3.1255 \(\int x^2 (d+e x^2)^2 (a+b \tan ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=580 \[ -\frac{i b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^3}+\frac{2 i b^2 d e \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{5 c^5}-\frac{i b^2 e^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{7 c^7}-\frac{i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac{2 b d^2 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}+\frac{2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}+\frac{4 b d e \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{5 c^5}+\frac{b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac{i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}-\frac{2 b e^2 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{7 c^7}+\frac{1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac{1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}+\frac{b^2 d^2 x}{3 c^2}-\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 d e x^3}{15 c^2}-\frac{3 b^2 d e x}{5 c^4}+\frac{3 b^2 d e \tan ^{-1}(c x)}{5 c^5}+\frac{b^2 e^2 x^5}{105 c^2}-\frac{5 b^2 e^2 x^3}{126 c^4}+\frac{11 b^2 e^2 x}{42 c^6}-\frac{11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7} \]

[Out]

(b^2*d^2*x)/(3*c^2) - (3*b^2*d*e*x)/(5*c^4) + (11*b^2*e^2*x)/(42*c^6) + (b^2*d*e*x^3)/(15*c^2) - (5*b^2*e^2*x^
3)/(126*c^4) + (b^2*e^2*x^5)/(105*c^2) - (b^2*d^2*ArcTan[c*x])/(3*c^3) + (3*b^2*d*e*ArcTan[c*x])/(5*c^5) - (11
*b^2*e^2*ArcTan[c*x])/(42*c^7) - (b*d^2*x^2*(a + b*ArcTan[c*x]))/(3*c) + (2*b*d*e*x^2*(a + b*ArcTan[c*x]))/(5*
c^3) - (b*e^2*x^2*(a + b*ArcTan[c*x]))/(7*c^5) - (b*d*e*x^4*(a + b*ArcTan[c*x]))/(5*c) + (b*e^2*x^4*(a + b*Arc
Tan[c*x]))/(14*c^3) - (b*e^2*x^6*(a + b*ArcTan[c*x]))/(21*c) - ((I/3)*d^2*(a + b*ArcTan[c*x])^2)/c^3 + (((2*I)
/5)*d*e*(a + b*ArcTan[c*x])^2)/c^5 - ((I/7)*e^2*(a + b*ArcTan[c*x])^2)/c^7 + (d^2*x^3*(a + b*ArcTan[c*x])^2)/3
+ (2*d*e*x^5*(a + b*ArcTan[c*x])^2)/5 + (e^2*x^7*(a + b*ArcTan[c*x])^2)/7 - (2*b*d^2*(a + b*ArcTan[c*x])*Log[
2/(1 + I*c*x)])/(3*c^3) + (4*b*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) - (2*b*e^2*(a + b*ArcTan[c*
x])*Log[2/(1 + I*c*x)])/(7*c^7) - ((I/3)*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 + (((2*I)/5)*b^2*d*e*PolyL
og[2, 1 - 2/(1 + I*c*x)])/c^5 - ((I/7)*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^7

________________________________________________________________________________________

Rubi [A]  time = 1.06815, antiderivative size = 580, normalized size of antiderivative = 1., number of steps used = 44, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {4980, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302} \[ -\frac{i b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^3}+\frac{2 i b^2 d e \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{5 c^5}-\frac{i b^2 e^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{7 c^7}-\frac{i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac{2 b d^2 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}+\frac{2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}+\frac{4 b d e \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{5 c^5}+\frac{b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac{i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}-\frac{2 b e^2 