3.7.61 \(\int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2+3 \cos (c+d x)}} \, dx\) [661]

Optimal. Leaf size=75 \[ -\frac {4 \cot (c+d x) \Pi \left (\frac {1}{3};\text {ArcSin}\left (\frac {\sqrt {-2+3 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 \sqrt {5} d} \]

[Out]

-4/15*cot(d*x+c)*EllipticPi((-2+3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),1/3,1/5*5^(1/2))*(-1+sec(d*x+c))^(1/2)*(1
+sec(d*x+c))^(1/2)/d*5^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 75, 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 = {2888} \begin {gather*} -\frac {4 \cot (c+d x) \sqrt {\sec (c+d x)-1} \sqrt {\sec (c+d x)+1} \Pi \left (\frac {1}{3};\text {ArcSin}\left (\frac {\sqrt {3 \cos (c+d x)-2}}{\sqrt {\cos (c+d x)}}\right )|\frac {1}{5}\right )}{3 \sqrt {5} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Cot[c + d*x]*EllipticPi[1/3, ArcSin[Sqrt[-2 + 3*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c +
d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*Sqrt[5]*d)

Rule 2888

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*b*(Tan
[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*El
lipticPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)],
x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]

Rubi steps

\begin {align*} \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2+3 \cos (c+d x)}} \, dx &=-\frac {4 \cot (c+d x) \Pi \left (\frac {1}{3};\sin ^{-1}\left (\frac {\sqrt {-2+3 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 \sqrt {5} d}\\ \end {align*}

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Mathematica [A]
time = 0.71, size = 140, normalized size = 1.87 \begin {gather*} -\frac {4 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {-2+3 \cos (c+d x)}{1+\cos (c+d x)}} \left (F\left (\text {ArcSin}\left (\sqrt {5} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {1}{5}\right )-2 \Pi \left (-\frac {1}{5};\text {ArcSin}\left (\sqrt {5} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {1}{5}\right )\right )}{\sqrt {5} d \sqrt {\cos (c+d x)} \sqrt {-2+3 \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(-2 + 3*Cos[c + d*x])/(1 + Cos[c + d*x])]*(E
llipticF[ArcSin[Sqrt[5]*Tan[(c + d*x)/2]], 1/5] - 2*EllipticPi[-1/5, ArcSin[Sqrt[5]*Tan[(c + d*x)/2]], 1/5]))/
(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. \(131\) vs. \(2(62)=124\).
time = 0.23, size = 132, normalized size = 1.76

method	result	size
default	\(-\frac {2 \left (\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \sqrt {5}\right )-2 \EllipticPi \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, -1, \sqrt {5}\right )\right ) \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{d \sqrt {-2+3 \cos \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\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(cos(c + d*x)^(1/2)/(3*cos(c + d*x) - 2)^(1/2),x)

[Out]

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

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