3.419 \(\int \sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx\)

Optimal. Leaf size=44 \[ -\frac{2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}} \]

[Out]

(-2*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])

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Rubi [A]  time = 0.0182359, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {3112} \[ -\frac{2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]

[Out]

(-2*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])

Rule 3112

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

Rubi steps

\begin{align*} \int \sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx &=-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)}}\\ \end{align*}

Mathematica [A]  time = 0.0383478, size = 75, normalized size = 1.7 \[ -\frac{2 \left (\cos \left (\frac{1}{2} (d+e x)\right )-3 \sin \left (\frac{1}{2} (d+e x)\right )\right ) \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}}{e \left (\sin \left (\frac{1}{2} (d+e x)\right )+3 \cos \left (\frac{1}{2} (d+e x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]

[Out]

(-2*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2])*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(e*(3*Cos[(d + e*x)/2]
+ Sin[(d + e*x)/2]))

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Maple [A]  time = 0.89, size = 50, normalized size = 1.1 \begin{align*} 10\,{\frac{ \left ( \sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) -1 \right ) \left ( 1+\sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) \right ) }{\cos \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) \sqrt{5+5\,\sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) }e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x)

[Out]

10*(sin(e*x+d+arctan(4/3))-1)*(1+sin(e*x+d+arctan(4/3)))/cos(e*x+d+arctan(4/3))/(5+5*sin(e*x+d+arctan(4/3)))^(
1/2)/e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5), x)

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Fricas [A]  time = 1.74993, size = 167, normalized size = 3.8 \begin{align*} -\frac{2 \, \sqrt{4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}{\left (\cos \left (e x + d\right ) - 3 \, \sin \left (e x + d\right ) + 1\right )}}{3 \, e \cos \left (e x + d\right ) + e \sin \left (e x + d\right ) + 3 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5)*(cos(e*x + d) - 3*sin(e*x + d) + 1)/(3*e*cos(e*x + d) + e*sin(e*x
+ d) + 3*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{3 \sin{\left (d + e x \right )} + 4 \cos{\left (d + e x \right )} + 5}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))**(1/2),x)

[Out]

Integral(sqrt(3*sin(d + e*x) + 4*cos(d + e*x) + 5), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5), x)