\(\int \csc ^{\frac {3}{2}}(a+b x) \, dx\) [11]

Optimal result

Integrand size = 10, antiderivative size = 63 \[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\frac {2 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3853, 3856, 2719} \[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \]

[In]

Int[Csc[a + b*x]^(3/2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \sqrt {\csc (a+b x)} \left (\cos (a+b x)-E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {\sin (a+b x)}\right )}{b} \]

[In]

Integrate[Csc[a + b*x]^(3/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.10

method	result	size
default	\(\frac {2 \sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (x b +a \right )^{2}}{\cos \left (x b +a \right ) \sqrt {\sin \left (x b +a \right )}\, b}\)	\(132\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14 \[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=-\frac {\sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \frac {2 \, \cos \left (b x + a\right )}{\sqrt {\sin \left (b x + a\right )}}}{b} \]

[In]

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

[Out]

-(sqrt(2*I)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a))) + sqrt(-2*I)*weier
strassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a))) + 2*cos(b*x + a)/sqrt(sin(b*x + a))
)/b

Sympy [F]

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

[In]

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

[Out]

Integral(csc(a + b*x)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(csc(b*x + a)^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(csc(b*x + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=\int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{3/2} \,d x \]

[In]

int((1/sin(a + b*x))^(3/2),x)

[Out]

int((1/sin(a + b*x))^(3/2), x)