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

Optimal. Leaf size=225 \[ -\frac{8 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{105 a^3}+\frac{1}{105} a^2 c^2 x^5+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2-\frac{c^2 x}{210 a^2}-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{c^2 \tan ^{-1}(a x)}{210 a^3}-\frac{16 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{105 a^3}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}+\frac{17 c^2 x^3}{630} \]

[Out]

-(c^2*x)/(210*a^2) + (17*c^2*x^3)/630 + (a^2*c^2*x^5)/105 + (c^2*ArcTan[a*x])/(210*a^3) - (8*c^2*x^2*ArcTan[a*
x])/(105*a) - (9*a*c^2*x^4*ArcTan[a*x])/70 - (a^3*c^2*x^6*ArcTan[a*x])/21 - (((8*I)/105)*c^2*ArcTan[a*x]^2)/a^
3 + (c^2*x^3*ArcTan[a*x]^2)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^2)/5 + (a^4*c^2*x^7*ArcTan[a*x]^2)/7 - (16*c^2*ArcT
an[a*x]*Log[2/(1 + I*a*x)])/(105*a^3) - (((8*I)/105)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3

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Rubi [A]  time = 0.752474, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 44, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302} \[ -\frac{8 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{105 a^3}+\frac{1}{105} a^2 c^2 x^5+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2-\frac{c^2 x}{210 a^2}-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{c^2 \tan ^{-1}(a x)}{210 a^3}-\frac{16 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{105 a^3}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}+\frac{17 c^2 x^3}{630} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*x)/(210*a^2) + (17*c^2*x^3)/630 + (a^2*c^2*x^5)/105 + (c^2*ArcTan[a*x])/(210*a^3) - (8*c^2*x^2*ArcTan[a*
x])/(105*a) - (9*a*c^2*x^4*ArcTan[a*x])/70 - (a^3*c^2*x^6*ArcTan[a*x])/21 - (((8*I)/105)*c^2*ArcTan[a*x]^2)/a^
3 + (c^2*x^3*ArcTan[a*x]^2)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^2)/5 + (a^4*c^2*x^7*ArcTan[a*x]^2)/7 - (16*c^2*ArcT
an[a*x]*Log[2/(1 + I*a*x)])/(105*a^3) - (((8*I)/105)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3

Rule 4948

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e,
c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])

Rule 4852

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

Rule 4916

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin{align*} \int x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx &=\int \left (c^2 x^2 \tan ^{-1}(a x)^2+2 a^2 c^2 x^4 \tan ^{-1}(a x)^2+a^4 c^2 x^6 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^2 \int x^2 \tan ^{-1}(a x)^2 \, dx+\left (2 a^2 c^2\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\left (a^4 c^2\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{1}{3} \left (2 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (4 a^3 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{7} \left (2 a^5 c^2\right ) \int \frac{x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{\left (2 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{3 a}+\frac{\left (2 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}-\frac{1}{5} \left (4 a c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac{1}{5} \left (4 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{7} \left (2 a^3 c^2\right ) \int x^5 \tan ^{-1}(a x) \, dx+\frac{1}{7} \left (2 a^3 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{c^2 x^2 \tan ^{-1}(a x)}{3 a}-\frac{1}{5} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac{i c^2 \tan ^{-1}(a x)^2}{3 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2+\frac{1}{3} c^2 \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^2}+\frac{\left (4 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{5 a}-\frac{\left (4 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a}+\frac{1}{7} \left (2 a c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac{1}{7} \left (2 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{5} \left (a^2 c^2\right ) \int \frac{x^4}{1+a^2 x^2} \, dx+\frac{1}{21} \left (a^4 c^2\right ) \int \frac{x^6}{1+a^2 x^2} \, dx\\ &=\frac{c^2 x}{3 a^2}+\frac{c^2 x^2 \tan ^{-1}(a x)}{15 a}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)+\frac{i c^2 \tan ^{-1}(a x)^2}{15 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{2 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{3 a^3}-\frac{1}{5} \left (2 c^2\right ) \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{3 a^2}+\frac{\left (2 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac{\left (4 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^2}-\frac{\left (2 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{7 a}+\frac{\left (2 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a}-\frac{1}{14} \left (a^2 c^2\right ) \int \frac{x^4}{1+a^2 x^2} \, dx+\frac{1}{5} \left (a^2 c^2\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx+\frac{1}{21} \left (a^4 c^2\right ) \int \left (\frac{1}{a^6}-\frac{x^2}{a^4}+\frac{x^4}{a^2}-\frac{1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac{23 c^2 x}{105 a^2}+\frac{16 c^2 x^3}{315}+\frac{1}{105} a^2 c^2 x^5-\frac{c^2 \tan ^{-1}(a x)}{3 a^3}-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2+\frac{2 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^3}+\frac{1}{7} c^2 \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{\left (2 i c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{3 a^3}-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{21 a^2}+\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{7 a^2}+\frac{\left (2 c^2\right ) \int \frac{1}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\left (4 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}-\frac{1}{14} \left (a^2 c^2\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac{c^2 x}{210 a^2}+\frac{17 c^2 x^3}{630}+\frac{1}{105} a^2 c^2 x^5+\frac{23 c^2 \tan ^{-1}(a x)}{105 a^3}-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{16 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{105 a^3}-\frac{i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{3 a^3}+\frac{\left (4 i c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{5 a^3}-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{14 a^2}-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{7 a^2}+\frac{\left (2 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}\\ &=-\frac{c^2 x}{210 a^2}+\frac{17 c^2 x^3}{630}+\frac{1}{105} a^2 c^2 x^5+\frac{c^2 \tan ^{-1}(a x)}{210 a^3}-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{16 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{105 a^3}+\frac{i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{15 a^3}-\frac{\left (2 i c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{7 a^3}\\ &=-\frac{c^2 x}{210 a^2}+\frac{17 c^2 x^3}{630}+\frac{1}{105} a^2 c^2 x^5+\frac{c^2 \tan ^{-1}(a x)}{210 a^3}-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{16 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{105 a^3}-\frac{8 i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{105 a^3}\\ \end{align*}

