\(\int (c+a^2 c x^2)^{3/2} \arctan (a x)^3 \, dx\) [423]

Optimal result

Integrand size = 21, antiderivative size = 760 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=-\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {9 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}} \]

[Out]

-1/4*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2/a+1/4*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^3-5/2*I*c^2*polylog(2,I*(1+I*a*
x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-9/8*I*c^2*arctan(a*x)^2*polylog(2,I*(1+I*a*x
)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)+5/2*I*c^2*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^
(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)+9/8*I*c^2*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2
))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)+9/4*I*c^2*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2
)/a/(a^2*c*x^2+c)^(1/2)-5*I*c^2*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c
*x^2+c)^(1/2)-9/4*c^2*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^
(1/2)+9/4*c^2*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-3/4
*I*c^2*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^3*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-9/4*I*c^2*pol
ylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-1/4*c*(a^2*c*x^2+c)^(1/2)/a+1/4
*c*x*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-9/8*c*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a+3/8*c*x*arctan(a*x)^3*(a^2*c*x^
2+c)^(1/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5000, 5010, 5008, 4266, 2611, 6744, 2320, 6724, 5006, 4998} \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\frac {9 i c^2 \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {9 i c^2 \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {9 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {9 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {9 i c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {9 i c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^2 \sqrt {a^2 x^2+1} \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \arctan (a x)}{a \sqrt {a^2 c x^2+c}}+\frac {1}{4} x \arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}+\frac {3}{8} c x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{4 a}-\frac {9 c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{8 a}+\frac {1}{4} c x \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {5 i c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {a^2 c x^2+c}}-\frac {c \sqrt {a^2 c x^2+c}}{4 a} \]

[In]

Int[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]

[Out]

-1/4*(c*Sqrt[c + a^2*c*x^2])/a + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 - (9*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x
]^2)/(8*a) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(4*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (x*(c
+ a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/4 - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^
3)/(a*Sqrt[c + a^2*c*x^2]) - ((5*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])
/(a*Sqrt[c + a^2*c*x^2]) + (((9*I)/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/
(a*Sqrt[c + a^2*c*x^2]) - (((9*I)/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*S
qrt[c + a^2*c*x^2]) + (((5*I)/2)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*
Sqrt[c + a^2*c*x^2]) - (((5*I)/2)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sq
rt[c + a^2*c*x^2]) - (9*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a*Sqrt[c + a^
2*c*x^2]) + (9*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a*Sqrt[c + a^2*c*x^2]) -
(((9*I)/4)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (((9*I)/4)*c^2*
Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])

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 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 4998

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(-b)*((d + e*x^2)^q/(2*c
*q*(2*q + 1))), x] + (Dist[2*d*(q/(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x] + Simp[x*(d
+ e*x^2)^q*((a + b*ArcTan[c*x])/(2*q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]

Rule 5000

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(-b)*p*(d + e*x^2)^
q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2*q + 1))), x] + (Dist[2*d*(q/(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a +
b*ArcTan[c*x])^p, x], x] + Dist[b^2*d*p*((p - 1)/(2*q*(2*q + 1))), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])
^(p - 2), x], x] + Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]

Rule 5006

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1
- I*c*x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x])]/(c*Sqrt[d])), x]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rule 5008

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c*Sqrt[d]), Subst
[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] &
& GtQ[d, 0]

Rule 5010

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*
d] && IGtQ[p, 0] &&  !GtQ[d, 0]

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*} \text {integral}& = -\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{2} c \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx \\ & = -\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{4} c^2 \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (3 c^2\right ) \int \frac {\arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{4} \left (9 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 \sqrt {c+a^2 c x^2}} \\ & = -\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {5 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\arctan (a x)\right )}{8 a \sqrt {c+a^2 c x^2}} \\ & = -\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{8 a \sqrt {c+a^2 c x^2}} \\ & = -\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a \sqrt {c+a^2 c x^2}} \\ & = -\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {9 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a \sqrt {c+a^2 c x^2}} \\ & = -\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {9 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}} \\ & = -\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {9 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2105\) vs. \(2(760)=1520\).

