\(\int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx\) [646]

Optimal result

Integrand size = 25, antiderivative size = 56 \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {-\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {\cos (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2893, 2892} \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {-\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {\cos (c+d x)}} \]

[In]

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

[Out]

(-2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 1/5])/(Sqrt[5]*d*Sqrt[Cos[c + d*x]]
)

Rule 2892

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

Rule 2893

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx}{\sqrt {\cos (c+d x)}} \\ & = -\frac {2 \sqrt {-\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {\cos (c+d x)}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(56)=112\).

Time = 1.71 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.55 \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(2-3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),-4\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{d \sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs. \(2(51)=102\).

Time = 7.70 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.91

method	result	size
default	\(\frac {2 \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2-3 \cos \left (d x +c \right )}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )}{d \left (-2+3 \cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}\)	\(107\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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