\(\int (c \csc (a+b x))^{3/2} \, dx\) [19]

Optimal result

Integrand size = 12, antiderivative size = 71 \[ \int (c \csc (a+b x))^{3/2} \, dx=-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}-\frac {2 c^2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, 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, 2719} \[ \int (c \csc (a+b x))^{3/2} \, dx=-\frac {2 c^2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b} \]

[In]

Int[(c*Csc[a + b*x])^(3/2),x]

[Out]

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

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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) \sqrt {c \csc (a+b x)}}{b}-c^2 \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx \\ & = -\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}-\frac {c^2 \int \sqrt {\sin (a+b x)} \, dx}{\sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ & = -\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}-\frac {2 c^2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int (c \csc (a+b x))^{3/2} \, dx=\frac {(c \csc (a+b x))^{3/2} \left (2 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {3}{2}}(a+b x)-\sin (2 (a+b x))\right )}{b} \]

[In]

Integrate[(c*Csc[a + b*x])^(3/2),x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.31 (sec) , antiderivative size = 413, normalized size of antiderivative = 5.82

method	result	size
default	\(\frac {c \sqrt {c \csc \left (x b +a \right )}\, \left (2 \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x b +a \right )-\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x b +a \right )+2 \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\right ) \sqrt {2}}{b}\)	\(413\)

[In]

int((c*csc(b*x+a))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/b*c*(c*csc(b*x+a))^(1/2)*(2*(-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2)*(I*(-I-cot(b*x+a)+csc(b*x+a)))^(1/2)*(I*(cs
c(b*x+a)-cot(b*x+a)))^(1/2)*EllipticE((-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2),1/2*2^(1/2))*cos(b*x+a)-(-I*(I-cot(
b*x+a)+csc(b*x+a)))^(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))*cos(b*x+a)+2*(-I*(I-cot(b*x+a)+csc(b*x+a)))^(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)*EllipticE((-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2),1/2*2
^(1/2))-(-I*(I-cot(b*x+a)+csc(b*x+a)))^(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))-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 = 83, normalized size of antiderivative = 1.17 \[ \int (c \csc (a+b x))^{3/2} \, dx=-\frac {2 \, c \sqrt {\frac {c}{\sin \left (b x + a\right )}} \cos \left (b x + a\right ) + \sqrt {2 i \, c} c {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i \, c} c {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b} \]

[In]

integrate((c*csc(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

-(2*c*sqrt(c/sin(b*x + a))*cos(b*x + a) + sqrt(2*I*c)*c*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*
x + a) + I*sin(b*x + a))) + sqrt(-2*I*c)*c*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) - I*si
n(b*x + a))))/b

Sympy [F]

\[ \int (c \csc (a+b x))^{3/2} \, dx=\int \left (c \csc {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((c*csc(b*x+a))**(3/2),x)

[Out]

Integral((c*csc(a + b*x))**(3/2), x)

Maxima [F]

\[ \int (c \csc (a+b x))^{3/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((c*csc(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate((c*csc(b*x + a))^(3/2), x)

Giac [F]

\[ \int (c \csc (a+b x))^{3/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((c*csc(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((c*csc(b*x + a))^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int (c \csc (a+b x))^{3/2} \, dx=\int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{3/2} \,d x \]

[In]

int((c/sin(a + b*x))^(3/2),x)

[Out]

int((c/sin(a + b*x))^(3/2), x)