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

Optimal. Leaf size=56 \[ -\frac {2 \sqrt {-\cos (c+d x)} F\left (\text {ArcSin}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )|\frac {1}{5}\right )}{\sqrt {5} d \sqrt {\cos (c+d x)}} \]

[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)

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Rubi [A]
time = 0.08, antiderivative size = 56, 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 = {2893, 2892} \begin {gather*} -\frac {2 \sqrt {-\cos (c+d x)} F\left (\text {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 {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx &=\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)} F\left (\sin ^{-1}\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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(56)=112\).
time = 1.16, size = 143, normalized size = 2.55 \begin {gather*} -\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) F\left (\left .\text {ArcSin}\left (\frac {1}{2} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\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)}} \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[(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]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(118\) vs. \(2(51)=102\).
time = 0.67, size = 119, normalized size = 2.12

method	result	size
default	\(\frac {2 \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {2-3 \cos \left (d x +c \right )}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \sqrt {5}\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{\frac {3}{2}} \left (-2+3 \cos \left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{2}}\)	\(119\)

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]

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

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

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

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

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {2-3\,\cos \left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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