3.29 \(\int \frac {1}{\sqrt {c \sin (a+b x)}} \, dx\)

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

[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))*
sin(b*x+a)^(1/2)/b/(c*sin(b*x+a))^(1/2)

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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 = {2642, 2641} \[ \frac {2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {c \sin (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c*Sin[a + b*x]],x]

[Out]

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

Rule 2641

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

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {c \sin (a+b x)}} \, dx &=\frac {\sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{\sqrt {c \sin (a+b x)}}\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b \sqrt {c \sin (a+b x)}}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c*Sin[a + b*x]],x]

[Out]

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

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \sin \left (b x + a\right )}}{c \sin \left (b x + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*sin(b*x + a))/(c*sin(b*x + a)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \sin \left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(c*sin(b*x + a)), x)

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maple [A]  time = 0.05, size = 74, normalized size = 1.72 \[ -\frac {\sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) \EllipticF \left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{\cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*sin(b*x+a))^(1/2),x)

[Out]

-(-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))/co
s(b*x+a)/(c*sin(b*x+a))^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \sin \left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(c*sin(b*x + a)), x)

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mupad [B]  time = 0.48, size = 36, normalized size = 0.84 \[ -\frac {2\,\sqrt {\sin \left (a+b\,x\right )}\,\mathrm {F}\left (\frac {\pi }{4}-\frac {a}{2}-\frac {b\,x}{2}\middle |2\right )}{b\,\sqrt {c\,\sin \left (a+b\,x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*sin(a + b*x))^(1/2),x)

[Out]

-(2*sin(a + b*x)^(1/2)*ellipticF(pi/4 - a/2 - (b*x)/2, 2))/(b*(c*sin(a + b*x))^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \sin {\left (a + b x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(c*sin(a + b*x)), x)

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