3.156 \(\int \frac{\sqrt{a+a \cos (x)}}{x^3} \, dx\)

Optimal. Leaf size=67 \[ -\frac{1}{8} \text{CosIntegral}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}-\frac{\sqrt{a \cos (x)+a}}{2 x^2}+\frac{\tan \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}}{4 x} \]

[Out]

-Sqrt[a + a*Cos[x]]/(2*x^2) - (Sqrt[a + a*Cos[x]]*CosIntegral[x/2]*Sec[x/2])/8 + (Sqrt[a + a*Cos[x]]*Tan[x/2])
/(4*x)

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Rubi [A]  time = 0.106145, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3319, 3297, 3302} \[ -\frac{1}{8} \text{CosIntegral}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}-\frac{\sqrt{a \cos (x)+a}}{2 x^2}+\frac{\tan \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Cos[x]]/x^3,x]

[Out]

-Sqrt[a + a*Cos[x]]/(2*x^2) - (Sqrt[a + a*Cos[x]]*CosIntegral[x/2]*Sec[x/2])/8 + (Sqrt[a + a*Cos[x]]*Tan[x/2])
/(4*x)

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 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+a \cos (x)}}{x^3} \, dx &=\left (\sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \frac{\cos \left (\frac{x}{2}\right )}{x^3} \, dx\\ &=-\frac{\sqrt{a+a \cos (x)}}{2 x^2}-\frac{1}{4} \left (\sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \frac{\sin \left (\frac{x}{2}\right )}{x^2} \, dx\\ &=-\frac{\sqrt{a+a \cos (x)}}{2 x^2}+\frac{\sqrt{a+a \cos (x)} \tan \left (\frac{x}{2}\right )}{4 x}-\frac{1}{8} \left (\sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \frac{\cos \left (\frac{x}{2}\right )}{x} \, dx\\ &=-\frac{\sqrt{a+a \cos (x)}}{2 x^2}-\frac{1}{8} \sqrt{a+a \cos (x)} \text{Ci}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right )+\frac{\sqrt{a+a \cos (x)} \tan \left (\frac{x}{2}\right )}{4 x}\\ \end{align*}

Mathematica [A]  time = 0.0749154, size = 44, normalized size = 0.66 \[ -\frac{\sqrt{a (\cos (x)+1)} \left (x^2 \text{CosIntegral}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right )-2 x \tan \left (\frac{x}{2}\right )+4\right )}{8 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Cos[x]]/x^3,x]

[Out]

-(Sqrt[a*(1 + Cos[x])]*(4 + x^2*CosIntegral[x/2]*Sec[x/2] - 2*x*Tan[x/2]))/(8*x^2)

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Maple [F]  time = 0.152, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\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^3,x)

[Out]

int((a+a*cos(x))^(1/2)/x^3,x)

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Maxima [C]  time = 2.13404, size = 26, normalized size = 0.39 \begin{align*} \frac{1}{8} \, \sqrt{2} \sqrt{a}{\left (\Gamma \left (-2, \frac{1}{2} i \, x\right ) + \Gamma \left (-2, -\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^3,x, algorithm="maxima")

[Out]

1/8*sqrt(2)*sqrt(a)*(gamma(-2, 1/2*I*x) + gamma(-2, -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^3,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^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(x))**(1/2)/x**3,x)

[Out]

Integral(sqrt(a*(cos(x) + 1))/x**3, 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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(x))^(1/2)/x^3,x, algorithm="giac")

[Out]

integrate(sqrt(a*cos(x) + a)/x^3, x)