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{7 c^7}+\frac{1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac{1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}+\frac{b^2 d^2 x}{3 c^2}-\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 d e x^3}{15 c^2}-\frac{3 b^2 d e x}{5 c^4}+\frac{3 b^2 d e \tan ^{-1}(c x)}{5 c^5}+\frac{b^2 e^2 x^5}{105 c^2}-\frac{5 b^2 e^2 x^3}{126 c^4}+\frac{11 b^2 e^2 x}{42 c^6}-\frac{11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*d^2*x)/(3*c^2) - (3*b^2*d*e*x)/(5*c^4) + (11*b^2*e^2*x)/(42*c^6) + (b^2*d*e*x^3)/(15*c^2) - (5*b^2*e^2*x^
3)/(126*c^4) + (b^2*e^2*x^5)/(105*c^2) - (b^2*d^2*ArcTan[c*x])/(3*c^3) + (3*b^2*d*e*ArcTan[c*x])/(5*c^5) - (11
*b^2*e^2*ArcTan[c*x])/(42*c^7) - (b*d^2*x^2*(a + b*ArcTan[c*x]))/(3*c) + (2*b*d*e*x^2*(a + b*ArcTan[c*x]))/(5*
c^3) - (b*e^2*x^2*(a + b*ArcTan[c*x]))/(7*c^5) - (b*d*e*x^4*(a + b*ArcTan[c*x]))/(5*c) + (b*e^2*x^4*(a + b*Arc
Tan[c*x]))/(14*c^3) - (b*e^2*x^6*(a + b*ArcTan[c*x]))/(21*c) - ((I/3)*d^2*(a + b*ArcTan[c*x])^2)/c^3 + (((2*I)
/5)*d*e*(a + b*ArcTan[c*x])^2)/c^5 - ((I/7)*e^2*(a + b*ArcTan[c*x])^2)/c^7 + (d^2*x^3*(a + b*ArcTan[c*x])^2)/3
+ (2*d*e*x^5*(a + b*ArcTan[c*x])^2)/5 + (e^2*x^7*(a + b*ArcTan[c*x])^2)/7 - (2*b*d^2*(a + b*ArcTan[c*x])*Log[
2/(1 + I*c*x)])/(3*c^3) + (4*b*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) - (2*b*e^2*(a + b*ArcTan[c*
x])*Log[2/(1 + I*c*x)])/(7*c^7) - ((I/3)*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 + (((2*I)/5)*b^2*d*e*PolyL
og[2, 1 - 2/(1 + I*c*x)])/c^5 - ((I/7)*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^7

Rule 4980

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With
[{u = ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b,
c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || 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 (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+2 d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2+e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+e^2 \int x^6 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} \left (2 b c d^2\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{5} (4 b c d e) \int \frac{x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{7} \left (2 b c e^2\right ) \int \frac{x^7 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{\left (2 b d^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac{\left (2 b d^2\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}-\frac{(4 b d e) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c}+\frac{(4 b d e) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c}-\frac{\left (2 b e^2\right ) \int x^5 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{7 c}+\frac{\left (2 b e^2\right ) \int \frac{x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{7 c}\\ &=-\frac{b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}-\frac{b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac{i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} \left (b^2 d^2\right ) \int \frac{x^2}{1+c^2 x^2} \, dx-\frac{\left (2 b d^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}+\frac{1}{5} \left (b^2 d e\right ) \int \frac{x^4}{1+c^2 x^2} \, dx+\frac{(4 b d e) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c^3}-\frac{(4 b d e) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c^3}+\frac{1}{21} \left (b^2 e^2\right ) \int \frac{x^6}{1+c^2 x^2} \, dx+\frac{\left (2 b e^2\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{7 c^3}-\frac{\left (2 b e^2\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{7 c^3}\\ &=\frac{b^2 d^2 x}{3 c^2}-\frac{b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac{b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac{b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac{i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}+\frac{1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}-\frac{\left (b^2 d^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{3 c^2}+\frac{\left (2 b^2 d^2\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}+\frac{1}{5} \left (b^2 d e\right ) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx+\frac{(4 b d e) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{5 c^4}-\frac{\left (2 b^2 d e\right ) \int \frac{x^2}{1+c^2 x^2} \, dx}{5 c^2}+\frac{1}{21} \left (b^2 e^2\right ) \int \left (\frac{1}{c^6}-\frac{x^2}{c^4}+\frac{x^4}{c^2}-\frac{1}{c^6 \left (1+c^2 x^2\right )}\right ) \, dx-\frac{\left (2 b e^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{7 c^5}+\frac{\left (2 b e^2\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{7 c^5}-\frac{\left (b^2 e^2\right ) \int \frac{x^4}{1+c^2 x^2} \, dx}{14 c^2}\\ &=\frac{b^2 d^2 x}{3 c^2}-\frac{3 b^2 d e x}{5 c^4}+\frac{b^2 e^2 x}{21 c^6}+\frac{b^2 d e x^3}{15 c^2}-\frac{b^2 e^2 x^3}{63 c^4}+\frac{b^2 e^2 x^5}{105 c^2}-\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c^3}-\frac{b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac{b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac{b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac{b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac{i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}-\frac{i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}+\frac{1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}+\frac{4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{5 c^5}-\frac{\left (2 i b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{3 c^3}+\frac{\left (b^2 d e\right ) \int \frac{1}{1+c^2 x^2} \, dx}{5 c^4}+\frac{\left (2 b^2 d e\right ) \int \frac{1}{1+c^2 x^2} \, dx}{5 c^4}-\frac{\left (4 b^2 d e\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^4}-\frac{\left (2 b e^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{7 c^6}-\frac{\left (b^2 e^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{21 c^6}+\frac{\left (b^2 e^2\right ) \int \frac{x^2}{1+c^2 x^2} \, dx}{7 c^4}-\frac{\left (b^2 e^2\right ) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{14 c^2}\\ &=\frac{b^2 d^2 x}{3 c^2}-\frac{3 b^2 d e x}{5 c^4}+\frac{11 b^2 e^2 x}{42 c^6}+\frac{b^2 d e x^3}{15 c^2}-\frac{5 b^2 e^2 x^3}{126 c^4}+\frac{b^2 e^2 x^5}{105 c^2}-\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac{3 b^2 d e \tan ^{-1}(c x)}{5 c^5}-\frac{b^2 e^2 \tan ^{-1}(c x)}{21 c^7}-\frac{b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac{b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac{b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac{b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac{i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}-\frac{i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}+\frac{1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}+\frac{4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{5 