3.644 \(\int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx\)

Optimal. Leaf size=32 \[ \frac {2 F\left (\sin ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )|\frac {1}{5}\right )}{\sqrt {5} d} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 32, 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 = {2813} \[ \frac {2 F\left (\sin ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )|\frac {1}{5}\right )}{\sqrt {5} d} \]

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])], 1/5])/(Sqrt[5]*d)

Rule 2813

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*
d*EllipticF[ArcSin[Cos[e + f*x]/(1 + d*Sin[e + f*x])], -((a - b*d)/(a + b*d))])/(f*Sqrt[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 (\sin ^{-1}\left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right )|\frac {1}{5}\right )}{\sqrt {5} d}\\ \end {align*}

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Mathematica [B]  time = 3.66, size = 131, normalized size = 4.09 \[ \frac {2 \sqrt {\cos (c+d x)} \sqrt {3 \cos (c+d x)+2} \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac {1}{2} \sqrt {(3 \cos (c+d x)+2) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right |-4\right )}{d \sqrt {\frac {-3 \cos (c+d x)-2}{\cos (c+d x)-1}} \sqrt {\frac {\cos (c+d x)}{\cos (c+d x)-1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {3 \, \cos \left (d x + c\right ) + 2} \sqrt {\cos \left (d x + c\right )}}{3 \, \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right )}, x\right ) \]

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

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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maple [B]  time = 0.20, size = 115, normalized size = 3.59 \[ -\frac {\sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \left (\sin ^{4}\left (d x +c \right )\right ) \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )}{5 d \sqrt {2+3 \cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{\frac {3}{2}} \left (-1+\cos \left (d x +c \right )\right )^{2}} \]

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)

[Out]

-1/5/d*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)/(2+3*cos(d*x+c))^(1/2)*sin(d*x+c)^4*10^(1/2)*((2+3*cos(d*x+c)
)/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*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 \[ \int \frac {1}{\sqrt {3 \, \cos \left (d x + c\right ) + 2} \sqrt {\cos \left (d x + c\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {3\,\cos \left (c+d\,x\right )+2}} \,d x \]

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

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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