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

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

[Out]

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

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Rubi [A]  time = 0.0240906, antiderivative size = 27, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [C]  time = 0.109203, size = 88, normalized size = 3.26 \[ \frac{2 \sqrt{-\left (\sin \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )-1\right ) \sin \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )} \sqrt{\sin \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )+1} \sec \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )\right )}{\sqrt [4]{13} d} \]

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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Maple [A]  time = 0.687, size = 85, normalized size = 3.2 \begin{align*}{\frac{1}{\cos \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) d}\sqrt{1+\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) }\sqrt{-2\,\sin \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) +2}\sqrt{-\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) }{\it EllipticF} \left ( \sqrt{1+\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{\sqrt{13}\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) }}}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}, x\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \sin{\left (c + d x \right )} + 2 \cos{\left (c + d x \right )}}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}\,{d x} \end{align*}

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)