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

Optimal. Leaf size=27 \[ \frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{\sqrt [4]{13} d} \]

[Out]

2/13*13^(3/4)*(cos(1/2*c+1/2*d*x-1/2*arctan(3/2))^2)^(1/2)/cos(1/2*c+1/2*d*x-1/2*arctan(3/2))*EllipticF(sin(1/
2*c+1/2*d*x-1/2*arctan(3/2)),2^(1/2))/d

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3077, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{\sqrt [4]{13} d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]],x]

[Out]

(2*EllipticF[(c + d*x - ArcTan[3/2])/2, 2])/(13^(1/4)*d)

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 3077

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a^2 + b^2)^(n/2),
Int[Cos[c + d*x - ArcTan[a, b]]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] &&  !(GeQ[n, 1] || LeQ[n, -1]) && GtQ[
a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {2 \cos (c+d x)+3 \sin (c+d x)}} \, dx &=\frac {\int \frac {1}{\sqrt {\cos \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )}} \, dx}{\sqrt [4]{13}}\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{\sqrt [4]{13} d}\\ \end {align*}

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Mathematica [C]  time = 0.09, size = 88, normalized size = 3.26 \[ \frac {2 \sqrt {-\left (\left (\sin \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )-1\right ) \sin \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )\right )} \sqrt {\sin \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )+1} \sec \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )\right )}{\sqrt [4]{13} d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[2/3]]^2]*Sec[c + d*x + ArcTan[2/3]]*Sqrt[-((-1 +
Sin[c + d*x + ArcTan[2/3]])*Sin[c + d*x + ArcTan[2/3]])]*Sqrt[1 + Sin[c + d*x + ArcTan[2/3]]])/(13^(1/4)*d)

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fricas [F]  time = 1.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(2*cos(d*x + c) + 3*sin(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*cos(d*x+c)+3*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.24, size = 85, normalized size = 3.15 \[ \frac {\sqrt {1+\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \sqrt {-2 \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )+2}\, \sqrt {-\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \EllipticF \left (\sqrt {1+\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{\cos \left (d x +c +\arctan \left (\frac {2}{3}\right )\right ) \sqrt {\sqrt {13}\, \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1+sin(d*x+c+arctan(2/3)))^(1/2)*(-2*sin(d*x+c+arctan(2/3))+2)^(1/2)*(-sin(d*x+c+arctan(2/3)))^(1/2)*EllipticF
((1+sin(d*x+c+arctan(2/3)))^(1/2),1/2*2^(1/2))/cos(d*x+c+arctan(2/3))/(13^(1/2)*sin(d*x+c+arctan(2/3)))^(1/2)/
d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\sqrt {2\,\cos \left (c+d\,x\right )+3\,\sin \left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(3*sin(c + d*x) + 2*cos(c + d*x)), x)

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