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

Optimal. Leaf size=268 \[ \frac{16 i c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{35 a}+\frac{1}{105} a^4 c^3 x^5+\frac{19}{315} a^2 c^3 x^3+\frac{1}{7} c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2-\frac{c^3 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)}{21 a}-\frac{3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{35 a}-\frac{8 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^2+\frac{16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac{32 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{35 a}+\frac{38 c^3 x}{105} \]

[Out]

(38*c^3*x)/105 + (19*a^2*c^3*x^3)/315 + (a^4*c^3*x^5)/105 - (8*c^3*(1 + a^2*x^2)*ArcTan[a*x])/(35*a) - (3*c^3*
(1 + a^2*x^2)^2*ArcTan[a*x])/(35*a) - (c^3*(1 + a^2*x^2)^3*ArcTan[a*x])/(21*a) + (((16*I)/35)*c^3*ArcTan[a*x]^
2)/a + (16*c^3*x*ArcTan[a*x]^2)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTa
n[a*x]^2)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/7 + (32*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(35*a) + (((1
6*I)/35)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a

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Rubi [A]  time = 0.184195, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {4880, 4846, 4920, 4854, 2402, 2315, 8, 194} \[ \frac{16 i c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{35 a}+\frac{1}{105} a^4 c^3 x^5+\frac{19}{315} a^2 c^3 x^3+\frac{1}{7} c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2-\frac{c^3 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)}{21 a}-\frac{3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{35 a}-\frac{8 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^2+\frac{16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac{32 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{35 a}+\frac{38 c^3 x}{105} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(38*c^3*x)/105 + (19*a^2*c^3*x^3)/315 + (a^4*c^3*x^5)/105 - (8*c^3*(1 + a^2*x^2)*ArcTan[a*x])/(35*a) - (3*c^3*
(1 + a^2*x^2)^2*ArcTan[a*x])/(35*a) - (c^3*(1 + a^2*x^2)^3*ArcTan[a*x])/(21*a) + (((16*I)/35)*c^3*ArcTan[a*x]^
2)/a + (16*c^3*x*ArcTan[a*x]^2)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTa
n[a*x]^2)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/7 + (32*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(35*a) + (((1
6*I)/35)*c^3*PolyLog[2, 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 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 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2 \, dx &=-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac{1}{21} c \int \left (c+a^2 c x^2\right )^2 \, dx+\frac{1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx\\ &=-\frac{3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac{1}{21} c \int \left (c^2+2 a^2 c^2 x^2+a^4 c^2 x^4\right ) \, dx+\frac{1}{35} \left (3 c^2\right ) \int \left (c+a^2 c x^2\right ) \, dx+\frac{1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx\\ &=\frac{2 c^3 x}{15}+\frac{19}{315} a^2 c^3 x^3+\frac{1}{105} a^4 c^3 x^5-\frac{8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac{1}{35} \left (8 c^3\right ) \int 1 \, dx+\frac{1}{35} \left (16 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx\\ &=\frac{38 c^3 x}{105}+\frac{19}{315} a^2 c^3 x^3+\frac{1}{105} a^4 c^3 x^5-\frac{8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2-\frac{1}{35} \left (32 a c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{38 c^3 x}{105}+\frac{19}{315} a^2 c^3 x^3+\frac{1}{105} a^4 c^3 x^5-\frac{8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac{16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac{1}{35} \left (32 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx\\ &=\frac{38 c^3 x}{105}+\frac{19}{315} a^2 c^3 x^3+\frac{1}{105} a^4 c^3 x^5-\frac{8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac{16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac{32 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{35 a}-\frac{1}{35} \left (32 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac{38 c^3 x}{105}+\frac{19}{315} a^2 c^3 x^3+\frac{1}{105} a^4 c^3 x^5-\frac{8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac{16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac{32 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{35 a}+\frac{\left (32 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}\\ &=\frac{38 c^3 x}{105}+\frac{19}{315} a^2 c^3 x^3+\frac{1}{105} a^4 c^3 x^5-\frac{8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac{16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac{32 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{35 a}+\frac{16 i c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{35 a}\\ \end{align*}

Mathematica [A]  time = 1.16528, size = 137, normalized size = 0.51 \[ \frac{c^3 \left (-144 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+a x \left (3 a^4 x^4+19 a^2 x^2+114\right )+9 \left (5 a^7 x^7+21 a^5 x^5+35 a^3 x^3+35 a x-16 i\right ) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \left (5 a^6 x^6+24 a^4 x^4+57 a^2 x^2-96 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+38\right )\right )}{315 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + a^2*c*x^2)^3*ArcTan[a*x]^2,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 - 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)*P
olyLog[2, -E^((2*I)*ArcTan[a*x])]))/(315*a)

