3.7.55 \(\int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx\) [655]

Optimal. Leaf size=27 \[ -\frac {2 F\left (\left .\text {ArcSin}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2892} \begin {gather*} -\frac {2 F\left (\left .\text {ArcSin}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(155\) vs. \(2(27)=54\).
time = 0.50, size = 155, normalized size = 5.74 \begin {gather*} \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 {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) F\left (\text {ArcSin}\left (\sqrt {\frac {5}{2}} \sqrt {\frac {\cos (c+d x)}{-1+\cos (c+d x)}}\right )|\frac {4}{5}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{\sqrt {5} d \sqrt {-2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(128\) vs. \(2(28)=56\).
time = 0.20, size = 129, normalized size = 4.78

method	result	size
default	\(-\frac {\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right ) \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-2-3 \cos \left (d x +c \right )}\, \left (\sin ^{2}\left (d x +c \right )\right )}{5 d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-\cos \left (d x +c \right )-2\right ) \sqrt {-\cos \left (d x +c \right )}}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {-3\,\cos \left (c+d\,x\right )-2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________