3.525 \(\int \sqrt{a+a \sin (e+f x)} \, dx\)

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

[Out]

(-2*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]])

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Rubi [A]  time = 0.0138016, 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} \[ -\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(-2*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*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 \sin (e+f x)} \, dx &=-\frac{2 a \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}

Mathematica [B]  time = 0.0334138, size = 65, normalized size = 2.5 \[ \frac{2 \sqrt{a (\sin (e+f x)+1)} \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )\right )}{f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(2*(-Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])])/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])
)

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Maple [A]  time = 0.467, size = 43, normalized size = 1.7 \begin{align*} 2\,{\frac{a \left ( 1+\sin \left ( fx+e \right ) \right ) \left ( -1+\sin \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) \sqrt{a+a\sin \left ( fx+e \right ) }f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(1/2),x)

[Out]

2*(1+sin(f*x+e))*a*(-1+sin(f*x+e))/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(f*x + e) + a), x)

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Fricas [B]  time = 1.77519, size = 136, normalized size = 5.23 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (f x + e\right ) + a}{\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/(f*cos(f*x + e) + f*sin(f*x + e) + f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(a*sin(e + f*x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sin(f*x + e) + a), x)