3.1.20 \(\int \sqrt {c \csc (a+b x)} \, dx\) [20]

Optimal. Leaf size=43 \[ \frac {2 \sqrt {c \csc (a+b x)} F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b} \]

[Out]

-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

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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3856, 2720} \begin {gather*} \frac {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)}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*Csc[a + b*x]],x]

[Out]

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

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Mathematica [A]
time = 0.03, size = 42, normalized size = 0.98 \begin {gather*} -\frac {2 \sqrt {c \csc (a+b x)} F\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {\sin (a+b x)}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*Csc[a + b*x]],x]

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 1.10, size = 165, normalized size = 3.84

method	result	size
default	\(-\frac {i \sqrt {2}\, \sqrt {\frac {c}{\sin \left (x b +a \right )}}\, \left (-1+\cos \left (x b +a \right )\right ) \sqrt {\frac {i \cos \left (x b +a \right )-i+\sin \left (x b +a \right )}{\sin \left (x b +a \right )}}\, \sqrt {-\frac {i \cos \left (x b +a \right )-i-\sin \left (x b +a \right )}{\sin \left (x b +a \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (x b +a \right )\right )}{\sin \left (x b +a \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x b +a \right )-i+\sin \left (x b +a \right )}{\sin \left (x b +a \right )}}, \frac {\sqrt {2}}{2}\right ) \left (\cos \left (x b +a \right )+1\right )^{2}}{b \sin \left (x b +a \right )^{2}}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*csc(b*x+a))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*csc(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*csc(b*x + a)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.65, size = 55, normalized size = 1.28 \begin {gather*} \frac {-i \, \sqrt {2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {-2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*csc(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c \csc {\left (a + b x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*csc(b*x+a))**(1/2),x)

[Out]

Integral(sqrt(c*csc(a + b*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*csc(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*csc(b*x + a)), x)

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Mupad [B]
time = 0.20, size = 63, normalized size = 1.47 \begin {gather*} -\frac {2\,\sqrt {\sin \left (a+b\,x\right )}\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (a+b\,x\right )}}{2}\right )\middle |2\right )\,\sqrt {\frac {c}{\sin \left (a+b\,x\right )}}\,\sqrt {{\cos \left (a+b\,x\right )}^2}}{b\,\cos \left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/sin(a + b*x))^(1/2),x)

[Out]

-(2*sin(a + b*x)^(1/2)*ellipticF(asin((2^(1/2)*(1 - sin(a + b*x))^(1/2))/2), 2)*(c/sin(a + b*x))^(1/2)*(cos(a
+ b*x)^2)^(1/2))/(b*cos(a + b*x))

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