3.366 \(\int (c+a^2 c x^2) \tan ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=172 \[ \frac{c \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{a}+\frac{2 i c \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a}-\frac{c \log \left (a^2 x^2+1\right )}{2 a}+\frac{1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3-\frac{c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{2 a}+\frac{2 i c \tan ^{-1}(a x)^3}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^3+c x \tan ^{-1}(a x)+\frac{2 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{a} \]

[Out]

c*x*ArcTan[a*x] - (c*(1 + a^2*x^2)*ArcTan[a*x]^2)/(2*a) + (((2*I)/3)*c*ArcTan[a*x]^3)/a + (2*c*x*ArcTan[a*x]^3
)/3 + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/3 + (2*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/a - (c*Log[1 + a^2*x^2])/(2
*a) + ((2*I)*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/a

________________________________________________________________________________________

Rubi [A]  time = 0.182432, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {4880, 4846, 4920, 4854, 4884, 4994, 6610, 260} \[ \frac{c \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{a}+\frac{2 i c \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a}-\frac{c \log \left (a^2 x^2+1\right )}{2 a}+\frac{1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3-\frac{c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{2 a}+\frac{2 i c \tan ^{-1}(a x)^3}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^3+c x \tan ^{-1}(a x)+\frac{2 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{a} \]

Antiderivative was successfully verified.

[In]

Int[(c + a^2*c*x^2)*ArcTan[a*x]^3,x]

[Out]

c*x*ArcTan[a*x] - (c*(1 + a^2*x^2)*ArcTan[a*x]^2)/(2*a) + (((2*I)/3)*c*ArcTan[a*x]^3)/a + (2*c*x*ArcTan[a*x]^3
)/3 + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/3 + (2*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/a - (c*Log[1 + a^2*x^2])/(2
*a) + ((2*I)*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/a

Rule 4880

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] && E
qQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]

Rule 4846

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

Rule 4920

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

Rule 4854

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

Rule 4884

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

Rule 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u
])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*
I)/(I - c*x))^2, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3 \, dx &=-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{2 a}+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac{1}{3} (2 c) \int \tan ^{-1}(a x)^3 \, dx+c \int \tan ^{-1}(a x) \, dx\\ &=c x \tan ^{-1}(a x)-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{2 a}+\frac{2}{3} c x \tan ^{-1}(a x)^3+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3-(a c) \int \frac{x}{1+a^2 x^2} \, dx-(2 a c) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=c x \tan ^{-1}(a x)-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{2 a}+\frac{2 i c \tan ^{-1}(a x)^3}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^3+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3-\frac{c \log \left (1+a^2 x^2\right )}{2 a}+(2 c) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx\\ &=c x \tan ^{-1}(a x)-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{2 a}+\frac{2 i c \tan ^{-1}(a x)^3}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^3+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac{2 c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a}-\frac{c \log \left (1+a^2 x^2\right )}{2 a}-(4 c) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=c x \tan ^{-1}(a x)-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{2 a}+\frac{2 i c \tan ^{-1}(a x)^3}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^3+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac{2 c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a}-\frac{c \log \left (1+a^2 x^2\right )}{2 a}+\frac{2 i c \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a}-(2 i c) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=c x \tan ^{-1}(a x)-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{2 a}+\frac{2 i c \tan ^{-1}(a x)^3}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^3+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac{2 c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a}-\frac{c \log \left (1+a^2 x^2\right )}{2 a}+\frac{2 i c \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a}+\frac{c \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0487358, size = 144, normalized size = 0.84 \[ \frac{c \left (-12 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+6 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-3 \log \left (a^2 x^2+1\right )+2 a^3 x^3 \tan ^{-1}(a x)^3-3 a^2 x^2 \tan ^{-1}(a x)^2+6 a x \tan ^{-1}(a x)^3-4 i \tan ^{-1}(a x)^3-3 \tan ^{-1}(a x)^2+6 a x \tan ^{-1}(a x)+12 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{6 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*(6*a*x*ArcTan[a*x] - 3*ArcTan[a*x]^2 - 3*a^2*x^2*ArcTan[a*x]^2 - (4*I)*ArcTan[a*x]^3 + 6*a*x*ArcTan[a*x]^3
+ 2*a^3*x^3*ArcTan[a*x]^3 + 12*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 3*Log[1 + a^2*x^2] - (12*I)*ArcT
an[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 6*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(6*a)

