3.13.55 \(\int x^2 (d+e x^2)^2 (a+b \text {ArcTan}(c x))^2 \, dx\) [1255]

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

[Out]

-2/3*b*d^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^3+2/5*I*d*e*(a+b*arctan(c*x))^2/c^5-1/3*b*d^2*x^2*(a+b*arctan(c
*x))/c+1/3*b^2*d^2*x/c^2-1/3*b^2*d^2*arctan(c*x)/c^3+2/5*b*d*e*x^2*(a+b*arctan(c*x))/c^3-1/5*b*d*e*x^4*(a+b*ar
ctan(c*x))/c-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*b*e^2*x^2*(a+b*arctan(c*x)
)/c^5+1/14*b*e^2*x^4*(a+b*arctan(c*x))/c^3-1/21*b*e^2*x^6*(a+b*arctan(c*x))/c-1/3*I*b^2*d^2*polylog(2,1-2/(1+I
*c*x))/c^3-1/7*I*b^2*e^2*polylog(2,1-2/(1+I*c*x))/c^7+4/5*b*d*e*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^5+1/3*d^2*
x^3*(a+b*arctan(c*x))^2-2/7*b*e^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^7+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+2/5*d*e*x^5*(a+b*arctan(c*x))^2-1/3*I*d^2*(a+b*arctan(
c*x))^2/c^3-1/7*I*e^2*(a+b*arctan(c*x))^2/c^7+1/7*e^2*x^7*(a+b*arctan(c*x))^2+2/5*I*b^2*d*e*polylog(2,1-2/(1+I
*c*x))/c^5

________________________________________________________________________________________

Rubi [A]
time = 0.76, antiderivative size = 580, normalized size of antiderivative = 1.00, 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 = {5100, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352, 308} \begin {gather*} -\frac {i e^2 (a+b \text {ArcTan}(c x))^2}{7 c^7}-\frac {2 b e^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))}{7 c^7}+\frac {2 i d e (a+b \text {ArcTan}(c x))^2}{5 c^5}+\frac {4 b d e \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))}{5 c^5}-\frac {b e^2 x^2 (a+b \text {ArcTan}(c x))}{7 c^5}-\frac {i d^2 (a+b \text {ArcTan}(c x))^2}{3 c^3}-\frac {2 b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))}{3 c^3}+\frac {2 b d e x^2 (a+b \text {ArcTan}(c x))}{5 c^3}+\frac {b e^2 x^4 (a+b \text {ArcTan}(c x))}{14 c^3}+\frac {1}{3} d^2 x^3 (a+b \text {ArcTan}(c x))^2-\frac {b d^2 x^2 (a+b \text {ArcTan}(c x))}{3 c}+\frac {2}{5} d e x^5 (a+b \text {ArcTan}(c x))^2-\frac {b d e x^4 (a+b \text {ArcTan}(c x))}{5 c}+\frac {1}{7} e^2 x^7 (a+b \text {ArcTan}(c x))^2-\frac {b e^2 x^6 (a+b \text {ArcTan}(c x))}{21 c}-\frac {11 b^2 e^2 \text {ArcTan}(c x)}{42 c^7}+\frac {3 b^2 d e \text {ArcTan}(c x)}{5 c^5}-\frac {b^2 d^2 \text {ArcTan}(c x)}{3 c^3}-\frac {i b^2 e^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{7 c^7}+\frac {11 b^2 e^2 x}{42 c^6}+\frac {2 i b^2 d e \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{5 c^5}-\frac {3 b^2 d e x}{5 c^4}-\frac {5 b^2 e^2 x^3}{126 c^4}-\frac {i b^2 d^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {b^2 d^2 x}{3 c^2}+\frac {b^2 d e x^3}{15 c^2}+\frac {b^2 e^2 x^5}{105 c^2} \end {gather*}

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 209

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

Rule 308

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]

Rule 327

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 2352

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

Rule 2449

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 4946

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

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[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 5036

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 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[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 5100

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])

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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 1.18, size = 513, normalized size = 0.88 \begin {gather*} \frac {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 \left (35 i c^4 d^2-42 i c^2 d e+15 i e^2+c^7 \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right )\right ) \text {ArcTan}(c x)^2-3 b \text {ArcTan}(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 (1+c^2 x^2\right ) \left (55 e^2-c^2 e \left (126 d+25 e x^2\right )+2 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )+4 b \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )\right )+210 a b c^4 d^2 \log \left (1+c^2 x^2\right )-252 a b c^2 d e \log \left (1+c^2 x^2\right )+90 a b e^2 \log \left (1+c^2 x^2\right )+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 \text {ArcTan}(c x)}\right )}{630 c^7} \end {gather*}

Antiderivative was successfully verified.

[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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1136 vs. \(2 (514 ) = 1028\).
time = 0.98, size = 1137, normalized size = 1.96

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1137\)
default	\(\text {Expression too large to display}\)	\(1137\)
risch	\(\text {Expression too large to display}\)	\(1414\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*x^7*e^2 + 2/5*a^2*d*x^5*e + 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*x
^7*e^2 + 42*b^2*d*x^5*e + 35*b^2*d^2*x^3)*arctan(c*x)^2 - 1/1680*(15*b^2*x^7*e^2 + 42*b^2*d*x^5*e + 35*b^2*d^2
*x^3)*log(c^2*x^2 + 1)^2 + integrate(1/1680*(1260*(b^2*c^2*x^8*e^2 + (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*x^8*e^2 + (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*x^7*e^2 + 42*b^2*c*d*x^5*e + 35*b^2
*c*d^2*x^3)*arctan(c*x) + 4*(15*b^2*c^2*x^8*e^2 + 42*b^2*c^2*d*x^6*e + 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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{2}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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