\(\int x^2 (d+e x^2) (a+b \arctan (c x))^2 \, dx\) [1248]

Optimal result

Integrand size = 21, antiderivative size = 323 \[ \int x^2 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {b^2 d x}{3 c^2}-\frac {3 b^2 e x}{10 c^4}+\frac {b^2 e x^3}{30 c^2}-\frac {b^2 d \arctan (c x)}{3 c^3}+\frac {3 b^2 e \arctan (c x)}{10 c^5}-\frac {b d x^2 (a+b \arctan (c x))}{3 c}+\frac {b e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e x^4 (a+b \arctan (c x))}{10 c}-\frac {i d (a+b \arctan (c x))^2}{3 c^3}+\frac {i e (a+b \arctan (c x))^2}{5 c^5}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{5} e x^5 (a+b \arctan (c x))^2-\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 b e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^5} \]

[Out]

1/3*b^2*d*x/c^2-3/10*b^2*e*x/c^4+1/30*b^2*e*x^3/c^2-1/3*b^2*d*arctan(c*x)/c^3+3/10*b^2*e*arctan(c*x)/c^5-1/3*b
*d*x^2*(a+b*arctan(c*x))/c+1/5*b*e*x^2*(a+b*arctan(c*x))/c^3-1/10*b*e*x^4*(a+b*arctan(c*x))/c-1/3*I*b^2*d*poly
log(2,1-2/(1+I*c*x))/c^3+1/5*I*e*(a+b*arctan(c*x))^2/c^5+1/3*d*x^3*(a+b*arctan(c*x))^2+1/5*e*x^5*(a+b*arctan(c
*x))^2-2/3*b*d*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^3+2/5*b*e*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^5-1/3*I*d*(a+
b*arctan(c*x))^2/c^3+1/5*I*b^2*e*polylog(2,1-2/(1+I*c*x))/c^5

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5100, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352, 308} \[ \int x^2 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {i e (a+b \arctan (c x))^2}{5 c^5}+\frac {2 b e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{5 c^5}-\frac {i d (a+b \arctan (c x))^2}{3 c^3}-\frac {2 b d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^3}+\frac {b e x^2 (a+b \arctan (c x))}{5 c^3}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2-\frac {b d x^2 (a+b \arctan (c x))}{3 c}+\frac {1}{5} e x^5 (a+b \arctan (c x))^2-\frac {b e x^4 (a+b \arctan (c x))}{10 c}+\frac {3 b^2 e \arctan (c x)}{10 c^5}-\frac {b^2 d \arctan (c x)}{3 c^3}+\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{5 c^5}-\frac {3 b^2 e x}{10 c^4}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {b^2 d x}{3 c^2}+\frac {b^2 e x^3}{30 c^2} \]

[In]

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

[Out]

(b^2*d*x)/(3*c^2) - (3*b^2*e*x)/(10*c^4) + (b^2*e*x^3)/(30*c^2) - (b^2*d*ArcTan[c*x])/(3*c^3) + (3*b^2*e*ArcTa
n[c*x])/(10*c^5) - (b*d*x^2*(a + b*ArcTan[c*x]))/(3*c) + (b*e*x^2*(a + b*ArcTan[c*x]))/(5*c^3) - (b*e*x^4*(a +
b*ArcTan[c*x]))/(10*c) - ((I/3)*d*(a + b*ArcTan[c*x])^2)/c^3 + ((I/5)*e*(a + b*ArcTan[c*x])^2)/c^5 + (d*x^3*(
a + b*ArcTan[c*x])^2)/3 + (e*x^5*(a + b*ArcTan[c*x])^2)/5 - (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*
c^3) + (2*b*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) - ((I/3)*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^
3 + ((I/5)*b^2*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^5

