3.4 \(\int \sqrt{a+a \cos (c+d x)} \, dx\)

Optimal. Leaf size=26 \[ \frac{2 a \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}} \]

[Out]

(2*a*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])

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Rubi [A]  time = 0.0127332, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2646} \[ \frac{2 a \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Cos[c + d*x]],x]

[Out]

(2*a*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])

Rule 2646

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

Rubi steps

\begin{align*} \int \sqrt{a+a \cos (c+d x)} \, dx &=\frac{2 a \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0300953, size = 29, normalized size = 1.12 \[ \frac{2 \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Cos[c + d*x]],x]

[Out]

(2*Sqrt[a*(1 + Cos[c + d*x])]*Tan[(c + d*x)/2])/d

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Maple [A]  time = 0.491, size = 43, normalized size = 1.7 \begin{align*} 2\,{\frac{a\cos \left ( 1/2\,dx+c/2 \right ) \sin \left ( 1/2\,dx+c/2 \right ) \sqrt{2}}{\sqrt{ \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+cos(d*x+c)*a)^(1/2),x)

[Out]

2*a*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)*2^(1/2)/(cos(1/2*d*x+1/2*c)^2*a)^(1/2)/d

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Maxima [A]  time = 2.01981, size = 27, normalized size = 1.04 \begin{align*} \frac{2 \, \sqrt{2} \sqrt{a} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(2)*sqrt(a)*sin(1/2*d*x + 1/2*c)/d

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Fricas [A]  time = 1.56153, size = 84, normalized size = 3.23 \begin{align*} \frac{2 \, \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/(d*cos(d*x + c) + d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(a*cos(c + d*x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*cos(d*x + c) + a), x)