Time = 12.73 (sec) , antiderivative size = 2105, normalized size of antiderivative = 2.77 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\text {Result too large to show} \]

[In]

Integrate[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]

[Out]

((-1/2*I)*c*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a
*x]^2 + I*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[a*x]
^2)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[
a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I
)*E^(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(a*Sqrt[1 + a^2*x^2]) + (c*((Sqrt[c*(1 + a^2*x^2)]*
(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x
])] - Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])]
)))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-1/8*(Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]]) - (3*Pi^2*((P
i/2 - ArcTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + I*(PolyLog[2,
-E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4 + (3*Pi*((Pi/2 - ArcTan[a*x])^2*(L
og[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(Pi/2 - ArcTan[a*x])*(PolyLo
g[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - Arc
Tan[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2 - ArcTan[a*x])^4 + (I/4)*(Pi/2 + (
-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2
+ (-1/2*Pi + ArcTan[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi +
ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcTan[a*x])^2*Pol
yLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2 - (Pi/2 + (-1/2*Pi
+ ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (
-1/2*Pi + ArcTan[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 +
(-1/2*Pi + ArcTan[a*x])/2))] - (3*(Pi/2 - ArcTan[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))])/4 - (3*Pi*((I/
3)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*
Pi + ArcTan[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a
*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]/2))/2 - (3*(Pi/2 + (-1/2*Pi + ArcTan[a*x
])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcTan
[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])))/(8*Sqrt[1 + a^2*x^2]) + (Sqr
t[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^4) + (Sqrt[c
*(1 + a^2*x^2)]*(2*ArcTan[a*x] - ArcTan[a*x]^2 - ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - S
in[ArcTan[a*x]/2])^2) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[Arc
Tan[a*x]/2] - Sin[ArcTan[a*x]/2])^3) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan
[a*x]/2] + Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x
^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-2*ArcTan[a*x] - ArcTan[a*x]^2 + Ar
cTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2) + (Sqrt[c*(1 + a^2*x^2)]*(Sin
[ArcTan[a*x]/2] - ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x
]/2])) + (Sqrt[c*(1 + a^2*x^2)]*(-Sin[ArcTan[a*x]/2] + ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]
*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]))))/a

Maple [A] (verified)

Time = 3.75 (sec) , antiderivative size = 466, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right )^{3} a^{3} x^{3}-2 x^{2} \arctan \left (a x \right )^{2} a^{2}+5 \arctan \left (a x \right )^{3} a x +2 x \arctan \left (a x \right ) a -11 \arctan \left (a x \right )^{2}-2\right )}{8 a}-\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-9 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+9 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+20 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-20 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 i \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 i \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-20 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+20 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8 a \sqrt {a^{2} x^{2}+1}}\)	\(466\)

[In]

int((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/8*c/a*(c*(a*x-I)*(I+a*x))^(1/2)*(2*arctan(a*x)^3*a^3*x^3-2*x^2*arctan(a*x)^2*a^2+5*arctan(a*x)^3*a*x+2*x*arc
tan(a*x)*a-11*arctan(a*x)^2-2)-1/8*c*(c*(a*x-I)*(I+a*x))^(1/2)*(3*arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(
1/2))-3*arctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-9*I*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)
^(1/2))+9*I*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+20*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)
^(1/2))+18*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-20*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(
1/2))-18*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+18*I*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-1
8*I*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-20*I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+20*I*dilog(1-I*(1+I*a
*x)/(a^2*x^2+1)^(1/2)))/a/(a^2*x^2+1)^(1/2)

Fricas [F]

\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

integral((a^2*c*x^2 + c)^(3/2)*arctan(a*x)^3, x)

Sympy [F]

\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]

[In]

integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)**3,x)

[Out]

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

Maxima [F]

\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]

[In]

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

[Out]

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