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

Optimal. Leaf size=205 \[ \frac{8 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{15 a}+\frac{1}{30} a^2 c^2 x^3+\frac{1}{5} c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2-\frac{c^2 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{10 a}-\frac{4 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{15 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^2+\frac{8 i c^2 \tan ^{-1}(a x)^2}{15 a}+\frac{16 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a}+\frac{11 c^2 x}{30} \]

[Out]

(11*c^2*x)/30 + (a^2*c^2*x^3)/30 - (4*c^2*(1 + a^2*x^2)*ArcTan[a*x])/(15*a) - (c^2*(1 + a^2*x^2)^2*ArcTan[a*x]
)/(10*a) + (((8*I)/15)*c^2*ArcTan[a*x]^2)/a + (8*c^2*x*ArcTan[a*x]^2)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^
2)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/5 + (16*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a) + (((8*I)/15)
*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a

________________________________________________________________________________________

Rubi [A]  time = 0.138938, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {4880, 4846, 4920, 4854, 2402, 2315, 8} \[ \frac{8 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{15 a}+\frac{1}{30} a^2 c^2 x^3+\frac{1}{5} c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2-\frac{c^2 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{10 a}-\frac{4 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{15 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^2+\frac{8 i c^2 \tan ^{-1}(a x)^2}{15 a}+\frac{16 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a}+\frac{11 c^2 x}{30} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*c^2*x)/30 + (a^2*c^2*x^3)/30 - (4*c^2*(1 + a^2*x^2)*ArcTan[a*x])/(15*a) - (c^2*(1 + a^2*x^2)^2*ArcTan[a*x]
)/(10*a) + (((8*I)/15)*c^2*ArcTan[a*x]^2)/a + (8*c^2*x*ArcTan[a*x]^2)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^
2)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/5 + (16*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a) + (((8*I)/15)
*c^2*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]

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx &=-\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{10 a}+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{1}{10} c \int \left (c+a^2 c x^2\right ) \, dx+\frac{1}{5} (4 c) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx\\ &=\frac{c^2 x}{10}+\frac{1}{30} a^2 c^2 x^3-\frac{4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a}-\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{10 a}+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{1}{15} \left (4 c^2\right ) \int 1 \, dx+\frac{1}{15} \left (8 c^2\right ) \int \tan ^{-1}(a x)^2 \, dx\\ &=\frac{11 c^2 x}{30}+\frac{1}{30} a^2 c^2 x^3-\frac{4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a}-\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{10 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2-\frac{1}{15} \left (16 a c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{11 c^2 x}{30}+\frac{1}{30} a^2 c^2 x^3-\frac{4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a}-\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{10 a}+\frac{8 i c^2 \tan ^{-1}(a x)^2}{15 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{1}{15} \left (16 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx\\ &=\frac{11 c^2 x}{30}+\frac{1}{30} a^2 c^2 x^3-\frac{4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a}-\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{10 a}+\frac{8 i c^2 \tan ^{-1}(a x)^2}{15 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{16 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a}-\frac{1}{15} \left (16 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac{11 c^2 x}{30}+\frac{1}{30} a^2 c^2 x^3-\frac{4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a}-\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{10 a}+\frac{8 i c^2 \tan ^{-1}(a x)^2}{15 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{16 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a}+\frac{\left (16 i c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{15 a}\\ &=\frac{11 c^2 x}{30}+\frac{1}{30} a^2 c^2 x^3-\frac{4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a}-\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{10 a}+\frac{8 i c^2 \tan ^{-1}(a x)^2}{15 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac{16 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a}+\frac{8 i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{15 a}\\ \end{align*}

Mathematica [A]  time = 0.654809, size = 112, normalized size = 0.55 \[ \frac{c^2 \left (-16 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+a x \left (a^2 x^2+11\right )+2 \left (3 a^5 x^5+10 a^3 x^3+15 a x-8 i\right ) \tan ^{-1}(a x)^2-\tan ^{-1}(a x) \left (3 a^4 x^4+14 a^2 x^2-32 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+11\right )\right )}{30 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*(a*x*(11 + a^2*x^2) + 2*(-8*I + 15*a*x + 10*a^3*x^3 + 3*a^5*x^5)*ArcTan[a*x]^2 - ArcTan[a*x]*(11 + 14*a^2
*x^2 + 3*a^4*x^4 - 32*Log[1 + E^((2*I)*ArcTan[a*x])]) - (16*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(30*a)

