3.163 \(\int \frac{\sqrt{a-a \cos (x)}}{x^3} \, dx\)

Optimal. Leaf size=70 \[ -\frac{1}{8} \text{Si}\left (\frac{x}{2}\right ) \csc \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)}-\frac{\sqrt{a-a \cos (x)}}{2 x^2}-\frac{\cot \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)}}{4 x} \]

[Out]

-Sqrt[a - a*Cos[x]]/(2*x^2) - (Sqrt[a - a*Cos[x]]*Cot[x/2])/(4*x) - (Sqrt[a - a*Cos[x]]*Csc[x/2]*SinIntegral[x
/2])/8

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Rubi [A]  time = 0.105748, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3319, 3297, 3299} \[ -\frac{1}{8} \text{Si}\left (\frac{x}{2}\right ) \csc \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)}-\frac{\sqrt{a-a \cos (x)}}{2 x^2}-\frac{\cot \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)}}{4 x} \]

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]]*Cot[x/2])/(4*x) - (Sqrt[a - a*Cos[x]]*Csc[x/2]*SinIntegral[x
/2])/8

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^3} \, dx &=\left (\sqrt{a-a \cos (x)} \csc \left (\frac{x}{2}\right )\right ) \int \frac{\sin \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)} \csc \left (\frac{x}{2}\right )\right ) \int \frac{\cos \left (\frac{x}{2}\right )}{x^2} \, dx\\ &=-\frac{\sqrt{a-a \cos (x)}}{2 x^2}-\frac{\sqrt{a-a \cos (x)} \cot \left (\frac{x}{2}\right )}{4 x}-\frac{1}{8} \left (\sqrt{a-a \cos (x)} \csc \left (\frac{x}{2}\right )\right ) \int \frac{\sin \left (\frac{x}{2}\right )}{x} \, dx\\ &=-\frac{\sqrt{a-a \cos (x)}}{2 x^2}-\frac{\sqrt{a-a \cos (x)} \cot \left (\frac{x}{2}\right )}{4 x}-\frac{1}{8} \sqrt{a-a \cos (x)} \csc \left (\frac{x}{2}\right ) \text{Si}\left (\frac{x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0569305, size = 45, normalized size = 0.64 \[ -\frac{\sqrt{a-a \cos (x)} \left (x^2 \text{Si}\left (\frac{x}{2}\right ) \csc \left (\frac{x}{2}\right )+2 x \cot \left (\frac{x}{2}\right )+4\right )}{8 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - a*Cos[x]]/x^3,x]

[Out]

-(Sqrt[a - a*Cos[x]]*(4 + 2*x*Cot[x/2] + x^2*Csc[x/2]*SinIntegral[x/2]))/(8*x^2)

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Maple [F]  time = 0.096, 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 [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="maxima")

[Out]

integrate(sqrt(-a*cos(x) + a)/x^3, 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)