3.2.81 \(\int \frac {x^2}{(a+a \cos (x))^{3/2}} \, dx\) [181]

Optimal. Leaf size=257 \[ -\frac {2 x}{a \sqrt {a+a \cos (x)}}-\frac {i x^2 \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {4 \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {2 i x \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {2 i x \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {4 \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {4 \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^2 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}} \]

[Out]

-2*x/a/(a+a*cos(x))^(1/2)-I*x^2*arctan(exp(1/2*I*x))*cos(1/2*x)/a/(a+a*cos(x))^(1/2)+4*arctanh(sin(1/2*x))*cos
(1/2*x)/a/(a+a*cos(x))^(1/2)+2*I*x*cos(1/2*x)*polylog(2,-I*exp(1/2*I*x))/a/(a+a*cos(x))^(1/2)-2*I*x*cos(1/2*x)
*polylog(2,I*exp(1/2*I*x))/a/(a+a*cos(x))^(1/2)-4*cos(1/2*x)*polylog(3,-I*exp(1/2*I*x))/a/(a+a*cos(x))^(1/2)+4
*cos(1/2*x)*polylog(3,I*exp(1/2*I*x))/a/(a+a*cos(x))^(1/2)+1/2*x^2*tan(1/2*x)/a/(a+a*cos(x))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3400, 4271, 3855, 4266, 2611, 2320, 6724} \begin {gather*} -\frac {i x^2 \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {2 i x \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {2 i x \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {4 \text {Li}_3\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {4 \text {Li}_3\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {x^2 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a \cos (x)+a}}-\frac {2 x}{a \sqrt {a \cos (x)+a}}+\frac {4 \cos \left (\frac {x}{2}\right ) \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right )}{a \sqrt {a \cos (x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + a*Cos[x])^(3/2),x]

[Out]

(-2*x)/(a*Sqrt[a + a*Cos[x]]) - (I*x^2*ArcTan[E^((I/2)*x)]*Cos[x/2])/(a*Sqrt[a + a*Cos[x]]) + (4*ArcTanh[Sin[x
/2]]*Cos[x/2])/(a*Sqrt[a + a*Cos[x]]) + ((2*I)*x*Cos[x/2]*PolyLog[2, (-I)*E^((I/2)*x)])/(a*Sqrt[a + a*Cos[x]])
- ((2*I)*x*Cos[x/2]*PolyLog[2, I*E^((I/2)*x)])/(a*Sqrt[a + a*Cos[x]]) - (4*Cos[x/2]*PolyLog[3, (-I)*E^((I/2)*
x)])/(a*Sqrt[a + a*Cos[x]]) + (4*Cos[x/2]*PolyLog[3, I*E^((I/2)*x)])/(a*Sqrt[a + a*Cos[x]]) + (x^2*Tan[x/2])/(
2*a*Sqrt[a + a*Cos[x]])

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 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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 185, normalized size = 0.72 \begin {gather*} \frac {\cos \left (\frac {x}{2}\right ) \left (-4 x \cos \left (\frac {x}{2}\right )-2 i x^2 \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos ^2\left (\frac {x}{2}\right )+8 \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right ) \cos ^2\left (\frac {x}{2}\right )+4 i x \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{\frac {i x}{2}}\right )-4 i x \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{\frac {i x}{2}}\right )-8 \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (3,-i e^{\frac {i x}{2}}\right )+8 \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (3,i e^{\frac {i x}{2}}\right )+x^2 \sin \left (\frac {x}{2}\right )\right )}{(a (1+\cos (x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + a*Cos[x])^(3/2),x]