c^5}-\frac{2 b e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{7 c^7}-\frac{i b^2 d^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{3 c^3}+\frac{\left (4 i b^2 d e\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{5 c^5}-\frac{\left (b^2 e^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{14 c^6}-\frac{\left (b^2 e^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{7 c^6}+\frac{\left (2 b^2 e^2\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{7 c^6}\\ &=\frac{b^2 d^2 x}{3 c^2}-\frac{3 b^2 d e x}{5 c^4}+\frac{11 b^2 e^2 x}{42 c^6}+\frac{b^2 d e x^3}{15 c^2}-\frac{5 b^2 e^2 x^3}{126 c^4}+\frac{b^2 e^2 x^5}{105 c^2}-\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac{3 b^2 d e \tan ^{-1}(c x)}{5 c^5}-\frac{11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7}-\frac{b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac{b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac{b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac{b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac{i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}-\frac{i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}+\frac{1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}+\frac{4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{5 c^5}-\frac{2 b e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{7 c^7}-\frac{i b^2 d^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{3 c^3}+\frac{2 i b^2 d e \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{5 c^5}-\frac{\left (2 i b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{7 c^7}\\ &=\frac{b^2 d^2 x}{3 c^2}-\frac{3 b^2 d e x}{5 c^4}+\frac{11 b^2 e^2 x}{42 c^6}+\frac{b^2 d e x^3}{15 c^2}-\frac{5 b^2 e^2 x^3}{126 c^4}+\frac{b^2 e^2 x^5}{105 c^2}-\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac{3 b^2 d e \tan ^{-1}(c x)}{5 c^5}-\frac{11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7}-\frac{b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac{b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac{b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac{b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac{i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}-\frac{i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}+\frac{1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}+\frac{4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{5 c^5}-\frac{2 b e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{7 c^7}-\frac{i b^2 d^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{3 c^3}+\frac{2 i b^2 d e \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{5 c^5}-\frac{i b^2 e^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{7 c^7}\\ \end{align*}

