3.6.11 \(\int \sqrt {d \csc (e+f x)} \, dx\) [511]

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

[Out]

-2*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2))*
(d*csc(f*x+e))^(1/2)*sin(f*x+e)^(1/2)/f

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Rubi [A]
time = 0.02, 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 (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*Csc[e + f*x]],x]

[Out]

(2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/f

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 {d \csc (e+f x)} \, dx &=\left (\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx\\ &=\frac {2 \sqrt {d \csc (e+f x)} F\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {\sin (e+f x)}}{f}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*Csc[e + f*x]],x]

[Out]

(-2*Sqrt[d*Csc[e + f*x]]*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]])/f

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Maple [C] Result contains complex when optimal does not.
time = 0.12, size = 162, normalized size = 3.77

method	result	size
default	\(-\frac {i \sqrt {2}\, \sqrt {\frac {d}{\sin \left (f x +e \right )}}\, \left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-i \cos \left (f x +e \right )+\sin \left (f x +e \right )+i}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \left (\cos \left (f x +e \right )+1\right )^{2}}{f \sin \left (f x +e \right )^{2}}\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*csc(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-I/f*2^(1/2)*(d/sin(f*x+e))^(1/2)*(-1+cos(f*x+e))*((I*cos(f*x+e)-I+sin(f*x+e))/sin(f*x+e))^(1/2)*((-I*cos(f*x+
e)+sin(f*x+e)+I)/sin(f*x+e))^(1/2)*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF(((I*cos(f*x+e)-I+sin(f*x+e)
)/sin(f*x+e))^(1/2),1/2*2^(1/2))/sin(f*x+e)^2*(cos(f*x+e)+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((d*csc(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*csc(f*x + e)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 59, normalized size = 1.37 \begin {gather*} \frac {-i \, \sqrt {2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {-2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

(-I*sqrt(2*I*d)*weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e)) + I*sqrt(-2*I*d)*weierstrassPInverse(
4, 0, cos(f*x + e) - I*sin(f*x + e)))/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(d*csc(e + f*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((d*csc(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*csc(f*x + e)), x)

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Mupad [B]
time = 0.62, size = 63, normalized size = 1.47 \begin {gather*} -\frac {2\,\sqrt {\sin \left (e+f\,x\right )}\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |2\right )\,\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}\,\sqrt {{\cos \left (e+f\,x\right )}^2}}{f\,\cos \left (e+f\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d/sin(e + f*x))^(1/2),x)

[Out]

-(2*sin(e + f*x)^(1/2)*ellipticF(asin((2^(1/2)*(1 - sin(e + f*x))^(1/2))/2), 2)*(d/sin(e + f*x))^(1/2)*(cos(e
+ f*x)^2)^(1/2))/(f*cos(e + f*x))

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