________________________________________________________________________________________

Maple [C]  time = 1.237, size = 1635, normalized size = 9.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*c*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+1/a*c*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-1/2/a*c*arctan(a*x)^2-1/2*a*c*ar
ctan(a*x)^2*x^2+1/3*a^2*c*arctan(a*x)^3*x^3-1/a*c*arctan(a*x)^2*ln(a^2*x^2+1)+2/a*c*arctan(a*x)^2*ln(2)+2/a*c*
arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-I/a*c*arctan(a*x)-2/3*I/a*c*arctan(a*x)^3-1/2*I/a*c*arctan(a*x)^
2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+
I*a*x)^2/(a^2*x^2+1)+1)^2)-1/4*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*csgn(I*(1+I*a*x)^4/(a^2*
x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)*x+1/2*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)*csgn(I*(1+I
*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*x+1/4*I/a*c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x
^2+1)+1)^2)^3+1/4*I/a*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^3-1/2
*I/a*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3-1/2*I/a*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x
^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+c*x*arctan(a*x)^3-1/2*I/a*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^
2+1)+I)*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2-1/2*I/a*c*arctan(a*x)^2*Pi*csgn(I*(1
+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))+1/2*I/a*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^
2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+1/2*I/a*c*arctan(a*x)^2*Pi*csgn(I/((
1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+c*x*arctan(a*x)+1
/4*c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*x-1/2*c*arct
an(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*x+I/a*c*arctan(a*x)^2
*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2+1/4*I/a*c*arctan(a*x)^2*Pi*csgn(I*((
1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-1/2*I/a*c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x
)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+1/4*I/a*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2
*x^2+1)+I)^2*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)+1/4*c*arctan(a*x)^2*Pi*csgn(I*((1
+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*x-1/4*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x
^2+1)+I)^3*x-2*I/a*c*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 28 \, a^{4} c \int \frac{x^{4} \arctan \left (a x\right )^{3}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 3 \, a^{4} c \int \frac{x^{4} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 4 \, a^{4} c \int \frac{x^{4} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - 4 \, a^{3} c \int \frac{x^{3} \arctan \left (a x\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + a^{3} c \int \frac{x^{3} \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + \frac{1}{24} \,{\left (a^{2} c x^{3} + 3 \, c x\right )} \arctan \left (a x\right )^{3} + \frac{7 \, c \arctan \left (a x\right )^{4}}{32 \, a} + 56 \, a^{2} c \int \frac{x^{2} \arctan \left (a x\right )^{3}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 6 \, a^{2} c \int \frac{x^{2} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a^{2} c \int \frac{x^{2} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - \frac{1}{32} \,{\left (a^{2} c x^{3} + 3 \, c x\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} - 12 \, a c \int \frac{x \arctan \left (a x\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 3 \, a c \int \frac{x \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 3 \, c \int \frac{\arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

28*a^4*c*integrate(1/32*x^4*arctan(a*x)^3/(a^2*x^2 + 1), x) + 3*a^4*c*integrate(1/32*x^4*arctan(a*x)*log(a^2*x
^2 + 1)^2/(a^2*x^2 + 1), x) + 4*a^4*c*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 4*a^
3*c*integrate(1/32*x^3*arctan(a*x)^2/(a^2*x^2 + 1), x) + a^3*c*integrate(1/32*x^3*log(a^2*x^2 + 1)^2/(a^2*x^2
+ 1), x) + 1/24*(a^2*c*x^3 + 3*c*x)*arctan(a*x)^3 + 7/32*c*arctan(a*x)^4/a + 56*a^2*c*integrate(1/32*x^2*arcta
n(a*x)^3/(a^2*x^2 + 1), x) + 6*a^2*c*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 12*
a^2*c*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 1/32*(a^2*c*x^3 + 3*c*x)*arctan(a*x)
*log(a^2*x^2 + 1)^2 - 12*a*c*integrate(1/32*x*arctan(a*x)^2/(a^2*x^2 + 1), x) + 3*a*c*integrate(1/32*x*log(a^2
*x^2 + 1)^2/(a^2*x^2 + 1), x) + 3*c*integrate(1/32*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c \left (\int a^{2} x^{2} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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