3.2.80 \(\int \frac {x^3}{(a+a \cos (x))^{3/2}} \, dx\) [180]

Optimal. Leaf size=423 \[ -\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {12 x \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {12 x \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (4,-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (4,i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}} \]

[Out]

-3*x^2/a/(a+a*cos(x))^(1/2)-24*I*x*arctan(exp(1/2*I*x))*cos(1/2*x)/a/(a+a*cos(x))^(1/2)-I*x^3*arctan(exp(1/2*I
*x))*cos(1/2*x)/a/(a+a*cos(x))^(1/2)+24*I*cos(1/2*x)*polylog(2,-I*exp(1/2*I*x))/a/(a+a*cos(x))^(1/2)+3*I*x^2*c
os(1/2*x)*polylog(2,-I*exp(1/2*I*x))/a/(a+a*cos(x))^(1/2)-24*I*cos(1/2*x)*polylog(2,I*exp(1/2*I*x))/a/(a+a*cos
(x))^(1/2)-3*I*x^2*cos(1/2*x)*polylog(2,I*exp(1/2*I*x))/a/(a+a*cos(x))^(1/2)-12*x*cos(1/2*x)*polylog(3,-I*exp(
1/2*I*x))/a/(a+a*cos(x))^(1/2)+12*x*cos(1/2*x)*polylog(3,I*exp(1/2*I*x))/a/(a+a*cos(x))^(1/2)-24*I*cos(1/2*x)*
polylog(4,-I*exp(1/2*I*x))/a/(a+a*cos(x))^(1/2)+24*I*cos(1/2*x)*polylog(4,I*exp(1/2*I*x))/a/(a+a*cos(x))^(1/2)
+1/2*x^3*tan(1/2*x)/a/(a+a*cos(x))^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3400, 4271, 4266, 2317, 2438, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {i x^3 \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {24 i x \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {3 i x^2 \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {3 i x^2 \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {12 x \text {Li}_3\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {12 x \text {Li}_3\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {24 i \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {24 i \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {24 i \text {Li}_4\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {24 i \text {Li}_4\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a \cos (x)+a}}-\frac {3 x^2}{a \sqrt {a \cos (x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + a*Cos[x])^(3/2),x]

[Out]

(-3*x^2)/(a*Sqrt[a + a*Cos[x]]) - ((24*I)*x*ArcTan[E^((I/2)*x)]*Cos[x/2])/(a*Sqrt[a + a*Cos[x]]) - (I*x^3*ArcT
an[E^((I/2)*x)]*Cos[x/2])/(a*Sqrt[a + a*Cos[x]]) + ((24*I)*Cos[x/2]*PolyLog[2, (-I)*E^((I/2)*x)])/(a*Sqrt[a +
a*Cos[x]]) + ((3*I)*x^2*Cos[x/2]*PolyLog[2, (-I)*E^((I/2)*x)])/(a*Sqrt[a + a*Cos[x]]) - ((24*I)*Cos[x/2]*PolyL
og[2, I*E^((I/2)*x)])/(a*Sqrt[a + a*Cos[x]]) - ((3*I)*x^2*Cos[x/2]*PolyLog[2, I*E^((I/2)*x)])/(a*Sqrt[a + a*Co
s[x]]) - (12*x*Cos[x/2]*PolyLog[3, (-I)*E^((I/2)*x)])/(a*Sqrt[a + a*Cos[x]]) + (12*x*Cos[x/2]*PolyLog[3, I*E^(
(I/2)*x)])/(a*Sqrt[a + a*Cos[x]]) - ((24*I)*Cos[x/2]*PolyLog[4, (-I)*E^((I/2)*x)])/(a*Sqrt[a + a*Cos[x]]) + ((
24*I)*Cos[x/2]*PolyLog[4, I*E^((I/2)*x)])/(a*Sqrt[a + a*Cos[x]]) + (x^3*Tan[x/2])/(2*a*Sqrt[a + a*Cos[x]])

Rule 2317

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2438

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]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 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])

Rule 4266

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 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps

\begin {align*} \int \frac {x^3}{(a+a \cos (x))^{3/2}} \, dx &=\frac {\cos \left (\frac {x}{2}\right ) \int x^3 \sec ^3\left (\frac {x}{2}\right ) \, dx}{2 a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}+\frac {\cos \left (\frac {x}{2}\right ) \int x^3 \sec \left (\frac {x}{2}\right ) \, dx}{4 a \sqrt {a+a \cos (x)}}+\frac {\left (6 \cos \left (\frac {x}{2}\right )\right ) \int x \sec \left (\frac {x}{2}\right ) \, dx}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}-\frac {\left (3 \cos \left (\frac {x}{2}\right )\right ) \int x^2 \log \left (1-i e^{\frac {i x}{2}}\right ) \, dx}{2 a \sqrt {a+a \cos (x)}}+\frac {\left (3 \cos \left (\frac {x}{2}\right )\right ) \int x^2 \log \left (1+i e^{\frac {i x}{2}}\right ) \, dx}{2 a \sqrt {a+a \cos (x)}}-\frac {\left (12 \cos \left (\frac {x}{2}\right )\right ) \int \log \left (1-i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}+\frac {\left (12 \cos \left (\frac {x}{2}\right )\right ) \int \log \left (1+i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}-\frac {\left (6 i \cos \left (\frac {x}{2}\right )\right ) \int x \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}+\frac {\left (6 i \cos \left (\frac {x}{2}\right )\right ) \int x \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}+\frac {\left (24 i \cos \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {\left (24 i \cos \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}+\frac {\left (12 \cos \left (\frac {x}{2}\right )\right ) \int \text {Li}_3\left (-i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}-\frac {\left (12 \cos \left (\frac {x}{2}\right )\right ) \int \text {Li}_3\left (i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}-\frac {\left (24 i \cos \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {\left (24 i \cos \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_4\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_4\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 257, normalized size = 0.61 \begin {gather*} -\frac {i \cos \left (\frac {x}{2}\right ) \left (-6 i x^2 \cos \left (\frac {x}{2}\right )+48 x \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos ^2\left (\frac {x}{2}\right )+2 x^3 \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos ^2\left (\frac {x}{2}\right )-6 \left (8+x^2\right ) \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{\frac {i x}{2}}\right )+6 \left (8+x^2\right ) \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{\frac {i x}{2}}\right )-24 i x \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (3,-i e^{\frac {i x}{2}}\right )+24 i x \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (3,i e^{\frac {i x}{2}}\right )+48 \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (4,-i e^{\frac {i x}{2}}\right )-48 \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (4,i e^{\frac {i x}{2}}\right )+i x^3 \sin \left (\frac {x}{2}\right )\right )}{(a (1+\cos (x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + a*Cos[x])^(3/2),x]

[Out]

((-I)*Cos[x/2]*((-6*I)*x^2*Cos[x/2] + 48*x*ArcTan[E^((I/2)*x)]*Cos[x/2]^2 + 2*x^3*ArcTan[E^((I/2)*x)]*Cos[x/2]
^2 - 6*(8 + x^2)*Cos[x/2]^2*PolyLog[2, (-I)*E^((I/2)*x)] + 6*(8 + x^2)*Cos[x/2]^2*PolyLog[2, I*E^((I/2)*x)] -
(24*I)*x*Cos[x/2]^2*PolyLog[3, (-I)*E^((I/2)*x)] + (24*I)*x*Cos[x/2]^2*PolyLog[3, I*E^((I/2)*x)] + 48*Cos[x/2]
^2*PolyLog[4, (-I)*E^((I/2)*x)] - 48*Cos[x/2]^2*PolyLog[4, I*E^((I/2)*x)] + I*x^3*Sin[x/2]))/(a*(1 + Cos[x]))^
(3/2)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (a +a \cos \left (x \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+a*cos(x))^(3/2),x)

[Out]

int(x^3/(a+a*cos(x))^(3/2),x)

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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(x^3/(a+a*cos(x))^(3/2),x, algorithm="maxima")

[Out]

