Integrand size = 19, antiderivative size = 199 \[ \int x \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=-\frac {a b d x}{c}+\frac {a b e x}{2 c^3}+\frac {b^2 e x^2}{12 c^2}-\frac {b^2 d x \arctan (c x)}{c}+\frac {b^2 e x \arctan (c x)}{2 c^3}-\frac {b e x^3 (a+b \arctan (c x))}{6 c}+\frac {d (a+b \arctan (c x))^2}{2 c^2}-\frac {e (a+b \arctan (c x))^2}{4 c^4}+\frac {1}{2} d x^2 (a+b \arctan (c x))^2+\frac {1}{4} e x^4 (a+b \arctan (c x))^2+\frac {b^2 d \log \left (1+c^2 x^2\right )}{2 c^2}-\frac {b^2 e \log \left (1+c^2 x^2\right )}{3 c^4} \]
-a*b*d*x/c+1/2*a*b*e*x/c^3+1/12*b^2*e*x^2/c^2-b^2*d*x*arctan(c*x)/c+1/2*b^ 2*e*x*arctan(c*x)/c^3-1/6*b*e*x^3*(a+b*arctan(c*x))/c+1/2*d*(a+b*arctan(c* x))^2/c^2-1/4*e*(a+b*arctan(c*x))^2/c^4+1/2*d*x^2*(a+b*arctan(c*x))^2+1/4* e*x^4*(a+b*arctan(c*x))^2+1/2*b^2*d*ln(c^2*x^2+1)/c^2-1/3*b^2*e*ln(c^2*x^2 +1)/c^4
Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.90 \[ \int x \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {c x \left (6 a b e+b^2 c e x+3 a^2 c^3 x \left (2 d+e x^2\right )-2 a b c^2 \left (6 d+e x^2\right )\right )+2 b \left (6 a c^2 d-3 a e+3 b c e x-b c^3 x \left (6 d+e x^2\right )+3 a c^4 \left (2 d x^2+e x^4\right )\right ) \arctan (c x)+3 b^2 \left (2 c^2 d-e+c^4 \left (2 d x^2+e x^4\right )\right ) \arctan (c x)^2+2 b^2 \left (3 c^2 d-2 e\right ) \log \left (1+c^2 x^2\right )}{12 c^4} \]
(c*x*(6*a*b*e + b^2*c*e*x + 3*a^2*c^3*x*(2*d + e*x^2) - 2*a*b*c^2*(6*d + e *x^2)) + 2*b*(6*a*c^2*d - 3*a*e + 3*b*c*e*x - b*c^3*x*(6*d + e*x^2) + 3*a* c^4*(2*d*x^2 + e*x^4))*ArcTan[c*x] + 3*b^2*(2*c^2*d - e + c^4*(2*d*x^2 + e *x^4))*ArcTan[c*x]^2 + 2*b^2*(3*c^2*d - 2*e)*Log[1 + c^2*x^2])/(12*c^4)
Time = 0.58 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5515, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5515 |
| \(\displaystyle \int \left (d x (a+b \arctan (c x))^2+e x^3 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {e (a+b \arctan (c x))^2}{4 c^4}+\frac {d (a+b \arctan (c x))^2}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))^2+\frac {1}{4} e x^4 (a+b \arctan (c x))^2-\frac {b e x^3 (a+b \arctan (c x))}{6 c}+\frac {a b e x}{2 c^3}-\frac {a b d x}{c}+\frac {b^2 e x \arctan (c x)}{2 c^3}-\frac {b^2 d x \arctan (c x)}{c}+\frac {b^2 d \log \left (c^2 x^2+1\right )}{2 c^2}+\frac {b^2 e x^2}{12 c^2}-\frac {b^2 e \log \left (c^2 x^2+1\right )}{3 c^4}\) |
-((a*b*d*x)/c) + (a*b*e*x)/(2*c^3) + (b^2*e*x^2)/(12*c^2) - (b^2*d*x*ArcTa n[c*x])/c + (b^2*e*x*ArcTan[c*x])/(2*c^3) - (b*e*x^3*(a + b*ArcTan[c*x]))/ (6*c) + (d*(a + b*ArcTan[c*x])^2)/(2*c^2) - (e*(a + b*ArcTan[c*x])^2)/(4*c ^4) + (d*x^2*(a + b*ArcTan[c*x])^2)/2 + (e*x^4*(a + b*ArcTan[c*x])^2)/4 + (b^2*d*Log[1 + c^2*x^2])/(2*c^2) - (b^2*e*Log[1 + c^2*x^2])/(3*c^4)
3.13.49.3.1 Defintions of rubi rules used
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])
