\(\int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx\) [659]

Optimal result

Integrand size = 27, antiderivative size = 60 \[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=-\frac {2 \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),-5\right ) \sqrt {-\tan ^2(c+d x)}}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 60, 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} \[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=-\frac {2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-2 \cos (c+d x)-3}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),-5\right )}{d} \]

[In]

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

[Out]

(-2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[-3 - 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], -5]*Sqrt[-Tan[c +
d*x]^2])/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 {5} \sqrt {-\cos (c+d x)}}\right ),-5\right ) \sqrt {-\tan ^2(c+d x)}}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(155\) vs. \(2(60)=120\).

Time = 0.44 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{3}} \sqrt {\frac {\cos (c+d x)}{-1+\cos (c+d x)}}\right ),\frac {6}{5}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{\sqrt {5} d \sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (57 ) = 114\).

Time = 6.84 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.97

method	result	size
default	\(\frac {\left (1+\cos \left (d x +c \right )\right ) F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-3-2 \cos \left (d x +c \right )}}{5 d \sqrt {-\cos \left (d x +c \right )}\, \left (3+2 \cos \left (d x +c \right )\right )}\)	\(118\)

[In]

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

[Out]

1/5/d*(1+cos(d*x+c))*EllipticF(cot(d*x+c)-csc(d*x+c),1/5*I*5^(1/2))*10^(1/2)*((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)/(3+2*cos(d*x+c))

Fricas [F]

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

[In]

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

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

[In]

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

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

[In]

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

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

[In]

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

[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)