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

Optimal. Leaf size=308 \[ -\frac{6 i c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{35 a^2}-\frac{1}{280} a^3 c^3 x^5-\frac{3 c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2}{56 a}-\frac{9 c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{35 a}+\frac{c^3 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^3}{8 a^2}+\frac{c^3 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)}{56 a^2}+\frac{9 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac{3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{35 a^2}-\frac{6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac{12 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{35 a^2}-\frac{19}{840} a c^3 x^3-\frac{19 c^3 x}{140 a}-\frac{6 c^3 x \tan ^{-1}(a x)^2}{35 a} \]

[Out]

(-19*c^3*x)/(140*a) - (19*a*c^3*x^3)/840 - (a^3*c^3*x^5)/280 + (3*c^3*(1 + a^2*x^2)*ArcTan[a*x])/(35*a^2) + (9
*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])/(280*a^2) + (c^3*(1 + a^2*x^2)^3*ArcTan[a*x])/(56*a^2) - (((6*I)/35)*c^3*Arc
Tan[a*x]^2)/a^2 - (6*c^3*x*ArcTan[a*x]^2)/(35*a) - (3*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/(35*a) - (9*c^3*x*(1
+ a^2*x^2)^2*ArcTan[a*x]^2)/(140*a) - (3*c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(56*a) + (c^3*(1 + a^2*x^2)^4*Ar
cTan[a*x]^3)/(8*a^2) - (12*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(35*a^2) - (((6*I)/35)*c^3*PolyLog[2, 1 - 2/(1
+ I*a*x)])/a^2

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Rubi [A]  time = 0.254217, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {4930, 4880, 4846, 4920, 4854, 2402, 2315, 8, 194} \[ -\frac{6 i c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{35 a^2}-\frac{1}{280} a^3 c^3 x^5-\frac{3 c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2}{56 a}-\frac{9 c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{35 a}+\frac{c^3 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^3}{8 a^2}+\frac{c^3 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)}{56 a^2}+\frac{9 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac{3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{35 a^2}-\frac{6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac{12 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{35 a^2}-\frac{19}{840} a c^3 x^3-\frac{19 c^3 x}{140 a}-\frac{6 c^3 x \tan ^{-1}(a x)^2}{35 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-19*c^3*x)/(140*a) - (19*a*c^3*x^3)/840 - (a^3*c^3*x^5)/280 + (3*c^3*(1 + a^2*x^2)*ArcTan[a*x])/(35*a^2) + (9
*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])/(280*a^2) + (c^3*(1 + a^2*x^2)^3*ArcTan[a*x])/(56*a^2) - (((6*I)/35)*c^3*Arc
Tan[a*x]^2)/a^2 - (6*c^3*x*ArcTan[a*x]^2)/(35*a) - (3*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/(35*a) - (9*c^3*x*(1
+ a^2*x^2)^2*ArcTan[a*x]^2)/(140*a) - (3*c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(56*a) + (c^3*(1 + a^2*x^2)^4*Ar
cTan[a*x]^3)/(8*a^2) - (12*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(35*a^2) - (((6*I)/35)*c^3*PolyLog[2, 1 - 2/(1
+ I*a*x)])/a^2