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

-1/9*(4*(3*sqrt(2)*x^2*sin(3/2*x) - 4*sqrt(2)*x*cos(3/2*x))*cos(3*x)^3 - 4*(3*sqrt(2)*x^2*cos(3/2*x) + 4*sqrt(
2)*x*sin(3/2*x))*sin(3*x)^3 + 96*sqrt(2)*cos(2*x)^2*sin(3/2*x) + 96*sqrt(2)*sin(2*x)^2*sin(3/2*x) - 4*((12*sqr
t(2)*x*cos(3/2*x) - (9*sqrt(2)*x^2 + 8*sqrt(2))*sin(3/2*x))*cos(2*x) + (12*sqrt(2)*x*cos(x) + (9*sqrt(2)*x^2 -
8*sqrt(2))*sin(x) + 12*sqrt(2)*x)*cos(3/2*x) + (12*sqrt(2)*x*sin(3/2*x) + (9*sqrt(2)*x^2 - 8*sqrt(2))*cos(3/2
*x))*sin(2*x) - (9*sqrt(2)*x^2 - 12*sqrt(2)*x*sin(x) + (9*sqrt(2)*x^2 + 8*sqrt(2))*cos(x))*sin(3/2*x))*cos(3*x
)^2 - 12*((12*sqrt(2)*x*cos(3/2*x) - (9*sqrt(2)*x^2 - 8*sqrt(2))*sin(3/2*x))*cos(3*x) + 3*(12*sqrt(2)*x*cos(3/
2*x) - (9*sqrt(2)*x^2 - 8*sqrt(2))*sin(3/2*x))*cos(2*x) + 3*(12*sqrt(2)*x*cos(x) + (9*sqrt(2)*x^2 - 8*sqrt(2))
*sin(x) + 4*sqrt(2)*x)*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)*c
os(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(4*x)*cos(3/2*x) + 4*x^2*cos(3*x)*cos(3/2*x) + 6*x^2*cos(2*x)*cos(3/2*x) + x^2*sin
(4*x)*sin(3/2*x) + 4*x^2*sin(3*x)*sin(3/2*x) + 6*x^2*sin(2*x)*sin(3/2*x) + 4*x^2*sin(3/2*x)*sin(x) + (4*x^2*co
s(x) + x^2)*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 + 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)*c
os(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) - 324*(sqrt(2)*a^2*cos(3*x)^2 + 9*sqrt(2)*a^2*cos(2*x)^2 + 9*sqr
t(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*sqr
t(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) + sq
rt(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*cos(3/2*x)*sin(4*x) + 4*x*cos(3/2*x)*sin(3*x) + 6*x*cos(3/2*x)*sin(2*x) - x
*cos(4*x)*sin(3/2*x) - 4*x*cos(3*x)*sin(3/2*x) - 6*x*cos(2*x)*sin(3/2*x) + 4*x*cos(3/2*x)*sin(x) - (4*x*cos(x)
+ x)*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) + (12*sqrt(2)*x*sin(3/2*x) + (9*sqrt(2)*x^2 - 8*sqrt(2))*cos(3/2*x))*s
in(3*x) + 3*(12*sqrt(2)*x*sin(3/2*x) + (9*sqrt(2)*x^2 - 8*sqrt(2))*cos(3/2*x))*sin(2*x) - (9*sqrt(2)*x^2 - 36*
sqrt(2)*x*sin(x) + 3*(9*sqrt(2)*x^2 - 8*sqrt(2))*cos(x) - 8*sqrt(2))*sin(3/2*x))*cos(4/3*arctan2(sin(3/2*x), c
os(3/2*x)))^2 - 12*((12*sqrt(2)*x*cos(3/2*x) - (9*sqrt(2)*x^2 - 8*sqrt(2))*sin(3/2*x))*cos(3*x) + 3*(12*sqrt(2
)*x*cos(3/2*x) - (9*sqrt(2)*x^2 - 8*sqrt(2))*sin(3/2*x))*cos(2*x) + 3*(12*sqrt(2)*x*cos(x) + (9*sqrt(2)*x^2 -
8*sqrt(2))*sin(x) + 4*sqrt(2)*x)*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*si
n(x))*sin(3*x))*integrate(1/9*(x^2*cos(4*x)*cos(3/2*x) + 4*x^2*cos(3*x)*cos(3/2*x) + 6*x^2*cos(2*x)*cos(3/2*x)
+ x^2*sin(4*x)*sin(3/2*x) + 4*x^2*sin(3*x)*sin(3/2*x) + 6*x^2*sin(2*x)*sin(3/2*x) + 4*x^2*sin(3/2*x)*sin(x) +
(4*x^2*cos(x) + x^2)*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 + 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) - 324*(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*c
os(x) + sqrt(2)*a^2)*cos(3*x) + 6*(3*sqrt(2)*a^...

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

int(x^2/(a + a*cos(x))^(3/2), x)

________________________________________________________________________________________