3.155 \(\int \frac{\sqrt{a+a \cos (x)}}{x^2} \, dx\)

Optimal. Leaf size=42 \[ -\frac{1}{2} \text{Si}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}-\frac{\sqrt{a \cos (x)+a}}{x} \]

[Out]

-(Sqrt[a + a*Cos[x]]/x) - (Sqrt[a + a*Cos[x]]*Sec[x/2]*SinIntegral[x/2])/2

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Rubi [A]  time = 0.0906291, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3319, 3297, 3299} \[ -\frac{1}{2} \text{Si}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}-\frac{\sqrt{a \cos (x)+a}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Cos[x]]/x^2,x]

[Out]

-(Sqrt[a + a*Cos[x]]/x) - (Sqrt[a + a*Cos[x]]*Sec[x/2]*SinIntegral[x/2])/2

Rule 3319

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((2*a)^IntPart[n
]*(a + b*Sin[e + f*x])^FracPart[n])/Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*FracPart[n]), Int[(c + d*x)^m*Sin[e/2
+ (a*Pi)/(4*b) + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+a \cos (x)}}{x^2} \, dx &=\left (\sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \frac{\cos \left (\frac{x}{2}\right )}{x^2} \, dx\\ &=-\frac{\sqrt{a+a \cos (x)}}{x}-\frac{1}{2} \left (\sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \frac{\sin \left (\frac{x}{2}\right )}{x} \, dx\\ &=-\frac{\sqrt{a+a \cos (x)}}{x}-\frac{1}{2} \sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right ) \text{Si}\left (\frac{x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.057508, size = 33, normalized size = 0.79 \[ -\frac{\sqrt{a (\cos (x)+1)} \left (x \text{Si}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right )+2\right )}{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Cos[x]]/x^2,x]

[Out]

-(Sqrt[a*(1 + Cos[x])]*(2 + x*Sec[x/2]*SinIntegral[x/2]))/(2*x)

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Maple [F]  time = 0.149, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt{a+a\cos \left ( x \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 2.18569, size = 31, normalized size = 0.74 \begin{align*} -\frac{1}{4} \, \sqrt{2} \sqrt{a}{\left (i \, \Gamma \left (-1, \frac{1}{2} i \, x\right ) - i \, \Gamma \left (-1, -\frac{1}{2} i \, x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*sqrt(a)*(I*gamma(-1, 1/2*I*x) - I*gamma(-1, -1/2*I*x))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (\cos{\left (x \right )} + 1\right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(cos(x) + 1))/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \cos \left (x\right ) + a}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cos(x) + a)/x^2, x)