3.406 \(\int \frac{1}{\sqrt{2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx\)

Optimal. Leaf size=45 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right ),\frac{2}{15} \left (17-\sqrt{34}\right )\right )}{\sqrt{2+\sqrt{34}} e} \]

[Out]

(2*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e)

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Rubi [A]  time = 0.0372017, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3126, 2661} \[ \frac{2 F\left (\frac{1}{2} \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )|\frac{2}{15} \left (17-\sqrt{34}\right )\right )}{\sqrt{2+\sqrt{34}} e} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]],x]

[Out]

(2*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e)

Rule 3126

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Int[1/Sqrt[a +
Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a +
Sqrt[b^2 + c^2], 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx &=\int \frac{1}{\sqrt{2+\sqrt{34} \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )}} \, dx\\ &=\frac{2 F\left (\frac{1}{2} \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )|\frac{2}{15} \left (17-\sqrt{34}\right )\right )}{\sqrt{2+\sqrt{34}} e}\\ \end{align*}

Mathematica [C]  time = 0.261814, size = 128, normalized size = 2.84 \[ \frac{\sqrt{\frac{2}{15}} \sqrt{\sqrt{34} \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+2} \sqrt{\cos ^2\left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )} \sec \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right ) F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+\sqrt{34}}{17+\sqrt{34}}\right )}{e} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]],x]

[Out]

(Sqrt[2/15]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[34]), (Sqrt[34
] + 17*Sin[d + e*x + ArcTan[3/5]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcTan[3/5]]^2]*Sec[d + e*x + ArcTan[3/
5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]])/e

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Maple [C]  time = 2.134, size = 158, normalized size = 3.5 \begin{align*}{\frac{ \left ( 2\,\sqrt{34}-34 \right ) \sqrt{17}}{17\,\cos \left ( ex+d+\arctan \left ( 3/5 \right ) \right ) e}\sqrt{-{\frac{17\,\sin \left ( ex+d+\arctan \left ( 3/5 \right ) \right ) +\sqrt{34}}{-\sqrt{34}+17}}}\sqrt{-17\,{\frac{\sin \left ( ex+d+\arctan \left ( 3/5 \right ) \right ) -1}{\sqrt{34}+17}}}\sqrt{{\frac{1+\sin \left ( ex+d+\arctan \left ({\frac{3}{5}} \right ) \right ) }{-\sqrt{34}+17}}}{\it EllipticF} \left ( \sqrt{-{\frac{17\,\sin \left ( ex+d+\arctan \left ( 3/5 \right ) \right ) +\sqrt{34}}{-\sqrt{34}+17}}},i\sqrt{{\frac{-\sqrt{34}+17}{\sqrt{34}+17}}} \right ){\frac{1}{\sqrt{\sqrt{34}\sin \left ( ex+d+\arctan \left ({\frac{3}{5}} \right ) \right ) +2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2),x)

[Out]

2/17*(34^(1/2)-17)*(-(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2)*(-17*(sin(e*x+d+arctan(3/5))-1
)/(34^(1/2)+17))^(1/2)*17^(1/2)*((1+sin(e*x+d+arctan(3/5)))/(-34^(1/2)+17))^(1/2)*EllipticF((-(17*sin(e*x+d+ar
ctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2),I*((-34^(1/2)+17)/(34^(1/2)+17))^(1/2))/cos(e*x+d+arctan(3/5))/(34^
(1/2)*sin(e*x+d+arctan(3/5))+2)^(1/2)/e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(3*cos(e*x + d) + 5*sin(e*x + d) + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2),x, algorithm="fricas")

[Out]

integral(1/sqrt(3*cos(e*x + d) + 5*sin(e*x + d) + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{5 \sin{\left (d + e x \right )} + 3 \cos{\left (d + e x \right )} + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))**(1/2),x)

[Out]

Integral(1/sqrt(5*sin(d + e*x) + 3*cos(d + e*x) + 2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(3*cos(e*x + d) + 5*sin(e*x + d) + 2), x)