3.172 \(\int \frac{x}{\sqrt{a+a \cos (c+d x)}} \, dx\)

Optimal. Leaf size=156 \[ \frac{4 i \text{Li}_2\left (-i e^{\frac{1}{2} i (c+d x)}\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right )}{d^2 \sqrt{a \cos (c+d x)+a}}-\frac{4 i \text{Li}_2\left (i e^{\frac{1}{2} i (c+d x)}\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right )}{d^2 \sqrt{a \cos (c+d x)+a}}-\frac{4 i x \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \tan ^{-1}\left (e^{\frac{1}{2} i (c+d x)}\right )}{d \sqrt{a \cos (c+d x)+a}} \]

[Out]

((-4*I)*x*ArcTan[E^((I/2)*(c + d*x))]*Cos[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + ((4*I)*Cos[c/2 + (d*x
)/2]*PolyLog[2, (-I)*E^((I/2)*(c + d*x))])/(d^2*Sqrt[a + a*Cos[c + d*x]]) - ((4*I)*Cos[c/2 + (d*x)/2]*PolyLog[
2, I*E^((I/2)*(c + d*x))])/(d^2*Sqrt[a + a*Cos[c + d*x]])

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Rubi [A]  time = 0.0856533, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3319, 4181, 2279, 2391} \[ \frac{4 i \text{Li}_2\left (-i e^{\frac{1}{2} i (c+d x)}\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right )}{d^2 \sqrt{a \cos (c+d x)+a}}-\frac{4 i \text{Li}_2\left (i e^{\frac{1}{2} i (c+d x)}\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right )}{d^2 \sqrt{a \cos (c+d x)+a}}-\frac{4 i x \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \tan ^{-1}\left (e^{\frac{1}{2} i (c+d x)}\right )}{d \sqrt{a \cos (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + a*Cos[c + d*x]],x]

[Out]

((-4*I)*x*ArcTan[E^((I/2)*(c + d*x))]*Cos[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + ((4*I)*Cos[c/2 + (d*x
)/2]*PolyLog[2, (-I)*E^((I/2)*(c + d*x))])/(d^2*Sqrt[a + a*Cos[c + d*x]]) - ((4*I)*Cos[c/2 + (d*x)/2]*PolyLog[
2, I*E^((I/2)*(c + d*x))])/(d^2*Sqrt[a + a*Cos[c + d*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 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && 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 (c+d x)}} \, dx &=\frac{\sin \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right ) \int x \csc \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right ) \, dx}{\sqrt{a+a \cos (c+d x)}}\\ &=-\frac{4 i x \tan ^{-1}\left (e^{\frac{1}{2} i (c+d x)}\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right )}{d \sqrt{a+a \cos (c+d x)}}-\frac{\left (2 \sin \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )\right ) \int \log \left (1-i e^{i \left (\frac{c}{2}+\frac{d x}{2}\right )}\right ) \, dx}{d \sqrt{a+a \cos (c+d x)}}+\frac{\left (2 \sin \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )\right ) \int \log \left (1+i e^{i \left (\frac{c}{2}+\frac{d x}{2}\right )}\right ) \, dx}{d \sqrt{a+a \cos (c+d x)}}\\ &=-\frac{4 i x \tan ^{-1}\left (e^{\frac{1}{2} i (c+d x)}\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right )}{d \sqrt{a+a \cos (c+d x)}}+\frac{\left (4 i \sin \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \left (\frac{c}{2}+\frac{d x}{2}\right )}\right )}{d^2 \sqrt{a+a \cos (c+d x)}}-\frac{\left (4 i \sin \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \left (\frac{c}{2}+\frac{d x}{2}\right )}\right )}{d^2 \sqrt{a+a \cos (c+d x)}}\\ &=-\frac{4 i x \tan ^{-1}\left (e^{\frac{1}{2} i (c+d x)}\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right )}{d \sqrt{a+a \cos (c+d x)}}+\frac{4 i \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Li}_2\left (-i e^{\frac{1}{2} i (c+d x)}\right )}{d^2 \sqrt{a+a \cos (c+d x)}}-\frac{4 i \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Li}_2\left (i e^{\frac{1}{2} i (c+d x)}\right )}{d^2 \sqrt{a+a \cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0507512, size = 89, normalized size = 0.57 \[ -\frac{4 i \cos \left (\frac{1}{2} (c+d x)\right ) \left (-\text{Li}_2\left (-i e^{\frac{1}{2} i (c+d x)}\right )+\text{Li}_2\left (i e^{\frac{1}{2} i (c+d x)}\right )+d x \tan ^{-1}\left (e^{\frac{1}{2} i (c+d x)}\right )\right )}{d^2 \sqrt{a (\cos (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a + a*Cos[c + d*x]],x]

[Out]

((-4*I)*Cos[(c + d*x)/2]*(d*x*ArcTan[E^((I/2)*(c + d*x))] - PolyLog[2, (-I)*E^((I/2)*(c + d*x))] + PolyLog[2,
I*E^((I/2)*(c + d*x))]))/(d^2*Sqrt[a*(1 + Cos[c + d*x])])

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Maple [F]  time = 0.178, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt{a+\cos \left ( dx+c \right ) a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+cos(d*x+c)*a)^(1/2),x)

[Out]

int(x/(a+cos(d*x+c)*a)^(1/2),x)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/sqrt(a*cos(d*x + c) + 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 (c + d x \right )} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+a*cos(d*x+c))**(1/2),x)

[Out]

Integral(x/sqrt(a*(cos(c + d*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 (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/sqrt(a*cos(d*x + c) + a), x)