4/9*((6*sqrt(2)*x^2*cos(3/2*x) - (3*sqrt(2)*x^3 - 8*sqrt(2)*x)*sin(3/2*x))*cos(3*x)^3 + (6*sqrt(2)*x^2*sin(3/2
*x) + (3*sqrt(2)*x^3 - 8*sqrt(2)*x)*cos(3/2*x))*sin(3*x)^3 + 48*sqrt(2)*cos(2*x)^2*cos(3/2*x) + 48*sqrt(2)*cos
(3/2*x)*sin(2*x)^2 + ((2*(9*sqrt(2)*x^2 + 8*sqrt(2))*cos(3/2*x) - 3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*sin(3/2*x))*
cos(2*x) + (18*sqrt(2)*x^2 + 2*(9*sqrt(2)*x^2 + 8*sqrt(2))*cos(x) + 3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*sin(x))*co
s(3/2*x) + (3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*cos(3/2*x) + 2*(9*sqrt(2)*x^2 - 8*sqrt(2))*sin(3/2*x))*sin(2*x) -
(9*sqrt(2)*x^3 + 3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*cos(x) - 2*(9*sqrt(2)*x^2 - 8*sqrt(2))*sin(x) - 24*sqrt(2)*x)
*sin(3/2*x))*cos(3*x)^2 + 3*((2*(9*sqrt(2)*x^2 - 8*sqrt(2))*cos(3/2*x) - 3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*sin(3
/2*x))*cos(3*x) + 3*(2*(9*sqrt(2)*x^2 - 8*sqrt(2))*cos(3/2*x) - 3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*sin(3/2*x))*co
s(2*x) + (18*sqrt(2)*x^2 + 6*(9*sqrt(2)*x^2 - 8*sqrt(2))*cos(x) + 9*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*sin(x) - 16*
sqrt(2))*cos(3/2*x) + 243*(sqrt(2)*a^2*cos(3*x)^2 + 9*sqrt(2)*a^2*cos(2*x)^2 + 9*sqrt(2)*a^2*cos(x)^2 + sqrt(2
)*a^2*sin(3*x)^2 + 9*sqrt(2)*a^2*sin(2*x)^2 + 18*sqrt(2)*a^2*sin(2*x)*sin(x) + 9*sqrt(2)*a^2*sin(x)^2 + 6*sqrt
(2)*a^2*cos(x) + sqrt(2)*a^2 + 2*(3*sqrt(2)*a^2*cos(2*x) + 3*sqrt(2)*a^2*cos(x) + sqrt(2)*a^2)*cos(3*x) + 6*(3
*sqrt(2)*a^2*cos(x) + sqrt(2)*a^2)*cos(2*x) + 6*(sqrt(2)*a^2*sin(2*x) + sqrt(2)*a^2*sin(x))*sin(3*x))*integrat
e(1/9*(x^3*cos(4*x)*cos(3/2*x) + 4*x^3*cos(3*x)*cos(3/2*x) + 6*x^3*cos(2*x)*cos(3/2*x) + x^3*sin(4*x)*sin(3/2*
x) + 4*x^3*sin(3*x)*sin(3/2*x) + 6*x^3*sin(2*x)*sin(3/2*x) + 4*x^3*sin(3/2*x)*sin(x) + (4*x^3*cos(x) + x^3)*co
s(3/2*x))/(a^2*cos(4*x)^2 + 16*a^2*cos(3*x)^2 + 36*a^2*cos(2*x)^2 + 16*a^2*cos(x)^2 + a^2*sin(4*x)^2 + 16*a^2*
sin(3*x)^2 + 36*a^2*sin(2*x)^2 + 48*a^2*sin(2*x)*sin(x) + 16*a^2*sin(x)^2 + 8*a^2*cos(x) + a^2 + 2*(4*a^2*cos(
3*x) + 6*a^2*cos(2*x) + 4*a^2*cos(x) + a^2)*cos(4*x) + 8*(6*a^2*cos(2*x) + 4*a^2*cos(x) + a^2)*cos(3*x) + 12*(
4*a^2*cos(x) + a^2)*cos(2*x) + 4*(2*a^2*sin(3*x) + 3*a^2*sin(2*x) + 2*a^2*sin(x))*sin(4*x) + 16*(3*a^2*sin(2*x
) + 2*a^2*sin(x))*sin(3*x)), x) - 486*(sqrt(2)*a^2*cos(3*x)^2 + 9*sqrt(2)*a^2*cos(2*x)^2 + 9*sqrt(2)*a^2*cos(x
)^2 + sqrt(2)*a^2*sin(3*x)^2 + 9*sqrt(2)*a^2*sin(2*x)^2 + 18*sqrt(2)*a^2*sin(2*x)*sin(x) + 9*sqrt(2)*a^2*sin(x
)^2 + 6*sqrt(2)*a^2*cos(x) + sqrt(2)*a^2 + 2*(3*sqrt(2)*a^2*cos(2*x) + 3*sqrt(2)*a^2*cos(x) + sqrt(2)*a^2)*cos
(3*x) + 6*(3*sqrt(2)*a^2*cos(x) + sqrt(2)*a^2)*cos(2*x) + 6*(sqrt(2)*a^2*sin(2*x) + sqrt(2)*a^2*sin(x))*sin(3*
x))*integrate(1/9*(x^2*cos(3/2*x)*sin(4*x) + 4*x^2*cos(3/2*x)*sin(3*x) + 6*x^2*cos(3/2*x)*sin(2*x) - x^2*cos(4
*x)*sin(3/2*x) - 4*x^2*cos(3*x)*sin(3/2*x) - 6*x^2*cos(2*x)*sin(3/2*x) + 4*x^2*cos(3/2*x)*sin(x) - (4*x^2*cos(
x) + x^2)*sin(3/2*x))/(a^2*cos(4*x)^2 + 16*a^2*cos(3*x)^2 + 36*a^2*cos(2*x)^2 + 16*a^2*cos(x)^2 + a^2*sin(4*x)