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Maple [A]  time = 0.069, size = 346, normalized size = 1.3 \begin{align*}{\frac{{a}^{6}{c}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{7}}{7}}+{\frac{3\,{a}^{4}{c}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{5}}{5}}+{a}^{2}{c}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}+{c}^{3}x \left ( \arctan \left ( ax \right ) \right ) ^{2}-{\frac{{a}^{5}{c}^{3}\arctan \left ( ax \right ){x}^{6}}{21}}-{\frac{8\,{a}^{3}{c}^{3}\arctan \left ( ax \right ){x}^{4}}{35}}-{\frac{19\,a{c}^{3}\arctan \left ( ax \right ){x}^{2}}{35}}-{\frac{16\,{c}^{3}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{35\,a}}+{\frac{{a}^{4}{c}^{3}{x}^{5}}{105}}+{\frac{19\,{a}^{2}{c}^{3}{x}^{3}}{315}}+{\frac{38\,{c}^{3}x}{105}}-{\frac{38\,{c}^{3}\arctan \left ( ax \right ) }{105\,a}}-{\frac{{\frac{4\,i}{35}}{c}^{3} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{a}}+{\frac{{\frac{8\,i}{35}}{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{a}}+{\frac{{\frac{8\,i}{35}}{c}^{3}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{a}}+{\frac{{\frac{8\,i}{35}}{c}^{3}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{a}}-{\frac{{\frac{8\,i}{35}}{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{a}}-{\frac{{\frac{8\,i}{35}}{c}^{3}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{a}}-{\frac{{\frac{8\,i}{35}}{c}^{3}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{a}}+{\frac{{\frac{4\,i}{35}}{c}^{3} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^6*c^3*arctan(a*x)^2*x^7+3/5*a^4*c^3*arctan(a*x)^2*x^5+a^2*c^3*arctan(a*x)^2*x^3+c^3*x*arctan(a*x)^2-1/21
*a^5*c^3*arctan(a*x)*x^6-8/35*a^3*c^3*arctan(a*x)*x^4-19/35*a*c^3*arctan(a*x)*x^2-16/35/a*c^3*arctan(a*x)*ln(a
^2*x^2+1)+1/105*a^4*c^3*x^5+19/315*a^2*c^3*x^3+38/105*c^3*x-38/105/a*c^3*arctan(a*x)-4/35*I/a*c^3*ln(a*x+I)^2+
8/35*I/a*c^3*ln(a^2*x^2+1)*ln(a*x+I)+8/35*I/a*c^3*ln(a*x-I)*ln(-1/2*I*(a*x+I))+8/35*I/a*c^3*dilog(-1/2*I*(a*x+
I))-8/35*I/a*c^3*ln(a^2*x^2+1)*ln(a*x-I)-8/35*I/a*c^3*dilog(1/2*I*(a*x-I))-8/35*I/a*c^3*ln(a*x+I)*ln(1/2*I*(a*
x-I))+4/35*I/a*c^3*ln(a*x-I)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

420*a^8*c^3*integrate(1/560*x^8*arctan(a*x)^2/(a^2*x^2 + 1), x) + 35*a^8*c^3*integrate(1/560*x^8*log(a^2*x^2 +
1)^2/(a^2*x^2 + 1), x) + 20*a^8*c^3*integrate(1/560*x^8*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 40*a^7*c^3*integ
rate(1/560*x^7*arctan(a*x)/(a^2*x^2 + 1), x) + 1680*a^6*c^3*integrate(1/560*x^6*arctan(a*x)^2/(a^2*x^2 + 1), x
) + 140*a^6*c^3*integrate(1/560*x^6*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 84*a^6*c^3*integrate(1/560*x^6*log(
a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 168*a^5*c^3*integrate(1/560*x^5*arctan(a*x)/(a^2*x^2 + 1), x) + 2520*a^4*c^3*
integrate(1/560*x^4*arctan(a*x)^2/(a^2*x^2 + 1), x) + 210*a^4*c^3*integrate(1/560*x^4*log(a^2*x^2 + 1)^2/(a^2*
x^2 + 1), x) + 140*a^4*c^3*integrate(1/560*x^4*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 280*a^3*c^3*integrate(1/56
0*x^3*arctan(a*x)/(a^2*x^2 + 1), x) + 1680*a^2*c^3*integrate(1/560*x^2*arctan(a*x)^2/(a^2*x^2 + 1), x) + 140*a
^2*c^3*integrate(1/560*x^2*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 140*a^2*c^3*integrate(1/560*x^2*log(a^2*x^2
+ 1)/(a^2*x^2 + 1), x) + 1/4*c^3*arctan(a*x)^3/a - 280*a*c^3*integrate(1/560*x*arctan(a*x)/(a^2*x^2 + 1), x) +
35*c^3*integrate(1/560*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 1/140*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*
c^3*x^3 + 35*c^3*x)*arctan(a*x)^2 - 1/560*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)*log(a^2
*x^2 + 1)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

c**3*(Integral(3*a**2*x**2*atan(a*x)**2, x) + Integral(3*a**4*x**4*atan(a*x)**2, x) + Integral(a**6*x**6*atan(
a*x)**2, x) + Integral(atan(a*x)**2, 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} \arctan \left (a x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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