\(\int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx\) [658]

Optimal result

Integrand size = 27, antiderivative size = 62 \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=-\frac {2 \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-3+2 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right ),-\frac {1}{5}\right ) \sqrt {-\tan ^2(c+d x)}}{\sqrt {5} d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2894} \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=-\frac {2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 \cos (c+d x)-3}}{\sqrt {-\cos (c+d x)}}\right ),-\frac {1}{5}\right )}{\sqrt {5} d} \]

[In]

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

[Out]

(-2*Cot[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 2894

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \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*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(62)=124\).

Time = 0.56 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\frac {4 \sqrt {-\cot ^2\left (\frac {1}{2} (c+d x)\right )} \cot (c+d x) \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\left ((-3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )\right )} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {-3+2 \cos (c+d x)}{-1+\cos (c+d x)}}}{\sqrt {3}}\right ),\frac {6}{5}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{\sqrt {5} d (-\cos (c+d x))^{3/2} \sqrt {-3+2 \cos (c+d x)}} \]

[In]

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

[Out]

(4*Sqrt[-Cot[(c + d*x)/2]^2]*Cot[c + d*x]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[-((-3 + 2*Cos[c + d*x]
)*Csc[(c + d*x)/2]^2)]*EllipticF[ArcSin[Sqrt[(-3 + 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]/Sqrt[3]], 6/5]*Sin[(c
+ d*x)/2]^4)/(Sqrt[5]*d*(-Cos[c + d*x])^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]])

Maple [A] (verified)

Time = 4.67 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.55

method	result	size
default	\(\frac {2 F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-3+2 \cos \left (d x +c \right )}}{d \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\cos \left (d x +c \right )}}\)	\(96\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {2 \, \cos \left (d x + c\right ) - 3}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {- \cos {\left (c + d x \right )}} \sqrt {2 \cos {\left (c + d x \right )} - 3}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {2 \, \cos \left (d x + c\right ) - 3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {2 \, \cos \left (d x + c\right ) - 3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {2\,\cos \left (c+d\,x\right )-3}} \,d x \]

[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)