3.7.45 \(\int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-2+3 \cos (c+d x)}} \, dx\) [645]

Optimal. Leaf size=25 \[ \frac {2 F\left (\left .\text {ArcSin}\left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right )\right |5\right )}{d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2892} \begin {gather*} \frac {2 F\left (\left .\text {ArcSin}\left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )\right |5\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[-2 + 3*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 {\cos (c+d x)} \sqrt {-2+3 \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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(156\) vs. \(2(25)=50\).
time = 0.98, size = 156, normalized size = 6.24 \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 {-\left ((-2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )\right )} \csc (c+d x) F\left (\text {ArcSin}\left (\frac {1}{2} \sqrt {-\left ((-2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )\right )}\right )|\frac {4}{5}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{\sqrt {5} d \sqrt {\cos (c+d x)} \sqrt {-2+3 \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs. \(2(26)=52\).
time = 0.63, size = 107, normalized size = 4.28

method	result	size
default	\(-\frac {2 \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \left (\sin ^{4}\left (d x +c \right )\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \sqrt {5}\right )}{d \sqrt {-2+3 \cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{\frac {3}{2}} \left (-1+\cos \left (d x +c \right )\right )^{2}}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(d*x+c)^(1/2)/(-2+3*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)*sin(d*x+c)^4*((-2+3*cos(d*x+c))/(1+cos(d*x+c)))
^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),5^(1/2))/cos(d*x+c)^(3/2)/(-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/cos(d*x+c)^(1/2)/(-2+3*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/cos(d*x+c)^(1/2)/(-2+3*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 {3 \cos {\left (c + d x \right )} - 2} \sqrt {\cos {\left (c + d x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(3*cos(c + d*x) - 2)*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/cos(d*x+c)^(1/2)/(-2+3*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 = -0.04 \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)

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