3.6.27 \(\int \frac {\csc (e+f x)}{\sqrt {d \csc (e+f x)}} \, dx\) [527]

Optimal. Leaf size=46 \[ \frac {2 \sqrt {d \csc (e+f x)} F\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {\sin (e+f x)}}{d f} \]

[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)/d/f

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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {16, 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)}}{d f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]/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]])/(d*f)

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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

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Mathematica [A]
time = 0.01, size = 45, 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)}}{d f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]/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]])/(d*f)

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Maple [C] Result contains complex when optimal does not.
time = 0.19, size = 165, normalized size = 3.59

method	result	size
default	\(-\frac {i \left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \cos \left (f x +e \right )-i-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {2}}{f \sqrt {\frac {d}{\sin \left (f x +e \right )}}\, \sin \left (f x +e \right )^{3}}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I/f*(-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)-I-sin(f*x+e))/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))*(cos(f*x+e)+1)^2*2^(1/2)/(d/sin(f*x+e))^(1/2)/sin(f*x+e)^3

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

[Out]

integrate(csc(f*x + e)/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 = 0.16, size = 62, normalized size = 1.35 \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 )}{d f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)/(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)))/(d*f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(e + f*x)*(d/sin(e + f*x))^(1/2)),x)

[Out]

int(1/(sin(e + f*x)*(d/sin(e + f*x))^(1/2)), x)

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