3.2.62 \(\int \frac {\sqrt {a-a \cos (x)}}{x^2} \, dx\) [162]

Optimal. Leaf size=44 \[ -\frac {\sqrt {a-a \cos (x)}}{x}+\frac {1}{2} \sqrt {a-a \cos (x)} \text {CosIntegral}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \]

[Out]

-(a-a*cos(x))^(1/2)/x+1/2*Ci(1/2*x)*csc(1/2*x)*(a-a*cos(x))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3400, 3378, 3383} \begin {gather*} \frac {1}{2} \text {CosIntegral}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}-\frac {\sqrt {a-a \cos (x)}}{x} \end {gather*}

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]]*CosIntegral[x/2]*Csc[x/2])/2

Rule 3378

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 3383

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]

Rule 3400

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])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 34, normalized size = 0.77 \begin {gather*} \frac {\sqrt {a-a \cos (x)} \left (-2+x \text {CosIntegral}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right )\right )}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - a*Cos[x]]/x^2,x]

[Out]

(Sqrt[a - a*Cos[x]]*(-2 + x*CosIntegral[x/2]*Csc[x/2]))/(2*x)

________________________________________________________________________________________

Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a -a \cos \left (x \right )}}{x^{2}}\, dx\]

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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a*cos(x) + a)/x^2, x)

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- a \left (\cos {\left (x \right )} - 1\right )}}{x^{2}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.60, size = 34, normalized size = 0.77 \begin {gather*} \frac {\sqrt {2} {\left (x \operatorname {Ci}\left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) - 2 \, \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*(x*cos_integral(1/2*x)*sgn(sin(1/2*x)) - 2*sgn(sin(1/2*x))*sin(1/2*x))*sqrt(a)/x

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {a-a\,\cos \left (x\right )}}{x^2} \,d x \end {gather*}

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)

________________________________________________________________________________________