3.650 \(\int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx\)

Optimal. Leaf size=84 \[ -\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \sqrt {-\tan ^2(c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {2 \cos (c+d x)-3}}{\sqrt {-\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{\sqrt {5} d} \]

[Out]

-2/5*csc(d*x+c)*EllipticF((-3+2*cos(d*x+c))^(1/2)/(-cos(d*x+c))^(1/2),1/5*I*5^(1/2))*(-cos(d*x+c))^(1/2)*cos(d
*x+c)^(1/2)*(-tan(d*x+c)^2)^(1/2)/d*5^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2817, 2815} \[ -\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \sqrt {-\tan ^2(c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {2 \cos (c+d x)-3}}{\sqrt {-\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{\sqrt {5} d} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]),x]

[Out]

(-2*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c
+ d*x]]], -1/5]*Sqrt[-Tan[c + d*x]^2])/(Sqrt[5]*d)

Rule 2815

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*
Sqrt[a^2]*Sqrt[-Cot[e + f*x]^2]*Rt[(a + b)/d, 2]*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[d*Sin[e + f*x
]]*Rt[(a + b)/d, 2])], -((a + b)/(a - b))])/(a*f*Sqrt[a^2 - b^2]*Cot[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x
] && GtQ[a^2 - b^2, 0] && PosQ[(a + b)/d] && GtQ[a^2, 0]

Rule 2817

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt
[-(d*Sin[e + f*x])]/Sqrt[d*Sin[e + f*x]], Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[-(d*Sin[e + f*x])]), x], x] /;
FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && NegQ[(a + b)/d]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx &=\frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx}{\sqrt {\cos (c+d x)}}\\ &=-\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {-3+2 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|-\frac {1}{5}\right ) \sqrt {-\tan ^2(c+d x)}}{\sqrt {5} d}\\ \end {align*}

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Mathematica [A]  time = 1.26, size = 144, normalized size = 1.71 \[ \frac {2 \sqrt {\cos (c+d x)} \sqrt {\frac {2 \cos (c+d x)-3}{\cos (c+d x)-1}} \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {-\cot ^2\left (\frac {1}{2} (c+d x)\right )} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {2 \cos (c+d x)-3}{\cos (c+d x)-1}}}{\sqrt {3}}\right )|\frac {6}{5}\right )}{\sqrt {5} d \sqrt {\frac {\cos (c+d x)}{\cos (c+d x)-1}} \sqrt {2 \cos (c+d x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]),x]

[Out]

(2*Sqrt[Cos[c + d*x]]*Sqrt[(-3 + 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]*Sqrt[-Cot[(c + d*x)/2]^2]*EllipticF[ArcS
in[Sqrt[(-3 + 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]/Sqrt[3]], 6/5]*Tan[(c + d*x)/2])/(Sqrt[5]*d*Sqrt[Cos[c + d*
x]/(-1 + Cos[c + d*x])]*Sqrt[-3 + 2*Cos[c + d*x]])

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fricas [F]  time = 1.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2 \, \cos \left (d x + c\right ) - 3} \sqrt {\cos \left (d x + c\right )}}{2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(d*x+c)^(1/2)/(-3+2*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(2*cos(d*x + c) - 3)*sqrt(cos(d*x + c))/(2*cos(d*x + c)^2 - 3*cos(d*x + c)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) - 3} \sqrt {\cos \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(d*x+c)^(1/2)/(-3+2*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(2*cos(d*x + c) - 3)*sqrt(cos(d*x + c))), x)

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maple [A]  time = 0.20, size = 123, normalized size = 1.46 \[ \frac {i \sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \left (\sin ^{4}\left (d x +c \right )\right ) \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sqrt {5}}{5 d \sqrt {-3+2 \cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{\frac {3}{2}} \left (-1+\cos \left (d x +c \right )\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(d*x+c)^(1/2)/(-3+2*cos(d*x+c))^(1/2),x)

[Out]

1/5*I/d*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)/(-3+2*cos(d*x+c))^(1/2)*sin(d*x+c)^4*(-2*(-3+2*cos(d*x+c))/(
1+cos(d*x+c)))^(1/2)*EllipticF(I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5*I*5^(1/2))/cos(d*x+c)^(3/2)/(-1+cos(d*
x+c))^2*5^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) - 3} \sqrt {\cos \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(d*x+c)^(1/2)/(-3+2*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(2*cos(d*x + c) - 3)*sqrt(cos(d*x + c))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^(1/2)*(2*cos(c + d*x) - 3)^(1/2)),x)

[Out]

int(1/(cos(c + d*x)^(1/2)*(2*cos(c + d*x) - 3)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \cos {\left (c + d x \right )} - 3} \sqrt {\cos {\left (c + d x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(d*x+c)**(1/2)/(-3+2*cos(d*x+c))**(1/2),x)

[Out]

Integral(1/(sqrt(2*cos(c + d*x) - 3)*sqrt(cos(c + d*x))), x)

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