3.7.75 \(\int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx\) [675]

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

[Out]

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

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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 = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2887} \begin {gather*} -\frac {3 \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \Pi \left (\frac {5}{2};\left .\text {ArcSin}\left (\frac {\sqrt {-2 \cos (c+d x)-3}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2887

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*c*Rt[b
*(c + d), 2]*Tan[e + f*x]*Sqrt[1 + Csc[e + f*x]]*(Sqrt[1 - Csc[e + f*x]]/(d*f*Sqrt[c^2 - d^2]))*EllipticPi[(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] && GtQ[c^2 - d^2, 0] && PosQ[(c + d)/b] && GtQ[c^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.33, size = 115, normalized size = 1.53 \begin {gather*} \frac {2 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (F\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|-\frac {1}{5}\right )-2 \Pi \left (-1;\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|-\frac {1}{5}\right )\right ) \sqrt {\cos (c+d x) (3+2 \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right )}}{\sqrt {5} d \sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (68 ) = 136\).
time = 0.21, size = 164, normalized size = 2.19

method	result	size
default	\(\frac {\sqrt {10}\, \sqrt {2}\, \left (\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right )-2 \EllipticPi \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, -1, \frac {i \sqrt {5}}{5}\right )\right ) \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {-3-2 \cos \left (d x +c \right )}\, \sqrt {-\cos \left (d x +c \right )}}{5 d \left (2 \left (\cos ^{2}\left (d x +c \right )\right )+\cos \left (d x +c \right )-3\right ) \cos \left (d x +c \right )}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

integral(-sqrt(-cos(d*x + c))*sqrt(-2*cos(d*x + c) - 3)/(2*cos(d*x + c) + 3), 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 {- 2 \cos {\left (c + d x \right )} - 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(sqrt(-cos(d*x + c))/sqrt(-2*cos(d*x + c) - 3), 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 {-2\,\cos \left (c+d\,x\right )-3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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