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*} \text {integral}& = \int \left (d x^2 (a+b \arctan (c x))^2+e x^4 (a+b \arctan (c x))^2\right ) \, dx \\ & = d \int x^2 (a+b \arctan (c x))^2 \, dx+e \int x^4 (a+b \arctan (c x))^2 \, dx \\ & = \frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{5} e x^5 (a+b \arctan (c x))^2-\frac {1}{3} (2 b c d) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{5} (2 b c e) \int \frac {x^5 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{5} e x^5 (a+b \arctan (c x))^2-\frac {(2 b d) \int x (a+b \arctan (c x)) \, dx}{3 c}+\frac {(2 b d) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c}-\frac {(2 b e) \int x^3 (a+b \arctan (c x)) \, dx}{5 c}+\frac {(2 b e) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c} \\ & = -\frac {b d x^2 (a+b \arctan (c x))}{3 c}-\frac {b e x^4 (a+b \arctan (c x))}{10 c}-\frac {i d (a+b \arctan (c x))^2}{3 c^3}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{5} e x^5 (a+b \arctan (c x))^2+\frac {1}{3} \left (b^2 d\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {(2 b d) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c^2}+\frac {1}{10} \left (b^2 e\right ) \int \frac {x^4}{1+c^2 x^2} \, dx+\frac {(2 b e) \int x (a+b \arctan (c x)) \, dx}{5 c^3}-\frac {(2 b e) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c^3} \\ & = \frac {b^2 d x}{3 c^2}-\frac {b d x^2 (a+b \arctan (c x))}{3 c}+\frac {b e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e x^4 (a+b \arctan (c x))}{10 c}-\frac {i d (a+b \arctan (c x))^2}{3 c^3}+\frac {i e (a+b \arctan (c x))^2}{5 c^5}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{5} e x^5 (a+b \arctan (c x))^2-\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}-\frac {\left (b^2 d\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}+\frac {1}{10} \left (b^2 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 {(2 b e) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{5 c^4}-\frac {\left (b^2 e\right ) \int \frac {x^2}{1+c^2 x^2} \, dx}{5 c^2} \\ & = \frac {b^2 d x}{3 c^2}-\frac {3 b^2 e x}{10 c^4}+\frac {b^2 e x^3}{30 c^2}-\frac {b^2 d \arctan (c x)}{3 c^3}-\frac {b d x^2 (a+b \arctan (c x))}{3 c}+\frac {b e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e x^4 (a+b \arctan (c x))}{10 c}-\frac {i d (a+b \arctan (c x))^2}{3 c^3}+\frac {i e (a+b \arctan (c x))^2}{5 c^5}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{5} e x^5 (a+b \arctan (c x))^2-\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 b e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {\left (2 i b^2 d\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 e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{10 c^4}+\frac {\left (b^2 e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^4}-\frac {\left (2 b^2 e\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^4} \\ & = \frac {b^2 d x}{3 c^2}-\frac {3 b^2 e x}{10 c^4}+\frac {b^2 e x^3}{30 c^2}-\frac {b^2 d \arctan (c x)}{3 c^3}+\frac {3 b^2 e \arctan (c x)}{10 c^5}-\frac {b d x^2 (a+b \arctan (c x))}{3 c}+\frac {b e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e x^4 (a+b \arctan (c x))}{10 c}-\frac {i d (a+b \arctan (c x))^2}{3 c^3}+\frac {i e (a+b \arctan (c x))^2}{5 c^5}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{5} e x^5 (a+b \arctan (c x))^2-\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 b e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {\left (2 i b^2 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 {b^2 d x}{3 c^2}-\frac {3 b^2 e x}{10 c^4}+\frac {b^2 e x^3}{30 c^2}-\frac {b^2 d \arctan (c x)}{3 c^3}+\frac {3 b^2 e \arctan (c x)}{10 c^5}-\frac {b d x^2 (a+b \arctan (c x))}{3 c}+\frac {b e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e x^4 (a+b \arctan (c x))}{10 c}-\frac {i d (a+b \arctan (c x))^2}{3 c^3}+\frac {i e (a+b \arctan (c x))^2}{5 c^5}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{5} e x^5 (a+b \arctan (c x))^2-\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 b e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.89 \[ \int x^2 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {9 a b e+10 b^2 c^3 d x-9 b^2 c e x-10 a b c^4 d x^2+6 a b c^2 e x^2+10 a^2 c^5 d x^3+b^2 c^3 e x^3-3 a b c^4 e x^4+6 a^2 c^5 e x^5+2 b^2 \left (5 i c^2 d-3 i e+c^5 \left (5 d x^3+3 e x^5\right )\right ) \arctan (c x)^2-b \arctan (c x) \left (-4 a c^5 x^3 \left (5 d+3 e x^2\right )+b \left (1+c^2 x^2\right ) \left (-9 e+c^2 \left (10 d+3 e x^2\right )\right )+4 b \left (5 c^2 d-3 e\right ) \log \left (1+e^{2 i \arctan (c x)}\right )\right )+10 a b c^2 d \log \left (1+c^2 x^2\right )-6 a b e \log \left (1+c^2 x^2\right )+2 i b^2 \left (5 c^2 d-3 e\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{30 c^5} \]