Mathematica [A]  time = 1.57, size = 513, normalized size = 0.88 \[ \frac{6 i b^2 \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+210 a^2 c^7 d^2 x^3+252 a^2 c^7 d e x^5+90 a^2 c^7 e^2 x^7-3 b \tan ^{-1}(c x) \left (-4 a c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+b \left (c^2 x^2+1\right ) \left (2 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )-c^2 e \left (126 d+25 e x^2\right )+55 e^2\right )+4 b \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-210 a b c^6 d^2 x^2+210 a b c^4 d^2 \log \left (c^2 x^2+1\right )-126 a b c^6 d e x^4+252 a b c^4 d e x^2-252 a b c^2 d e \log \left (c^2 x^2+1\right )+378 a b c^2 d e-30 a b c^6 e^2 x^6+45 a b c^4 e^2 x^4-90 a b c^2 e^2 x^2+90 a b e^2 \log \left (c^2 x^2+1\right )-165 a b e^2+6 b^2 \tan ^{-1}(c x)^2 \left (c^7 \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right )+35 i c^4 d^2-42 i c^2 d e+15 i e^2\right )+210 b^2 c^5 d^2 x+42 b^2 c^5 d e x^3-378 b^2 c^3 d e x+6 b^2 c^5 e^2 x^5-25 b^2 c^3 e^2 x^3+165 b^2 c e^2 x}{630 c^7} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(378*a*b*c^2*d*e - 165*a*b*e^2 + 210*b^2*c^5*d^2*x - 378*b^2*c^3*d*e*x + 165*b^2*c*e^2*x - 210*a*b*c^6*d^2*x^2
+ 252*a*b*c^4*d*e*x^2 - 90*a*b*c^2*e^2*x^2 + 210*a^2*c^7*d^2*x^3 + 42*b^2*c^5*d*e*x^3 - 25*b^2*c^3*e^2*x^3 -
126*a*b*c^6*d*e*x^4 + 45*a*b*c^4*e^2*x^4 + 252*a^2*c^7*d*e*x^5 + 6*b^2*c^5*e^2*x^5 - 30*a*b*c^6*e^2*x^6 + 90*a
^2*c^7*e^2*x^7 + 6*b^2*((35*I)*c^4*d^2 - (42*I)*c^2*d*e + (15*I)*e^2 + c^7*(35*d^2*x^3 + 42*d*e*x^5 + 15*e^2*x
^7))*ArcTan[c*x]^2 - 3*b*ArcTan[c*x]*(-4*a*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) + b*(1 + c^2*x^2)*(55*e^
2 - c^2*e*(126*d + 25*e*x^2) + 2*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4)) + 4*b*(35*c^4*d^2 - 42*c^2*d*e + 15*e^
2)*Log[1 + E^((2*I)*ArcTan[c*x])]) + 210*a*b*c^4*d^2*Log[1 + c^2*x^2] - 252*a*b*c^2*d*e*Log[1 + c^2*x^2] + 90*
a*b*e^2*Log[1 + c^2*x^2] + (6*I)*b^2*(35*c^4*d^2 - 42*c^2*d*e + 15*e^2)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(6
30*c^7)

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Maple [B]  time = 0.139, size = 1158, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/c*b^2*arctan(c*x)*x^2*d^2+1/3*a^2*d^2*x^3+1/7*a^2*e^2*x^7-1/14*I/c^7*b^2*ln(c*x-I)*ln(-1/2*I*(c*x+I))*e^2
+1/14*I/c^7*b^2*ln(c*x+I)*ln(1/2*I*(c*x-I))*e^2+1/14*I/c^7*b^2*ln(c*x-I)*ln(c^2*x^2+1)*e^2-1/10*I/c^5*b^2*ln(c
*x+I)^2*d*e-2/5/c^5*b^2*arctan(c*x)*ln(c^2*x^2+1)*d*e-1/5/c*b^2*arctan(c*x)*x^4*d*e-1/5/c*a*b*x^4*d*e+2/5/c^3*
a*b*x^2*d*e+11/42*b^2*e^2*x/c^6-5/126*b^2*e^2*x^3/c^4+1/105*b^2*e^2*x^5/c^2-11/42*b^2*e^2*arctan(c*x)/c^7+1/6*
I/c^3*b^2*ln(c*x+I)*ln(1/2*I*(c*x-I))*d^2-1/6*I/c^3*b^2*ln(c*x-I)*ln(-1/2*I*(c*x+I))*d^2+1/6*I/c^3*b^2*ln(c*x-
I)*ln(c^2*x^2+1)*d^2-1/5*I/c^5*b^2*dilog(1/2*I*(c*x-I))*d*e+1/10*I/c^5*b^2*ln(c*x-I)^2*d*e-1/14*I/c^7*b^2*ln(c
*x+I)*ln(c^2*x^2+1)*e^2+1/5*I/c^5*b^2*dilog(-1/2*I*(c*x+I))*d*e+2/5/c^3*b^2*arctan(c*x)*x^2*d*e+1/3*b^2*arctan
(c*x)^2*d^2*x^3+1/7*b^2*arctan(c*x)^2*e^2*x^7+2/5*a^2*e*d*x^5-1/6*I/c^3*b^2*dilog(-1/2*I*(c*x+I))*d^2+1/12*I/c
^3*b^2*ln(c*x+I)^2*d^2+1/14*I/c^7*b^2*dilog(1/2*I*(c*x-I))*e^2+1/6*I/c^3*b^2*dilog(1/2*I*(c*x-I))*d^2-1/14*I/c
^7*b^2*dilog(-1/2*I*(c*x+I))*e^2+1/28*I/c^7*b^2*ln(c*x+I)^2*e^2-1/28*I/c^7*b^2*ln(c*x-I)^2*e^2+1/3/c^3*b^2*arc
tan(c*x)*ln(c^2*x^2+1)*d^2-1/7/c^5*b^2*arctan(c*x)*x^2*e^2-1/21/c*b^2*arctan(c*x)*e^2*x^6+1/3/c^3*a*b*ln(c^2*x
^2+1)*d^2-1/12*I/c^3*b^2*ln(c*x-I)^2*d^2+1/14/c^3*b^2*arctan(c*x)*x^4*e^2+2/3*a*b*arctan(c*x)*d^2*x^3+2/7*a*b*
arctan(c*x)*e^2*x^7+2/5*b^2*arctan(c*x)^2*e*d*x^5+1/7/c^7*a*b*ln(c^2*x^2+1)*e^2+1/5*I/c^5*b^2*ln(c*x+I)*ln(c^2
