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

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

[Out]

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

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Rubi [A]  time = 0.0227896, 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, 2639} \[ \frac{2 \sqrt [4]{13} E\left (\left .\frac{1}{2} \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )\right |2\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*13^(1/4)*EllipticE[(c + d*x - ArcTan[3/2])/2, 2])/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 2639

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

Rubi steps

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

Mathematica [C]  time = 0.864092, size = 184, normalized size = 6.81 \[ \frac{-\frac{3 \sqrt [4]{13} \sin \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cos ^2\left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )\right )}{\sqrt{-\left (\cos \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )-1\right ) \cos \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )} \sqrt{\cos \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )+1}}+4 \sqrt{3 \sin (c+d x)+2 \cos (c+d x)}-4 \sqrt [4]{13} \sqrt{\cos \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )}+\frac{3 \sqrt [4]{13} \sin \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )}{\sqrt{\cos \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )}}}{3 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.935, size = 112, normalized size = 4.2 \begin{align*} -{\frac{\sqrt{13}}{\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 ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1+\sin \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1+\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) \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((2*cos(d*x+c)+3*sin(d*x+c))^(1/2),x)

[Out]

-13^(1/2)*(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)
*(2*EllipticE((1+sin(d*x+c+arctan(2/3)))^(1/2),1/2*2^(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 \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((2*cos(d*x+c)+3*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

[Out]

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