[In]

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

[Out]

(9*a*b*e + 10*b^2*c^3*d*x - 9*b^2*c*e*x - 10*a*b*c^4*d*x^2 + 6*a*b*c^2*e*x^2 + 10*a^2*c^5*d*x^3 + b^2*c^3*e*x^
3 - 3*a*b*c^4*e*x^4 + 6*a^2*c^5*e*x^5 + 2*b^2*((5*I)*c^2*d - (3*I)*e + c^5*(5*d*x^3 + 3*e*x^5))*ArcTan[c*x]^2
- b*ArcTan[c*x]*(-4*a*c^5*x^3*(5*d + 3*e*x^2) + b*(1 + c^2*x^2)*(-9*e + c^2*(10*d + 3*e*x^2)) + 4*b*(5*c^2*d -
3*e)*Log[1 + E^((2*I)*ArcTan[c*x])]) + 10*a*b*c^2*d*Log[1 + c^2*x^2] - 6*a*b*e*Log[1 + c^2*x^2] + (2*I)*b^2*(
5*c^2*d - 3*e)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(30*c^5)

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.26

method	result	size
parts	\(a^{2} \left (\frac {1}{5} e \,x^{5}+\frac {1}{3} d \,x^{3}\right )+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c^{3} e \,x^{5}}{5}+\frac {\arctan \left (c x \right )^{2} d \,c^{3} x^{3}}{3}-\frac {2 \left (\frac {5 \arctan \left (c x \right ) c^{4} d \,x^{2}}{2}+\frac {3 \arctan \left (c x \right ) c^{4} e \,x^{4}}{4}-\frac {3 \arctan \left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {5 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e}{2}-\frac {e \,c^{3} x^{3}}{4}-\frac {5 c^{3} x d}{2}+\frac {9 e c x}{4}-\frac {\left (-10 c^{2} d +9 e \right ) \arctan \left (c x \right )}{4}-\frac {\left (-10 c^{2} d +6 e \right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{4}\right )}{15 c^{2}}\right )}{c^{3}}+\frac {2 a b \left (\frac {c^{3} \arctan \left (c x \right ) e \,x^{5}}{5}+\frac {\arctan \left (c x \right ) d \,c^{3} x^{3}}{3}-\frac {\frac {5 d \,c^{4} x^{2}}{2}+\frac {3 e \,c^{4} x^{4}}{4}-\frac {3 e \,c^{2} x^{2}}{2}+\frac {\left (-5 c^{2} d +3 e \right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{15 c^{2}}\right )}{c^{3}}\)	\(406\)
derivativedivides	\(\frac {\frac {a^{2} \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} d \,c^{5} x^{3}}{3}+\frac {\arctan \left (c x \right )^{2} e \,c^{5} x^{5}}{5}-\frac {\arctan \left (c x \right ) c^{4} d \,x^{2}}{3}-\frac {\arctan \left (c x \right ) c^{4} e \,x^{4}}{10}+\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{5}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d}{3}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e}{5}+\frac {c^{3} x d}{3}+\frac {e \,c^{3} x^{3}}{30}-\frac {3 e c x}{10}+\frac {\left (-10 c^{2} d +9 e \right ) \arctan \left (c x \right )}{30}+\frac {\left (-10 c^{2} d +6 e \right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{30}\right )}{c^{2}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\arctan \left (c x \right ) e \,c^{5} x^{5}}{5}-\frac {d \,c^{4} x^{2}}{6}-\frac {e \,c^{4} x^{4}}{20}+\frac {e \,c^{2} x^{2}}{10}-\frac {\left (-5 c^{2} d +3 e \right ) \ln \left (c^{2} x^{2}+1\right )}{30}\right )}{c^{2}}}{c^{3}}\)	\(407\)
default	\(\frac {\frac {a^{2} \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} d \,c^{5} x^{3}}{3}+\frac {\arctan \left (c x \right )^{2} e \,c^{5} x^{5}}{5}-\frac {\arctan \left (c x \right ) c^{4} d \,x^{2}}{3}-\frac {\arctan \left (c x \right ) c^{4} e \,x^{4}}{10}+\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{5}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d}{3}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e}{5}+\frac {c^{3} x d}{3}+\frac {e \,c^{3} x^{3}}{30}-\frac {3 e c x}{10}+\frac {\left (-10 c^{2} d +9 e \right ) \arctan \left (c x \right )}{30}+\frac {\left (-10 c^{2} d +6 e \right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{30}\right )}{c^{2}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\arctan \left (c x \right ) e \,c^{5} x^{5}}{5}-\frac {d \,c^{4} x^{2}}{6}-\frac {e \,c^{4} x^{4}}{20}+\frac {e \,c^{2} x^{2}}{10}-\frac {\left (-5 c^{2} d +3 e \right ) \ln \left (c^{2} x^{2}+1\right )}{30}\right )}{c^{2}}}{c^{3}}\)	\(407\)