Time = 0.34 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.25
| method | result | size |
| parts | \(\frac {a^{2} \left (e \,x^{2}+d \right )^{2}}{4 e}+\frac {b^{2} \arctan \left (c x \right )^{2} e \,x^{4}}{4}+\frac {b^{2} \arctan \left (c x \right )^{2} x^{2} d}{2}-\frac {b^{2} e \arctan \left (c x \right ) x^{3}}{6 c}-\frac {b^{2} d x \arctan \left (c x \right )}{c}+\frac {b^{2} e x \arctan \left (c x \right )}{2 c^{3}}+\frac {b^{2} \arctan \left (c x \right )^{2} d}{2 c^{2}}-\frac {b^{2} e \arctan \left (c x \right )^{2}}{4 c^{4}}+\frac {b^{2} e \,x^{2}}{12 c^{2}}+\frac {b^{2} d \ln \left (c^{2} x^{2}+1\right )}{2 c^{2}}-\frac {b^{2} e \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}}+\frac {a b \arctan \left (c x \right ) e \,x^{4}}{2}+a b \arctan \left (c x \right ) x^{2} d -\frac {a b e \,x^{3}}{6 c}-\frac {a b d x}{c}+\frac {a b e x}{2 c^{3}}+\frac {a b d \arctan \left (c x \right )}{c^{2}}-\frac {a b e \arctan \left (c x \right )}{2 c^{4}}\) | \(248\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}{4 c^{2} e}+\frac {b^{2} \arctan \left (c x \right )^{2} d \,c^{2} x^{2}}{2}+\frac {b^{2} c^{2} \arctan \left (c x \right )^{2} e \,x^{4}}{4}-b^{2} \arctan \left (c x \right ) d c x -\frac {b^{2} c e \arctan \left (c x \right ) x^{3}}{6}+\frac {b^{2} e x \arctan \left (c x \right )}{2 c}+\frac {b^{2} \arctan \left (c x \right )^{2} d}{2}-\frac {b^{2} e \arctan \left (c x \right )^{2}}{4 c^{2}}+\frac {b^{2} e \,x^{2}}{12}+\frac {b^{2} \ln \left (c^{2} x^{2}+1\right ) d}{2}-\frac {b^{2} e \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}+a b \arctan \left (c x \right ) d \,c^{2} x^{2}+\frac {a b \,c^{2} \arctan \left (c x \right ) e \,x^{4}}{2}-a b d c x -\frac {a b c e \,x^{3}}{6}+\frac {a b e x}{2 c}+a b \arctan \left (c x \right ) d -\frac {a b e \arctan \left (c x \right )}{2 c^{2}}}{c^{2}}\) | \(254\) |
| default | \(\frac {\frac {a^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}{4 c^{2} e}+\frac {b^{2} \arctan \left (c x \right )^{2} d \,c^{2} x^{2}}{2}+\frac {b^{2} c^{2} \arctan \left (c x \right )^{2} e \,x^{4}}{4}-b^{2} \arctan \left (c x \right ) d c x -\frac {b^{2} c e \arctan \left (c x \right ) x^{3}}{6}+\frac {b^{2} e x \arctan \left (c x \right )}{2 c}+\frac {b^{2} \arctan \left (c x \right )^{2} d}{2}-\frac {b^{2} e \arctan \left (c x \right )^{2}}{4 c^{2}}+\frac {b^{2} e \,x^{2}}{12}+\frac {b^{2} \ln \left (c^{2} x^{2}+1\right ) d}{2}-\frac {b^{2} e \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}+a b \arctan \left (c x \right ) d \,c^{2} x^{2}+\frac {a b \,c^{2} \arctan \left (c x \right ) e \,x^{4}}{2}-a b d c x -\frac {a b c e \,x^{3}}{6}+\frac {a b e x}{2 c}+a b \arctan \left (c x \right ) d -\frac {a b e \arctan \left (c x \right )}{2 c^{2}}}{c^{2}}\) | \(254\) |