________________________________________________________________________________________

Maple [A]  time = 0.085, size = 304, normalized size = 1.5 \begin{align*}{\frac{{a}^{4}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{5}}{5}}+{\frac{2\,{a}^{2}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}}{3}}+{c}^{2}x \left ( \arctan \left ( ax \right ) \right ) ^{2}-{\frac{{a}^{3}{c}^{2}\arctan \left ( ax \right ){x}^{4}}{10}}-{\frac{7\,a{c}^{2}\arctan \left ( ax \right ){x}^{2}}{15}}-{\frac{8\,{c}^{2}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{15\,a}}+{\frac{{a}^{2}{c}^{2}{x}^{3}}{30}}+{\frac{11\,{c}^{2}x}{30}}-{\frac{11\,{c}^{2}\arctan \left ( ax \right ) }{30\,a}}+{\frac{{\frac{4\,i}{15}}{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{a}}+{\frac{{\frac{4\,i}{15}}{c}^{2}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{a}}+{\frac{{\frac{4\,i}{15}}{c}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{a}}-{\frac{{\frac{4\,i}{15}}{c}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{a}}-{\frac{{\frac{2\,i}{15}}{c}^{2} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{a}}+{\frac{{\frac{2\,i}{15}}{c}^{2} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{a}}-{\frac{{\frac{4\,i}{15}}{c}^{2}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{a}}-{\frac{{\frac{4\,i}{15}}{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^4*c^2*arctan(a*x)^2*x^5+2/3*a^2*c^2*arctan(a*x)^2*x^3+c^2*x*arctan(a*x)^2-1/10*a^3*c^2*arctan(a*x)*x^4-7
/15*a*c^2*arctan(a*x)*x^2-8/15/a*c^2*arctan(a*x)*ln(a^2*x^2+1)+1/30*a^2*c^2*x^3+11/30*c^2*x-11/30/a*c^2*arctan
(a*x)+4/15*I/a*c^2*ln(a^2*x^2+1)*ln(a*x+I)+4/15*I/a*c^2*ln(a*x-I)*ln(-1/2*I*(a*x+I))+4/15*I/a*c^2*dilog(-1/2*I
*(a*x+I))-4/15*I/a*c^2*dilog(1/2*I*(a*x-I))-2/15*I/a*c^2*ln(a*x+I)^2+2/15*I/a*c^2*ln(a*x-I)^2-4/15*I/a*c^2*ln(
a*x+I)*ln(1/2*I*(a*x-I))-4/15*I/a*c^2*ln(a^2*x^2+1)*ln(a*x-I)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 180 \, a^{6} c^{2} \int \frac{x^{6} \arctan \left (a x\right )^{2}}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 15 \, a^{6} c^{2} \int \frac{x^{6} \log \left (a^{2} x^{2} + 1\right )^{2}}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a^{6} c^{2} \int \frac{x^{6} \log \left (a^{2} x^{2} + 1\right )}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - 24 \, a^{5} c^{2} \int \frac{x^{5} \arctan \left (a x\right )}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 540 \, a^{4} c^{2} \int \frac{x^{4} \arctan \left (a x\right )^{2}}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 45 \, a^{4} c^{2} \int \frac{x^{4} \log \left (a^{2} x^{2} + 1\right )^{2}}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 40 \, a^{4} c^{2} \int \frac{x^{4} \log \left (a^{2} x^{2} + 1\right )}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - 80 \, a^{3} c^{2} \int \frac{x^{3} \arctan \left (a x\right )}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 540 \, a^{2} c^{2} \int \frac{x^{2} \arctan \left (a x\right )^{2}}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 45 \, a^{2} c^{2} \int \frac{x^{2} \log \left (a^{2} x^{2} + 1\right )^{2}}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 60 \, a^{2} c^{2} \int \frac{x^{2} \log \left (a^{2} x^{2} + 1\right )}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + \frac{c^{2} \arctan \left (a x\right )^{3}}{4 \, a} - 120 \, a c^{2} \int \frac{x \arctan \left (a x\right )}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + \frac{1}{60} \,{\left (3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \arctan \left (a x\right )^{2} + 15 \, c^{2} \int \frac{\log \left (a^{2} x^{2} + 1\right )^{2}}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - \frac{1}{240} \,{\left (3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \log \left (a^{2} x^{2} + 1\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

180*a^6*c^2*integrate(1/240*x^6*arctan(a*x)^2/(a^2*x^2 + 1), x) + 15*a^6*c^2*integrate(1/240*x^6*log(a^2*x^2 +
1)^2/(a^2*x^2 + 1), x) + 12*a^6*c^2*integrate(1/240*x^6*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 24*a^5*c^2*integ
rate(1/240*x^5*arctan(a*x)/(a^2*x^2 + 1), x) + 540*a^4*c^2*integrate(1/240*x^4*arctan(a*x)^2/(a^2*x^2 + 1), x)
+ 45*a^4*c^2*integrate(1/240*x^4*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 40*a^4*c^2*integrate(1/240*x^4*log(a^
2*x^2 + 1)/(a^2*x^2 + 1), x) - 80*a^3*c^2*integrate(1/240*x^3*arctan(a*x)/(a^2*x^2 + 1), x) + 540*a^2*c^2*inte
grate(1/240*x^2*arctan(a*x)^2/(a^2*x^2 + 1), x) + 45*a^2*c^2*integrate(1/240*x^2*log(a^2*x^2 + 1)^2/(a^2*x^2 +
1), x) + 60*a^2*c^2*integrate(1/240*x^2*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) + 1/4*c^2*arctan(a*x)^3/a - 120*a*
c^2*integrate(1/240*x*arctan(a*x)/(a^2*x^2 + 1), x) + 1/60*(3*a^4*c^2*x^5 + 10*a^2*c^2*x^3 + 15*c^2*x)*arctan(
a*x)^2 + 15*c^2*integrate(1/240*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) - 1/240*(3*a^4*c^2*x^5 + 10*a^2*c^2*x^3 +
15*c^2*x)*log(a^2*x^2 + 1)^2

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

c**2*(Integral(2*a**2*x**2*atan(a*x)**2, x) + Integral(a**4*x**4*atan(a*x)**2, x) + Integral(atan(a*x)**2, x))

________________________________________________________________________________________

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

[Out]

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