Rule 4930

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

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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3 \, dx &=\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac{3 \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2 \, dx}{8 a}\\ &=\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac{3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac{c \int \left (c+a^2 c x^2\right )^2 \, dx}{56 a}-\frac{(9 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx}{28 a}\\ &=\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac{9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac{c \int \left (c^2+2 a^2 c^2 x^2+a^4 c^2 x^4\right ) \, dx}{56 a}-\frac{\left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \, dx}{280 a}-\frac{\left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx}{35 a}\\ &=-\frac{c^3 x}{20 a}-\frac{19}{840} a c^3 x^3-\frac{1}{280} a^3 c^3 x^5+\frac{3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac{3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac{\left (3 c^3\right ) \int 1 \, dx}{35 a}-\frac{\left (6 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx}{35 a}\\ &=-\frac{19 c^3 x}{140 a}-\frac{19}{840} a c^3 x^3-\frac{1}{280} a^3 c^3 x^5+\frac{3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac{6 c^3 x \tan ^{-1}(a x)^2}{35 a}-\frac{3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}+\frac{1}{35} \left (12 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{19 c^3 x}{140 a}-\frac{19}{840} a c^3 x^3-\frac{1}{280} a^3 c^3 x^5+\frac{3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac{6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac{6 c^3 x \tan ^{-1}(a x)^2}{35 a}-\frac{3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac{\left (12 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{35 a}\\ &=-\frac{19 c^3 x}{140 a}-\frac{19}{840} a c^3 x^3-\frac{1}{280} a^3 c^3 x^5+\frac{3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac{6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac{6 c^3 x \tan ^{-1}(a x)^2}{35 a}-\frac{3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac{12 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{35 a^2}+\frac{\left (12 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{35 a}\\ &=-\frac{19 c^3 x}{140 a}-\frac{19}{840} a c^3 x^3-\frac{1}{280} a^3 c^3 x^5+\frac{3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac{6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac{6 c^3 x \tan ^{-1}(a x)^2}{35 a}-\frac{3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac{12 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{35 a^2}-\frac{\left (12 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{35 a^2}\\ &=-\frac{19 c^3 x}{140 a}-\frac{19}{840} a c^3 x^3-\frac{1}{280} a^3 c^3 x^5+\frac{3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac{6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac{6 c^3 x \tan ^{-1}(a x)^2}{35 a}-\frac{3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac{12 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{35 a^2}-\frac{6 i c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{35 a^2}\\ \end{align*}

Mathematica [A]  time = 1.31684, size = 157, normalized size = 0.51 \[ \frac{c^3 \left (144 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-a x \left (3 a^4 x^4+19 a^2 x^2+114\right )+105 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^3-9 \left (5 a^7 x^7+21 a^5 x^5+35 a^3 x^3+35 a x-16 i\right ) \tan ^{-1}(a x)^2+3 \tan ^{-1}(a x) \left (5 a^6 x^6+24 a^4 x^4+57 a^2 x^2-96 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+38\right )\right )}{840 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(-(a*x*(114 + 19*a^2*x^2 + 3*a^4*x^4)) - 9*(-16*I + 35*a*x + 35*a^3*x^3 + 21*a^5*x^5 + 5*a^7*x^7)*ArcTan[
a*x]^2 + 105*(1 + a^2*x^2)^4*ArcTan[a*x]^3 + 3*ArcTan[a*x]*(38 + 57*a^2*x^2 + 24*a^4*x^4 + 5*a^6*x^6 - 96*Log[
1 + E^((2*I)*ArcTan[a*x])]) + (144*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(840*a^2)

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Maple [A]  time = 0.096, size = 428, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-19/840*c^3*x^3*a+1/2*a^4*c^3*arctan(a*x)^3*x^6+3/4*a^2*c^3*arctan(a*x)^3*x^4+1/56*a^4*c^3*arctan(a*x)*x^6+3/3
5*a^2*c^3*arctan(a*x)*x^4-3/56*a^5*c^3*arctan(a*x)^2*x^7-9/40*a^3*c^3*arctan(a*x)^2*x^5-3/8*a*c^3*arctan(a*x)^
2*x^3+3/35*I/a^2*c^3*dilog(1/2*I*(a*x-I))+3/70*I/a^2*c^3*ln(a*x+I)^2-3/35*I/a^2*c^3*dilog(-1/2*I*(a*x+I))-3/70
*I/a^2*c^3*ln(a*x-I)^2+6/35/a^2*c^3*arctan(a*x)*ln(a^2*x^2+1)+1/8*a^6*c^3*arctan(a*x)^3*x^8-3/35*I/a^2*c^3*ln(
a*x-I)*ln(-1/2*I*(a*x+I))+3/35*I/a^2*c^3*ln(a*x-I)*ln(a^2*x^2+1)-3/35*I/a^2*c^3*ln(a*x+I)*ln(a^2*x^2+1)+3/35*I
/a^2*c^3*ln(a*x+I)*ln(1/2*I*(a*x-I))+1/2*c^3*arctan(a*x)^3*x^2+57/280*c^3*arctan(a*x)*x^2+1/8/a^2*c^3*arctan(a
*x)^3+19/140/a^2*c^3*arctan(a*x)-19/140*c^3*x/a-1/280*a^3*c^3*x^5-3/8*c^3*x*arctan(a*x)^2/a

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x)*arctan(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int x \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{3} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{5} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{7} \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(x*atan(a*x)**3, x) + Integral(3*a**2*x**3*atan(a*x)**3, x) + Integral(3*a**4*x**5*atan(a*x)**3,
x) + Integral(a**6*x**7*atan(a*x)**3, x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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