risch	\(\frac {i b^{2} e \ln \left (i c x +1\right ) x^{4}}{20 c}-\frac {i e \,b^{2} \ln \left (-i c x +1\right ) x^{4}}{20 c}+\frac {i e \,b^{2} \ln \left (-i c x +1\right ) x^{2}}{10 c^{3}}-\frac {i d \,b^{2} \ln \left (-i c x +1\right ) x^{2}}{6 c}+\frac {i b^{2} e \ln \left (i c x +1\right ) \ln \left (-i c x +1\right )}{10 c^{5}}-\frac {i b^{2} e \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{5 c^{5}}+\frac {i b^{2} e \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{5}}-\frac {b^{2} d \ln \left (i c x +1\right )^{2} x^{3}}{12}-\frac {b^{2} e \ln \left (i c x +1\right )^{2} x^{5}}{20}-\frac {e \,b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}-\frac {d \,b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}-\frac {i d \,a^{2}}{3 c^{3}}+\frac {i e \,a^{2}}{5 c^{5}}+\frac {413 i e \,b^{2}}{2250 c^{5}}-\frac {17 i b^{2} d}{54 c^{3}}+\frac {i e b a \ln \left (-i c x +1\right ) x^{5}}{5}+\frac {d \,a^{2} x^{3}}{3}+\frac {e \,a^{2} x^{5}}{5}+\frac {i a b d \ln \left (-i c x +1\right ) x^{3}}{3}-\frac {i b e a \ln \left (i c x +1\right ) x^{5}}{5}-\frac {i b a d \ln \left (i c x +1\right ) x^{3}}{3}-\frac {i b^{2} d \ln \left (i c x +1\right ) \ln \left (-i c x +1\right )}{6 c^{3}}+\frac {i b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{3 c^{3}}-\frac {i b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}-\frac {i b^{2} e \ln \left (i c x +1\right ) x^{2}}{10 c^{3}}+\frac {i b^{2} d \ln \left (i c x +1\right ) x^{2}}{6 c}-\frac {3 b^{2} e x}{10 c^{4}}+\frac {b^{2} e \,x^{3}}{30 c^{2}}+\frac {3 b^{2} e \arctan \left (c x \right )}{20 c^{5}}+\frac {b^{2} d x}{3 c^{2}}-\frac {b^{2} d \arctan \left (c x \right )}{6 c^{3}}-\frac {i e \,b^{2} \ln \left (-i c x +1\right )^{2}}{20 c^{5}}+\frac {i d \,b^{2} \ln \left (-i c x +1\right )^{2}}{12 c^{3}}-\frac {i b^{2} d \ln \left (i c x +1\right )^{2}}{12 c^{3}}+\frac {11 i b^{2} d \ln \left (i c x +1\right )}{36 c^{3}}+\frac {i b^{2} e \ln \left (i c x +1\right )^{2}}{20 c^{5}}-\frac {137 i b^{2} e \ln \left (i c x +1\right )}{600 c^{5}}+\frac {b^{2} d \ln \left (i c x +1\right ) \ln \left (-i c x +1\right ) x^{3}}{6}+\frac {b^{2} e \ln \left (i c x +1\right ) \ln \left (-i c x +1\right ) x^{5}}{10}-\frac {b e a \ln \left (c^{2} x^{2}+1\right )}{5 c^{5}}+\frac {b a d \ln \left (c^{2} x^{2}+1\right )}{3 c^{3}}+\frac {i b^{2} e \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{5}}-\frac {47 i b^{2} e \ln \left (-i c x +1\right )}{600 c^{5}}+\frac {5 i b^{2} d \ln \left (-i c x +1\right )}{36 c^{3}}-\frac {i b^{2} d \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}+\frac {23 i e \,b^{2} \ln \left (c^{2} x^{2}+1\right )}{150 c^{5}}-\frac {2 i b^{2} d \ln \left (c^{2} x^{2}+1\right )}{9 c^{3}}-\frac {11 a b d}{9 c^{3}}+\frac {137 e b a}{150 c^{5}}+\frac {e b a \,x^{2}}{5 c^{3}}-\frac {e b a \,x^{4}}{10 c}-\frac {a b d \,x^{2}}{3 c}\)	\(907\)