| parallelrisch | \(\frac {3 x^{4} \arctan \left (c x \right )^{2} b^{2} c^{4} e +6 x^{4} \arctan \left (c x \right ) a b \,c^{4} e +3 c^{4} a^{2} e \,x^{4}+6 x^{2} \arctan \left (c x \right )^{2} b^{2} c^{4} d -2 x^{3} \arctan \left (c x \right ) b^{2} c^{3} e +12 x^{2} \arctan \left (c x \right ) a b \,c^{4} d -2 a b \,c^{3} e \,x^{3}+6 c^{4} a^{2} d \,x^{2}-12 x \arctan \left (c x \right ) b^{2} c^{3} d +b^{2} c^{2} e \,x^{2}-12 a b \,c^{3} d x +6 \arctan \left (c x \right )^{2} b^{2} c^{2} d +6 b^{2} c^{2} d \ln \left (c^{2} x^{2}+1\right )+6 x \arctan \left (c x \right ) b^{2} c e +12 a b \,c^{2} d \arctan \left (c x \right )+6 a b c e x -3 \arctan \left (c x \right )^{2} b^{2} e -6 d \,a^{2} c^{2}-4 b^{2} e \ln \left (c^{2} x^{2}+1\right )-6 a b e \arctan \left (c x \right )-b^{2} e}{12 c^{4}}\) | \(275\) |
| risch | \(-\frac {b^{2} \left (e \,c^{4} x^{4}+2 d \,c^{4} x^{2}+2 c^{2} d -e \right ) \ln \left (i c x +1\right )^{2}}{16 c^{4}}-\frac {i b^{2} d x \ln \left (-i c x +1\right )}{2 c}-\frac {b^{2} e \,x^{4} \ln \left (-i c x +1\right )^{2}}{16}-\frac {i b \left (6 x^{4} a \,c^{4} e +3 i b \,c^{4} e \,x^{4} \ln \left (-i c x +1\right )+12 x^{2} a \,c^{4} d -2 b \,c^{3} e \,x^{3}+6 i b \,c^{4} d \,x^{2} \ln \left (-i c x +1\right )-12 b \,c^{3} d x +6 b c e x +6 i b \,c^{2} d \ln \left (-i c x +1\right )-3 i b e \ln \left (-i c x +1\right )\right ) \ln \left (i c x +1\right )}{24 c^{4}}-\frac {b^{2} d \,x^{2} \ln \left (-i c x +1\right )^{2}}{8}+\frac {i a b d \,x^{2} \ln \left (-i c x +1\right )}{2}+\frac {i a b e \,x^{4} \ln \left (-i c x +1\right )}{4}+\frac {x^{4} e \,a^{2}}{4}-\frac {i b^{2} e \,x^{3} \ln \left (-i c x +1\right )}{12 c}+\frac {x^{2} d \,a^{2}}{2}-\frac {a b e \,x^{3}}{6 c}-\frac {b^{2} d \ln \left (-i c x +1\right )^{2}}{8 c^{2}}+\frac {i b^{2} e x \ln \left (-i c x +1\right )}{4 c^{3}}-\frac {a b d x}{c}+\frac {b^{2} e \,x^{2}}{12 c^{2}}+\frac {a b d \arctan \left (c x \right )}{c^{2}}+\frac {b^{2} d \ln \left (c^{2} x^{2}+1\right )}{2 c^{2}}+\frac {b^{2} e \ln \left (-i c x +1\right )^{2}}{16 c^{4}}+\frac {a b e x}{2 c^{3}}-\frac {a b e \arctan \left (c x \right )}{2 c^{4}}-\frac {b^{2} e \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}}\) | \(463\) |
1/4*a^2*(e*x^2+d)^2/e+1/4*b^2*arctan(c*x)^2*e*x^4+1/2*b^2*arctan(c*x)^2*x^ 2*d-1/6*b^2/c*e*arctan(c*x)*x^3-b^2*d*x*arctan(c*x)/c+1/2*b^2*e*x*arctan(c *x)/c^3+1/2*b^2/c^2*arctan(c*x)^2*d-1/4*b^2/c^4*e*arctan(c*x)^2+1/12*b^2*e *x^2/c^2+1/2*b^2*d*ln(c^2*x^2+1)/c^2-1/3*b^2*e*ln(c^2*x^2+1)/c^4+1/2*a*b*a rctan(c*x)*e*x^4+a*b*arctan(c*x)*x^2*d-1/6/c*a*b*e*x^3-a*b*d*x/c+1/2*a*b*e *x/c^3+1/c^2*a*b*d*arctan(c*x)-1/2/c^4*a*b*e*arctan(c*x)
