\(\int \frac {\sqrt {a-a \cos (x)}}{x} \, dx\) [161]

Optimal result

Integrand size = 15, antiderivative size = 24 \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3400, 3380} \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\text {Si}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \]

[In]

Int[Sqrt[a - a*Cos[x]]/x,x]

[Out]

Sqrt[a - a*Cos[x]]*Csc[x/2]*SinIntegral[x/2]

Rule 3380

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]

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*} \text {integral}& = \left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \frac {\sin \left (\frac {x}{2}\right )}{x} \, dx \\ & = \sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right ) \]

[In]

Integrate[Sqrt[a - a*Cos[x]]/x,x]

[Out]

Sqrt[a - a*Cos[x]]*Csc[x/2]*SinIntegral[x/2]

Maple [F]

\[\int \frac {\sqrt {a -\cos \left (x \right ) a}}{x}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\int \frac {\sqrt {- a \left (\cos {\left (x \right )} - 1\right )}}{x}\, dx \]

[In]

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

[Out]

Integral(sqrt(-a*(cos(x) - 1))/x, x)

Maxima [F]

\[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\int { \frac {\sqrt {-a \cos \left (x\right ) + a}}{x} \,d x } \]

[In]

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

[Out]

integrate(sqrt(-a*cos(x) + a)/x, x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\sqrt {2} \sqrt {a} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, x\right ) \]

[In]

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

[Out]

sqrt(2)*sqrt(a)*sgn(sin(1/2*x))*sin_integral(1/2*x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\int \frac {\sqrt {a-a\,\cos \left (x\right )}}{x} \,d x \]

[In]

int((a - a*cos(x))^(1/2)/x,x)

[Out]

int((a - a*cos(x))^(1/2)/x, x)