[In]

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

[Out]

a^2*(1/5*e*x^5+1/3*d*x^3)+b^2/c^3*(1/5*arctan(c*x)^2*c^3*e*x^5+1/3*arctan(c*x)^2*d*c^3*x^3-2/15/c^2*(5/2*arcta
n(c*x)*c^4*d*x^2+3/4*arctan(c*x)*c^4*e*x^4-3/2*arctan(c*x)*e*c^2*x^2-5/2*arctan(c*x)*ln(c^2*x^2+1)*c^2*d+3/2*a
rctan(c*x)*ln(c^2*x^2+1)*e-1/4*e*c^3*x^3-5/2*c^3*x*d+9/4*e*c*x-1/4*(-10*c^2*d+9*e)*arctan(c*x)-1/4*(-10*c^2*d+
6*e)*(-1/2*I*(ln(c*x-I)*ln(c^2*x^2+1)-1/2*ln(c*x-I)^2-dilog(-1/2*I*(I+c*x))-ln(c*x-I)*ln(-1/2*I*(I+c*x)))+1/2*
I*(ln(I+c*x)*ln(c^2*x^2+1)-1/2*ln(I+c*x)^2-dilog(1/2*I*(c*x-I))-ln(I+c*x)*ln(1/2*I*(c*x-I))))))+2*a*b/c^3*(1/5
*c^3*arctan(c*x)*e*x^5+1/3*arctan(c*x)*d*c^3*x^3-1/15/c^2*(5/2*d*c^4*x^2+3/4*e*c^4*x^4-3/2*e*c^2*x^2+1/2*(-5*c
^2*d+3*e)*ln(c^2*x^2+1)))

Fricas [F]

\[ \int x^2 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int x^2 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^2 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]

[In]

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

[Out]

1/5*a^2*e*x^5 + 1/3*a^2*d*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 + 1/10*(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*e + 1/60*(3*b^2*e*x^5 + 5*b^2*d*x^3)*
arctan(c*x)^2 - 1/240*(3*b^2*e*x^5 + 5*b^2*d*x^3)*log(c^2*x^2 + 1)^2 + integrate(1/240*(180*(b^2*c^2*e*x^6 + b
^2*d*x^2 + (b^2*c^2*d + b^2*e)*x^4)*arctan(c*x)^2 + 15*(b^2*c^2*e*x^6 + b^2*d*x^2 + (b^2*c^2*d + b^2*e)*x^4)*l
og(c^2*x^2 + 1)^2 - 8*(3*b^2*c*e*x^5 + 5*b^2*c*d*x^3)*arctan(c*x) + 4*(3*b^2*c^2*e*x^6 + 5*b^2*c^2*d*x^4)*log(
c^2*x^2 + 1))/(c^2*x^2 + 1), x)

Giac [F]

\[ \int x^2 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \]

[In]

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

[Out]

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