Integrand size = 13, antiderivative size = 97 \[ \int \frac {x}{\sqrt {a-a \cos (x)}} \, dx=-\frac {4 x \text {arctanh}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}} \]
-4*x*arctanh(exp(1/2*I*x))*sin(1/2*x)/(a-a*cos(x))^(1/2)+4*I*polylog(2,-ex p(1/2*I*x))*sin(1/2*x)/(a-a*cos(x))^(1/2)-4*I*polylog(2,exp(1/2*I*x))*sin( 1/2*x)/(a-a*cos(x))^(1/2)
Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {x}{\sqrt {a-a \cos (x)}} \, dx=\frac {2 \left (x \left (\log \left (1-e^{\frac {i x}{2}}\right )-\log \left (1+e^{\frac {i x}{2}}\right )\right )+2 i \operatorname {PolyLog}\left (2,-e^{\frac {i x}{2}}\right )-2 i \operatorname {PolyLog}\left (2,e^{\frac {i x}{2}}\right )\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}} \]
(2*(x*(Log[1 - E^((I/2)*x)] - Log[1 + E^((I/2)*x)]) + (2*I)*PolyLog[2, -E^ ((I/2)*x)] - (2*I)*PolyLog[2, E^((I/2)*x)])*Sin[x/2])/Sqrt[a - a*Cos[x]]
Time = 0.35 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.66, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 3800, 3042, 4671, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x}{\sqrt {a-a \cos (x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {x}{\sqrt {a-a \sin \left (x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \frac {\sin \left (\frac {x}{2}\right ) \int x \csc \left (\frac {x}{2}\right )dx}{\sqrt {a-a \cos (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin \left (\frac {x}{2}\right ) \int x \csc \left (\frac {x}{2}\right )dx}{\sqrt {a-a \cos (x)}}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle \frac {\sin \left (\frac {x}{2}\right ) \left (-2 \int \log \left (1-e^{\frac {i x}{2}}\right )dx+2 \int \log \left (1+e^{\frac {i x}{2}}\right )dx-4 x \text {arctanh}\left (e^{\frac {i x}{2}}\right )\right )}{\sqrt {a-a \cos (x)}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\sin \left (\frac {x}{2}\right ) \left (4 i \int e^{-\frac {i x}{2}} \log \left (1-e^{\frac {i x}{2}}\right )de^{\frac {i x}{2}}-4 i \int e^{-\frac {i x}{2}} \log \left (1+e^{\frac {i x}{2}}\right )de^{\frac {i x}{2}}-4 x \text {arctanh}\left (e^{\frac {i x}{2}}\right )\right )}{\sqrt {a-a \cos (x)}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\sin \left (\frac {x}{2}\right ) \left (-4 x \text {arctanh}\left (e^{\frac {i x}{2}}\right )+4 i \operatorname {PolyLog}\left (2,-e^{\frac {i x}{2}}\right )-4 i \operatorname {PolyLog}\left (2,e^{\frac {i x}{2}}\right )\right )}{\sqrt {a-a \cos (x)}}\) |
((-4*x*ArcTanh[E^((I/2)*x)] + (4*I)*PolyLog[2, -E^((I/2)*x)] - (4*I)*PolyL og[2, E^((I/2)*x)])*Sin[x/2])/Sqrt[a - a*Cos[x]]
3.2.77.3.1 Defintions of rubi rules used
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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])
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
\[\int \frac {x}{\sqrt {a -\cos \left (x \right ) a}}d x\]
\[ \int \frac {x}{\sqrt {a-a \cos (x)}} \, dx=\int { \frac {x}{\sqrt {-a \cos \left (x\right ) + a}} \,d x } \]
\[ \int \frac {x}{\sqrt {a-a \cos (x)}} \, dx=\int \frac {x}{\sqrt {- a \left (\cos {\left (x \right )} - 1\right )}}\, dx \]
\[ \int \frac {x}{\sqrt {a-a \cos (x)}} \, dx=\int { \frac {x}{\sqrt {-a \cos \left (x\right ) + a}} \,d x } \]
\[ \int \frac {x}{\sqrt {a-a \cos (x)}} \, dx=\int { \frac {x}{\sqrt {-a \cos \left (x\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {x}{\sqrt {a-a \cos (x)}} \, dx=\int \frac {x}{\sqrt {a-a\,\cos \left (x\right )}} \,d x \]