right ) \sqrt {c \csc (a+b x)}}{3 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b} \]

[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 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3853

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 3856

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]

Rubi steps \begin{align*} \text {integral}& = -\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)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{3 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int (c \csc (a+b x))^{5/2} \, dx=-\frac {(c \csc (a+b x))^{5/2} \left (2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sin ^{\frac {5}{2}}(a+b x)+\sin (2 (a+b x))\right )}{3 b} \]

[In]

Integrate[(c*Csc[a + b*x])^(5/2),x]

[Out]

-1/3*((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)]))/b

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.64 (sec) , antiderivative size = 312, normalized size of antiderivative = 4.16

method	result	size
default	\(\frac {\csc \left (x b +a \right )^{2} {\left (\frac {c \left (\csc \left (x b +a \right ) \left (1-\cos \left (x b +a \right )\right )^{2}+\sin \left (x b +a \right )\right )}{1-\cos \left (x b +a \right )}\right )}^{\frac {5}{2}} \left (1-\cos \left (x b +a \right )\right )^{2} \left (2 i \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )+\csc \left (x b +a \right )^{4} \left (1-\cos \left (x b +a \right )\right )^{4}-1\right ) \sqrt {2}}{6 b \sqrt {\csc \left (x b +a \right )^{3} \left (1-\cos \left (x b +a \right )\right )^{3}+\csc \left (x b +a \right )-\cot \left (x b +a \right )}\, \sqrt {\csc \left (x b +a \right ) \left (1-\cos \left (x b +a \right )\right ) \left (\csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}+1\right )}\, \left (\csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}+1\right )^{2}}\)	\(312\)

[In]

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

[Out]

1/6/b*csc(b*x+a)^2*(c/(1-cos(b*x+a))*(csc(b*x+a)*(1-cos(b*x+a))^2+sin(b*x+a)))^(5/2)*(1-cos(b*x+a))^2*(2*I*(-I
*(I-cot(b*x+a)+csc(b*x+a)))^(1/2)*2^(1/2)*(-I*(I+cot(b*x+a)-csc(b*x+a)))^(1/2)*(I*(csc(b*x+a)-cot(b*x+a)))^(1/
2)*EllipticF((-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2),1/2*2^(1/2))*(csc(b*x+a)-cot(b*x+a))+csc(b*x+a)^4*(1-cos(b*x
+a))^4-1)/(csc(b*x+a)^3*(1-cos(b*x+a))^3+csc(b*x+a)-cot(b*x+a))^(1/2)/(csc(b*x+a)*(1-cos(b*x+a))*(csc(b*x+a)^2
*(1-cos(b*x+a))^2+1))^(1/2)/(csc(b*x+a)^2*(1-cos(b*x+a))^2+1)^2*2^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.40 \[ \int (c \csc (a+b x))^{5/2} \, dx=\frac {-i \, \sqrt {2 i \, c} c^{2} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {-2 i \, c} c^{2} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 2 \, c^{2} \sqrt {\frac {c}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )}{3 \, b \sin \left (b x + a\right )} \]

[In]

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

[Out]

1/3*(-I*sqrt(2*I*c)*c^2*sin(b*x + a)*weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a)) + I*sqrt(-2*I*c)
*c^2*sin(b*x + a)*weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a)) - 2*c^2*sqrt(c/sin(b*x + a))*cos(b*
x + a))/(b*sin(b*x + a))

Sympy [F]

\[ \int (c \csc (a+b x))^{5/2} \, dx=\int \left (c \csc {\left (a + b x \right )}\right )^{\frac {5}{2}}\, dx \]

[In]

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

[Out]

Integral((c*csc(a + b*x))**(5/2), x)

Maxima [F]

\[ \int (c \csc (a+b x))^{5/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]

[In]

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

[Out]

integrate((c*csc(b*x + a))^(5/2), x)

Giac [F]

\[ \int (c \csc (a+b x))^{5/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]

[In]

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

[Out]

integrate((c*csc(b*x + a))^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int (c \csc (a+b x))^{5/2} \, dx=\int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{5/2} \,d x \]

[In]

int((c/sin(a + b*x))^(5/2),x)

[Out]

int((c/sin(a + b*x))^(5/2), x)