^2 + 16*a^2*sin(3*x)^2 + 36*a^2*sin(2*x)^2 + 48*a^2*sin(2*x)*sin(x) + 16*a^2*sin(x)^2 + 8*a^2*cos(x) + a^2 + 2
*(4*a^2*cos(3*x) + 6*a^2*cos(2*x) + 4*a^2*cos(x) + a^2)*cos(4*x) + 8*(6*a^2*cos(2*x) + 4*a^2*cos(x) + a^2)*cos
(3*x) + 12*(4*a^2*cos(x) + a^2)*cos(2*x) + 4*(2*a^2*sin(3*x) + 3*a^2*sin(2*x) + 2*a^2*sin(x))*sin(4*x) + 16*(3
*a^2*sin(2*x) + 2*a^2*sin(x))*sin(3*x)), x) - 648*(sqrt(2)*a^2*cos(3*x)^2 + 9*sqrt(2)*a^2*cos(2*x)^2 + 9*sqrt(
2)*a^2*cos(x)^2 + sqrt(2)*a^2*sin(3*x)^2 + 9*sqrt(2)*a^2*sin(2*x)^2 + 18*sqrt(2)*a^2*sin(2*x)*sin(x) + 9*sqrt(
2)*a^2*sin(x)^2 + 6*sqrt(2)*a^2*cos(x) + sqrt(2)*a^2 + 2*(3*sqrt(2)*a^2*cos(2*x) + 3*sqrt(2)*a^2*cos(x) + sqrt
(2)*a^2)*cos(3*x) + 6*(3*sqrt(2)*a^2*cos(x) + sqrt(2)*a^2)*cos(2*x) + 6*(sqrt(2)*a^2*sin(2*x) + sqrt(2)*a^2*si
n(x))*sin(3*x))*integrate(1/9*(x*cos(4*x)*cos(3/2*x) + 4*x*cos(3*x)*cos(3/2*x) + 6*x*cos(2*x)*cos(3/2*x) + x*s
in(4*x)*sin(3/2*x) + 4*x*sin(3*x)*sin(3/2*x) + 6*x*sin(2*x)*sin(3/2*x) + 4*x*sin(3/2*x)*sin(x) + (4*x*cos(x) +
x)*cos(3/2*x))/(a^2*cos(4*x)^2 + 16*a^2*cos(3*x)^2 + 36*a^2*cos(2*x)^2 + 16*a^2*cos(x)^2 + a^2*sin(4*x)^2 + 1
6*a^2*sin(3*x)^2 + 36*a^2*sin(2*x)^2 + 48*a^2*sin(2*x)*sin(x) + 16*a^2*sin(x)^2 + 8*a^2*cos(x) + a^2 + 2*(4*a^
2*cos(3*x) + 6*a^2*cos(2*x) + 4*a^2*cos(x) + a^2)*cos(4*x) + 8*(6*a^2*cos(2*x) + 4*a^2*cos(x) + a^2)*cos(3*x)
+ 12*(4*a^2*cos(x) + a^2)*cos(2*x) + 4*(2*a^2*sin(3*x) + 3*a^2*sin(2*x) + 2*a^2*sin(x))*sin(4*x) + 16*(3*a^2*s
in(2*x) + 2*a^2*sin(x))*sin(3*x)), x) + (3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*cos(3/2*x) + 2*(9*sqrt(2)*x^2 - 8*sqr
t(2))*sin(3/2*x))*sin(3*x) + 3*(3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*cos(3/2*x) + 2*(9*sqrt(2)*x^2 - 8*sqrt(2))*sin
(3/2*x))*sin(2*x) - 3*(3*sqrt(2)*x^3 + 3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*cos(x) - 2*(9*sqrt(2)*x^2 - 8*sqrt(2))*
sin(x) - 8*sqrt(2)*x)*sin(3/2*x))*cos(4/3*arctan2(sin(3/2*x), cos(3/2*x)))^2 + 3*((2*(9*sqrt(2)*x^2 - 8*sqrt(2
))*cos(3/2*x) - 3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*sin(3/2*x))*cos(3*x) + 3*(2*(9*sqrt(2)*x^2 - 8*sqrt(2))*cos(3/
2*x) - 3*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*sin(3/2*x))*cos(2*x) + (18*sqrt(2)*x^2 + 6*(9*sqrt(2)*x^2 - 8*sqrt(2))*
cos(x) + 9*(3*sqrt(2)*x^3 - 8*sqrt(2)*x)*sin(x)...

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+a*cos(x))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*cos(x) + a)*x^3/(a^2*cos(x)^2 + 2*a^2*cos(x) + a^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a \left (\cos {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+a*cos(x))**(3/2),x)

[Out]

Integral(x**3/(a*(cos(x) + 1))**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+a*cos(x))^(3/2),x, algorithm="giac")

[Out]

integrate(x^3/(a*cos(x) + a)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{{\left (a+a\,\cos \left (x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + a*cos(x))^(3/2),x)

[Out]

int(x^3/(a + a*cos(x))^(3/2), x)

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