Mathematica [A]  time = 1.24216, size = 133, normalized size = 0.59 \[ \frac{c^2 \left (48 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+a x \left (6 a^4 x^4+17 a^2 x^2-3\right )+6 \left (15 a^7 x^7+42 a^5 x^5+35 a^3 x^3+8 i\right ) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \left (10 a^6 x^6+27 a^4 x^4+16 a^2 x^2+32 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-1\right )\right )}{630 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*(a*x*(-3 + 17*a^2*x^2 + 6*a^4*x^4) + 6*(8*I + 35*a^3*x^3 + 42*a^5*x^5 + 15*a^7*x^7)*ArcTan[a*x]^2 - 3*Arc
Tan[a*x]*(-1 + 16*a^2*x^2 + 27*a^4*x^4 + 10*a^6*x^6 + 32*Log[1 + E^((2*I)*ArcTan[a*x])]) + (48*I)*PolyLog[2, -
E^((2*I)*ArcTan[a*x])]))/(630*a^3)

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Maple [A]  time = 0.092, size = 333, normalized size = 1.5 \begin{align*}{\frac{{a}^{4}{c}^{2}{x}^{7} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{7}}+{\frac{2\,{a}^{2}{c}^{2}{x}^{5} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{5}}+{\frac{{c}^{2}{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3}}-{\frac{{a}^{3}{c}^{2}{x}^{6}\arctan \left ( ax \right ) }{21}}-{\frac{9\,a{c}^{2}{x}^{4}\arctan \left ( ax \right ) }{70}}-{\frac{8\,{c}^{2}{x}^{2}\arctan \left ( ax \right ) }{105\,a}}+{\frac{8\,{c}^{2}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{105\,{a}^{3}}}+{\frac{{a}^{2}{c}^{2}{x}^{5}}{105}}+{\frac{17\,{c}^{2}{x}^{3}}{630}}-{\frac{{c}^{2}x}{210\,{a}^{2}}}+{\frac{{c}^{2}\arctan \left ( ax \right ) }{210\,{a}^{3}}}+{\frac{{\frac{4\,i}{105}}{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{{a}^{3}}}+{\frac{{\frac{4\,i}{105}}{c}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}}+{\frac{{\frac{4\,i}{105}}{c}^{2}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}}-{\frac{{\frac{2\,i}{105}}{c}^{2} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{3}}}+{\frac{{\frac{2\,i}{105}}{c}^{2} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{3}}}-{\frac{{\frac{4\,i}{105}}{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{{a}^{3}}}-{\frac{{\frac{4\,i}{105}}{c}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}}-{\frac{{\frac{4\,i}{105}}{c}^{2}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^4*c^2*x^7*arctan(a*x)^2+2/5*a^2*c^2*x^5*arctan(a*x)^2+1/3*c^2*x^3*arctan(a*x)^2-1/21*a^3*c^2*x^6*arctan(
a*x)-9/70*a*c^2*x^4*arctan(a*x)-8/105*c^2*x^2*arctan(a*x)/a+8/105/a^3*c^2*arctan(a*x)*ln(a^2*x^2+1)+1/105*a^2*
c^2*x^5+17/630*c^2*x^3-1/210*c^2*x/a^2+1/210*c^2*arctan(a*x)/a^3+4/105*I/a^3*c^2*ln(a^2*x^2+1)*ln(a*x-I)+4/105
*I/a^3*c^2*dilog(1/2*I*(a*x-I))+4/105*I/a^3*c^2*ln(a*x+I)*ln(1/2*I*(a*x-I))-2/105*I/a^3*c^2*ln(a*x-I)^2+2/105*
I/a^3*c^2*ln(a*x+I)^2-4/105*I/a^3*c^2*ln(a^2*x^2+1)*ln(a*x+I)-4/105*I/a^3*c^2*dilog(-1/2*I*(a*x+I))-4/105*I/a^
3*c^2*ln(a*x-I)*ln(-1/2*I*(a*x+I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{420} \,{\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \arctan \left (a x\right )^{2} - \frac{1}{1680} \,{\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{1260 \,{\left (a^{6} c^{2} x^{8} + 3 \, a^{4} c^{2} x^{6} + 3 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 105 \,{\left (a^{6} c^{2} x^{8} + 3 \, a^{4} c^{2} x^{6} + 3 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} - 8 \,{\left (15 \, a^{5} c^{2} x^{7} + 42 \, a^{3} c^{2} x^{5} + 35 \, a c^{2} x^{3}\right )} \arctan \left (a x\right ) + 4 \,{\left (15 \, a^{6} c^{2} x^{8} + 42 \, a^{4} c^{2} x^{6} + 35 \, a^{2} c^{2} x^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{1680 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 + 35*c^2*x^3)*arctan(a*x)^2 - 1/1680*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 +
35*c^2*x^3)*log(a^2*x^2 + 1)^2 + integrate(1/1680*(1260*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x^4 + c^2*x^
2)*arctan(a*x)^2 + 105*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x^4 + c^2*x^2)*log(a^2*x^2 + 1)^2 - 8*(15*a^5*
c^2*x^7 + 42*a^3*c^2*x^5 + 35*a*c^2*x^3)*arctan(a*x) + 4*(15*a^6*c^2*x^8 + 42*a^4*c^2*x^6 + 35*a^2*c^2*x^4)*lo
g(a^2*x^2 + 1))/(a^2*x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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