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

Optimal. Leaf size=71 \[ -\frac {2 c^2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b} \]

[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]  time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3768, 3771, 2639} \[ -\frac {2 c^2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b} \]

Antiderivative was successfully verified.

[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 2639

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

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]

Rubi steps

\begin {align*} \int (c \csc (a+b x))^{3/2} \, dx &=-\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]  time = 0.11, size = 54, normalized size = 0.76 \[ \frac {(c \csc (a+b x))^{3/2} \left (2 \sin ^{\frac {3}{2}}(a+b x) E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )-\sin (2 (a+b x))\right )}{b} \]

Antiderivative was successfully verified.

[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

________________________________________________________________________________________

fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c \csc \left (b x + a\right )} c \csc \left (b x + a\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x + a\right )\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*csc(b*x + a))^(3/2), x)

________________________________________________________________________________________

maple [C]  time = 1.18, size = 505, normalized size = 7.11 \[ -\frac {\left (-2 \cos \left (b x +a \right ) \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-i \cos \left (b x +a \right )+\sin \left (b x +a \right )+i}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}+\cos \left (b x +a \right ) \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-i \cos \left (b x +a \right )+\sin \left (b x +a \right )+i}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}-2 \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-i \cos \left (b x +a \right )+\sin \left (b x +a \right )+i}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}+\sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-i \cos \left (b x +a \right )+\sin \left (b x +a \right )+i}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}+\sqrt {2}\right ) \left (\frac {c}{\sin \left (b x +a \right )}\right )^{\frac {3}{2}} \sin \left (b x +a \right ) \sqrt {2}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b*(-2*cos(b*x+a)*(-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)*Elli
pticE(((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/
2)+cos(b*x+a)*(-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)*EllipticF(
((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2)-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)*EllipticE(((I*cos(b*x+a)-
I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/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)*EllipticF(((I*cos(b*x+a)-I+sin(b*x+a))/sin
(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2)+2^(1/2))*(c/sin(b*x+a))^(3/2)*sin(b
*x+a)*2^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x + a\right )\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*csc(b*x + a))^(3/2), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/sin(a + b*x))^(3/2),x)

[Out]

int((c/sin(a + b*x))^(3/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*csc(a + b*x))**(3/2), x)

________________________________________________________________________________________