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

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

[Out]

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

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Rubi [A]  time = 0.0305556, 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 = {3118, 2653} \[ \frac{2 \sqrt{2+\sqrt{34}} E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )|\frac{2}{15} \left (17-\sqrt{34}\right )\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3118

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Int[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 2653

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

Rubi steps

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

Mathematica [C]  time = 2.30288, size = 326, normalized size = 7.24 \[ \frac{\sqrt{\sin ^2\left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )} \left (2 \sqrt{30} \sqrt{\sqrt{34} \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+2} \sqrt{\cos ^2\left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )} \sqrt{\sqrt{34} \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )+2} \sec \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right ) F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+\sqrt{34}}{17+\sqrt{34}}\right )+45 \sin (d+e x)-75 \cos (d+e x)\right )-15 \sqrt{30} \sin \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right ) F_1\left (-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};\frac{17 \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )+\sqrt{34}}{17+\sqrt{34}}\right )}{15 e \sqrt{\sin ^2\left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )} \sqrt{\sqrt{34} \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )+2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-15*Sqrt[30]*AppellF1[-1/2, -1/2, -1/2, 1/2, (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(-17 + Sqrt[34]), (Sq
rt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(17 + Sqrt[34])]*Sin[d + e*x - ArcTan[5/3]] + (-75*Cos[d + e*x] + 45*S
in[d + e*x] + 2*Sqrt[30]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[3
4]), (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcTan[3/5]]^2]*Sqrt[2 + S
qrt[34]*Cos[d + e*x - ArcTan[5/3]]]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]])*
Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])/(15*e*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]]*Sqrt[Sin[d + e*x - Arc
Tan[5/3]]^2])

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Maple [C]  time = 2.867, size = 316, normalized size = 7. \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}}} \left ({\it EllipticE} \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 ) \sqrt{34}-\sqrt{34}{\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 ) +2\,{\it EllipticE} \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 ) -2\,{\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 ) \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((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)*(EllipticE((-(17*sin(e*x+d+
arctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2),I*((-34^(1/2)+17)/(34^(1/2)+17))^(1/2))*34^(1/2)-34^(1/2)*Ellipti
cF((-(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2),I*((-34^(1/2)+17)/(34^(1/2)+17))^(1/2))+2*Elli
pticE((-(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2),I*((-34^(1/2)+17)/(34^(1/2)+17))^(1/2))-2*E
llipticF((-(17*sin(e*x+d+arctan(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 \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((2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

[Out]

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