3.177 \(\int \frac{x}{\sqrt{a-a \cos (x)}} \, dx\)

Optimal. Leaf size=97 \[ \frac{4 i \text{Li}_2\left (-e^{\frac{i x}{2}}\right ) \sin \left (\frac{x}{2}\right )}{\sqrt{a-a \cos (x)}}-\frac{4 i \text{Li}_2\left (e^{\frac{i x}{2}}\right ) \sin \left (\frac{x}{2}\right )}{\sqrt{a-a \cos (x)}}-\frac{4 x \sin \left (\frac{x}{2}\right ) \tanh ^{-1}\left (e^{\frac{i x}{2}}\right )}{\sqrt{a-a \cos (x)}} \]

[Out]

(-4*x*ArcTanh[E^((I/2)*x)]*Sin[x/2])/Sqrt[a - a*Cos[x]] + ((4*I)*PolyLog[2, -E^((I/2)*x)]*Sin[x/2])/Sqrt[a - a
*Cos[x]] - ((4*I)*PolyLog[2, E^((I/2)*x)]*Sin[x/2])/Sqrt[a - a*Cos[x]]

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Rubi [A]  time = 0.0784426, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3319, 4183, 2279, 2391} \[ \frac{4 i \text{Li}_2\left (-e^{\frac{i x}{2}}\right ) \sin \left (\frac{x}{2}\right )}{\sqrt{a-a \cos (x)}}-\frac{4 i \text{Li}_2\left (e^{\frac{i x}{2}}\right ) \sin \left (\frac{x}{2}\right )}{\sqrt{a-a \cos (x)}}-\frac{4 x \sin \left (\frac{x}{2}\right ) \tanh ^{-1}\left (e^{\frac{i x}{2}}\right )}{\sqrt{a-a \cos (x)}} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a - a*Cos[x]],x]

[Out]

(-4*x*ArcTanh[E^((I/2)*x)]*Sin[x/2])/Sqrt[a - a*Cos[x]] + ((4*I)*PolyLog[2, -E^((I/2)*x)]*Sin[x/2])/Sqrt[a - a
*Cos[x]] - ((4*I)*PolyLog[2, E^((I/2)*x)]*Sin[x/2])/Sqrt[a - a*Cos[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 4183

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] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2391

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]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{a-a \cos (x)}} \, dx &=\frac{\sin \left (\frac{x}{2}\right ) \int x \csc \left (\frac{x}{2}\right ) \, dx}{\sqrt{a-a \cos (x)}}\\ &=-\frac{4 x \tanh ^{-1}\left (e^{\frac{i x}{2}}\right ) \sin \left (\frac{x}{2}\right )}{\sqrt{a-a \cos (x)}}-\frac{\left (2 \sin \left (\frac{x}{2}\right )\right ) \int \log \left (1-e^{\frac{i x}{2}}\right ) \, dx}{\sqrt{a-a \cos (x)}}+\frac{\left (2 \sin \left (\frac{x}{2}\right )\right ) \int \log \left (1+e^{\frac{i x}{2}}\right ) \, dx}{\sqrt{a-a \cos (x)}}\\ &=-\frac{4 x \tanh ^{-1}\left (e^{\frac{i x}{2}}\right ) \sin \left (\frac{x}{2}\right )}{\sqrt{a-a \cos (x)}}+\frac{\left (4 i \sin \left (\frac{x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\frac{i x}{2}}\right )}{\sqrt{a-a \cos (x)}}-\frac{\left (4 i \sin \left (\frac{x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\frac{i x}{2}}\right )}{\sqrt{a-a \cos (x)}}\\ &=-\frac{4 x \tanh ^{-1}\left (e^{\frac{i x}{2}}\right ) \sin \left (\frac{x}{2}\right )}{\sqrt{a-a \cos (x)}}+\frac{4 i \text{Li}_2\left (-e^{\frac{i x}{2}}\right ) \sin \left (\frac{x}{2}\right )}{\sqrt{a-a \cos (x)}}-\frac{4 i \text{Li}_2\left (e^{\frac{i x}{2}}\right ) \sin \left (\frac{x}{2}\right )}{\sqrt{a-a \cos (x)}}\\ \end{align*}

Mathematica [A]  time = 0.0330293, size = 83, normalized size = 0.86 \[ \frac{2 \sin \left (\frac{x}{2}\right ) \left (2 i \text{Li}_2\left (-e^{\frac{i x}{2}}\right )-2 i \text{Li}_2\left (e^{\frac{i x}{2}}\right )+x \left (\log \left (1-e^{\frac{i x}{2}}\right )-\log \left (1+e^{\frac{i x}{2}}\right )\right )\right )}{\sqrt{a-a \cos (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a - a*Cos[x]],x]

[Out]

(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]]

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt{a-a\cos \left ( x \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{-a \cos \left (x\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/sqrt(-a*cos(x) + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a \cos \left (x\right ) + a} x}{a \cos \left (x\right ) - a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a*cos(x) + a)*x/(a*cos(x) - a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(-a*(cos(x) - 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{-a \cos \left (x\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/sqrt(-a*cos(x) + a), x)