Time = 0.28 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.10 \[ \int x \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {3 \, a^{2} c^{4} e x^{4} - 2 \, a b c^{3} e x^{3} + {\left (6 \, a^{2} c^{4} d + b^{2} c^{2} e\right )} x^{2} + 3 \, {\left (b^{2} c^{4} e x^{4} + 2 \, b^{2} c^{4} d x^{2} + 2 \, b^{2} c^{2} d - b^{2} e\right )} \arctan \left (c x\right )^{2} - 6 \, {\left (2 \, a b c^{3} d - a b c e\right )} x + 2 \, {\left (3 \, a b c^{4} e x^{4} + 6 \, a b c^{4} d x^{2} - b^{2} c^{3} e x^{3} + 6 \, a b c^{2} d - 3 \, a b e - 3 \, {\left (2 \, b^{2} c^{3} d - b^{2} c e\right )} x\right )} \arctan \left (c x\right ) + 2 \, {\left (3 \, b^{2} c^{2} d - 2 \, b^{2} e\right )} \log \left (c^{2} x^{2} + 1\right )}{12 \, c^{4}} \]
1/12*(3*a^2*c^4*e*x^4 - 2*a*b*c^3*e*x^3 + (6*a^2*c^4*d + b^2*c^2*e)*x^2 + 3*(b^2*c^4*e*x^4 + 2*b^2*c^4*d*x^2 + 2*b^2*c^2*d - b^2*e)*arctan(c*x)^2 - 6*(2*a*b*c^3*d - a*b*c*e)*x + 2*(3*a*b*c^4*e*x^4 + 6*a*b*c^4*d*x^2 - b^2*c ^3*e*x^3 + 6*a*b*c^2*d - 3*a*b*e - 3*(2*b^2*c^3*d - b^2*c*e)*x)*arctan(c*x ) + 2*(3*b^2*c^2*d - 2*b^2*e)*log(c^2*x^2 + 1))/c^4
Time = 0.40 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.49 \[ \int x \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\begin {cases} \frac {a^{2} d x^{2}}{2} + \frac {a^{2} e x^{4}}{4} + a b d x^{2} \operatorname {atan}{\left (c x \right )} + \frac {a b e x^{4} \operatorname {atan}{\left (c x \right )}}{2} - \frac {a b d x}{c} - \frac {a b e x^{3}}{6 c} + \frac {a b d \operatorname {atan}{\left (c x \right )}}{c^{2}} + \frac {a b e x}{2 c^{3}} - \frac {a b e \operatorname {atan}{\left (c x \right )}}{2 c^{4}} + \frac {b^{2} d x^{2} \operatorname {atan}^{2}{\left (c x \right )}}{2} + \frac {b^{2} e x^{4} \operatorname {atan}^{2}{\left (c x \right )}}{4} - \frac {b^{2} d x \operatorname {atan}{\left (c x \right )}}{c} - \frac {b^{2} e x^{3} \operatorname {atan}{\left (c x \right )}}{6 c} + \frac {b^{2} d \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{2}} + \frac {b^{2} d \operatorname {atan}^{2}{\left (c x \right )}}{2 c^{2}} + \frac {b^{2} e x^{2}}{12 c^{2}} + \frac {b^{2} e x \operatorname {atan}{\left (c x \right )}}{2 c^{3}} - \frac {b^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{3 c^{4}} - \frac {b^{2} e \operatorname {atan}^{2}{\left (c x \right )}}{4 c^{4}} & \text {for}\: c \neq 0 \\a^{2} \left (\frac {d x^{2}}{2} + \frac {e x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d*x**2/2 + a**2*e*x**4/4 + a*b*d*x**2*atan(c*x) + a*b*e*x* *4*atan(c*x)/2 - a*b*d*x/c - a*b*e*x**3/(6*c) + a*b*d*atan(c*x)/c**2 + a*b *e*x/(2*c**3) - a*b*e*atan(c*x)/(2*c**4) + b**2*d*x**2*atan(c*x)**2/2 + b* *2*e*x**4*atan(c*x)**2/4 - b**2*d*x*atan(c*x)/c - b**2*e*x**3*atan(c*x)/(6 *c) + b**2*d*log(x**2 + c**(-2))/(2*c**2) + b**2*d*atan(c*x)**2/(2*c**2) + b**2*e*x**2/(12*c**2) + b**2*e*x*atan(c*x)/(2*c**3) - b**2*e*log(x**2 + c **(-2))/(3*c**4) - b**2*e*atan(c*x)**2/(4*c**4), Ne(c, 0)), (a**2*(d*x**2/ 2 + e*x**4/4), True))
Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.24 \[ \int x \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {1}{4} \, b^{2} e x^{4} \arctan \left (c x\right )^{2} + \frac {1}{4} \, a^{2} e x^{4} + \frac {1}{2} \, b^{2} d x^{2} \arctan \left (c x\right )^{2} + \frac {1}{2} \, a^{2} d x^{2} + {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} a b d - \frac {1}{2} \, {\left (2 \, c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )} \arctan \left (c x\right ) + \frac {\arctan \left (c x\right )^{2} - \log \left (c^{2} x^{2} + 1\right )}{c^{2}}\right )} b^{2} d + \frac {1}{6} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b e - \frac {1}{12} \, {\left (2 \, c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )} \arctan \left (c x\right ) - \frac {c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )} b^{2} e \]
1/4*b^2*e*x^4*arctan(c*x)^2 + 1/4*a^2*e*x^4 + 1/2*b^2*d*x^2*arctan(c*x)^2 + 1/2*a^2*d*x^2 + (x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a*b*d - 1/2*(2*c*(x/c^2 - arctan(c*x)/c^3)*arctan(c*x) + (arctan(c*x)^2 - log(c^2* x^2 + 1))/c^2)*b^2*d + 1/6*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3 *arctan(c*x)/c^5))*a*b*e - 1/12*(2*c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/ c^5)*arctan(c*x) - (c^2*x^2 + 3*arctan(c*x)^2 - 4*log(c^2*x^2 + 1))/c^4)*b ^2*e
\[ \int x \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \]
Time = 1.09 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.25 \[ \int x \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {a^2\,d\,x^2}{2}+\frac {a^2\,e\,x^4}{4}+\frac {b^2\,d\,\ln \left (c^2\,x^2+1\right )}{2\,c^2}-\frac {b^2\,e\,\ln \left (c^2\,x^2+1\right )}{3\,c^4}+\frac {b^2\,e\,x^2}{12\,c^2}+\frac {b^2\,d\,{\mathrm {atan}\left (c\,x\right )}^2}{2\,c^2}-\frac {b^2\,e\,{\mathrm {atan}\left (c\,x\right )}^2}{4\,c^4}+\frac {b^2\,d\,x^2\,{\mathrm {atan}\left (c\,x\right )}^2}{2}+\frac {b^2\,e\,x^4\,{\mathrm {atan}\left (c\,x\right )}^2}{4}-\frac {a\,b\,e\,x^3}{6\,c}-\frac {b^2\,d\,x\,\mathrm {atan}\left (c\,x\right )}{c}+\frac {b^2\,e\,x\,\mathrm {atan}\left (c\,x\right )}{2\,c^3}-\frac {b^2\,e\,x^3\,\mathrm {atan}\left (c\,x\right )}{6\,c}-\frac {a\,b\,d\,x}{c}+\frac {a\,b\,e\,x}{2\,c^3}+\frac {a\,b\,d\,\mathrm {atan}\left (c\,x\right )}{c^2}-\frac {a\,b\,e\,\mathrm {atan}\left (c\,x\right )}{2\,c^4}+a\,b\,d\,x^2\,\mathrm {atan}\left (c\,x\right )+\frac {a\,b\,e\,x^4\,\mathrm {atan}\left (c\,x\right )}{2} \]
(a^2*d*x^2)/2 + (a^2*e*x^4)/4 + (b^2*d*log(c^2*x^2 + 1))/(2*c^2) - (b^2*e* log(c^2*x^2 + 1))/(3*c^4) + (b^2*e*x^2)/(12*c^2) + (b^2*d*atan(c*x)^2)/(2* c^2) - (b^2*e*atan(c*x)^2)/(4*c^4) + (b^2*d*x^2*atan(c*x)^2)/2 + (b^2*e*x^ 4*atan(c*x)^2)/4 - (a*b*e*x^3)/(6*c) - (b^2*d*x*atan(c*x))/c + (b^2*e*x*at an(c*x))/(2*c^3) - (b^2*e*x^3*atan(c*x))/(6*c) - (a*b*d*x)/c + (a*b*e*x)/( 2*c^3) + (a*b*d*atan(c*x))/c^2 - (a*b*e*atan(c*x))/(2*c^4) + a*b*d*x^2*ata n(c*x) + (a*b*e*x^4*atan(c*x))/2