*x^2+1)*d*e+1/5*I/c^5*b^2*ln(c*x-I)*ln(-1/2*I*(c*x+I))*d*e-1/5*I/c^5*b^2*ln(c*x-I)*ln(c^2*x^2+1)*d*e-1/5*I/c^5
*b^2*ln(c*x+I)*ln(1/2*I*(c*x-I))*d*e-3/5*b^2*d*e*x/c^4+1/15*b^2*d*e*x^3/c^2+3/5*b^2*d*e*arctan(c*x)/c^5+1/7/c^
7*b^2*arctan(c*x)*ln(c^2*x^2+1)*e^2-1/3/c*a*b*x^2*d^2-1/21/c*a*b*e^2*x^6+1/14/c^3*a*b*x^4*e^2-1/7/c^5*a*b*x^2*
e^2-2/5/c^5*a*b*ln(c^2*x^2+1)*d*e+4/5*a*b*arctan(c*x)*e*d*x^5-1/6*I/c^3*b^2*ln(c*x+I)*ln(c^2*x^2+1)*d^2+1/3*b^
2*d^2*x/c^2-1/3*b^2*d^2*arctan(c*x)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{7} \, a^{2} e^{2} x^{7} + \frac{2}{5} \, a^{2} d e x^{5} + \frac{1}{3} \, a^{2} d^{2} x^{3} + \frac{1}{3} \,{\left (2 \, x^{3} \arctan \left (c x\right ) - c{\left (\frac{x^{2}}{c^{2}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} a b d^{2} + \frac{1}{5} \,{\left (4 \, x^{5} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} a b d e + \frac{1}{42} \,{\left (12 \, x^{7} \arctan \left (c x\right ) - c{\left (\frac{2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac{6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} a b e^{2} + \frac{1}{420} \,{\left (15 \, b^{2} e^{2} x^{7} + 42 \, b^{2} d e x^{5} + 35 \, b^{2} d^{2} x^{3}\right )} \arctan \left (c x\right )^{2} - \frac{1}{1680} \,{\left (15 \, b^{2} e^{2} x^{7} + 42 \, b^{2} d e x^{5} + 35 \, b^{2} d^{2} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + \int \frac{1260 \,{\left (b^{2} c^{2} e^{2} x^{8} +{\left (2 \, b^{2} c^{2} d e + b^{2} e^{2}\right )} x^{6} + b^{2} d^{2} x^{2} +{\left (b^{2} c^{2} d^{2} + 2 \, b^{2} d e\right )} x^{4}\right )} \arctan \left (c x\right )^{2} + 105 \,{\left (b^{2} c^{2} e^{2} x^{8} +{\left (2 \, b^{2} c^{2} d e + b^{2} e^{2}\right )} x^{6} + b^{2} d^{2} x^{2} +{\left (b^{2} c^{2} d^{2} + 2 \, b^{2} d e\right )} x^{4}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 8 \,{\left (15 \, b^{2} c e^{2} x^{7} + 42 \, b^{2} c d e x^{5} + 35 \, b^{2} c d^{2} x^{3}\right )} \arctan \left (c x\right ) + 4 \,{\left (15 \, b^{2} c^{2} e^{2} x^{8} + 42 \, b^{2} c^{2} d e x^{6} + 35 \, b^{2} c^{2} d^{2} x^{4}\right )} \log \left (c^{2} x^{2} + 1\right )}{1680 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^2*e^2*x^7 + 2/5*a^2*d*e*x^5 + 1/3*a^2*d^2*x^3 + 1/3*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c
^4))*a*b*d^2 + 1/5*(4*x^5*arctan(c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*log(c^2*x^2 + 1)/c^6))*a*b*d*e + 1/42*(12
*x^7*arctan(c*x) - c*((2*c^4*x^6 - 3*c^2*x^4 + 6*x^2)/c^6 - 6*log(c^2*x^2 + 1)/c^8))*a*b*e^2 + 1/420*(15*b^2*e
^2*x^7 + 42*b^2*d*e*x^5 + 35*b^2*d^2*x^3)*arctan(c*x)^2 - 1/1680*(15*b^2*e^2*x^7 + 42*b^2*d*e*x^5 + 35*b^2*d^2
*x^3)*log(c^2*x^2 + 1)^2 + integrate(1/1680*(1260*(b^2*c^2*e^2*x^8 + (2*b^2*c^2*d*e + b^2*e^2)*x^6 + b^2*d^2*x
^2 + (b^2*c^2*d^2 + 2*b^2*d*e)*x^4)*arctan(c*x)^2 + 105*(b^2*c^2*e^2*x^8 + (2*b^2*c^2*d*e + b^2*e^2)*x^6 + b^2
*d^2*x^2 + (b^2*c^2*d^2 + 2*b^2*d*e)*x^4)*log(c^2*x^2 + 1)^2 - 8*(15*b^2*c*e^2*x^7 + 42*b^2*c*d*e*x^5 + 35*b^2
*c*d^2*x^3)*arctan(c*x) + 4*(15*b^2*c^2*e^2*x^8 + 42*b^2*c^2*d*e*x^6 + 35*b^2*c^2*d^2*x^4)*log(c^2*x^2 + 1))/(
c^2*x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(e*x**2+d)**2*(a+b*atan(c*x))**2,x)

[Out]

Integral(x**2*(a + b*atan(c*x))**2*(d + e*x**2)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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