Integrand size = 20, antiderivative size = 200 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^3 \left (1+a^2 x^2\right )}{35 a^2}+\frac {3 c^3 \left (1+a^2 x^2\right )^2}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3}{168 a^2}-\frac {4 c^3 x \arctan (a x)}{35 a}-\frac {2 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)}{70 a}-\frac {c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)}{28 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^2}{8 a^2}+\frac {2 c^3 \log \left (1+a^2 x^2\right )}{35 a^2} \] Output:
1/35*c^3*(a^2*x^2+1)/a^2+3/280*c^3*(a^2*x^2+1)^2/a^2+1/168*c^3*(a^2*x^2+1) ^3/a^2-4/35*c^3*x*arctan(a*x)/a-2/35*c^3*x*(a^2*x^2+1)*arctan(a*x)/a-3/70* c^3*x*(a^2*x^2+1)^2*arctan(a*x)/a-1/28*c^3*x*(a^2*x^2+1)^3*arctan(a*x)/a+1 /8*c^3*(a^2*x^2+1)^4*arctan(a*x)^2/a^2+2/35*c^3*ln(a^2*x^2+1)/a^2
Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.50 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^3 \left (57 a^2 x^2+24 a^4 x^4+5 a^6 x^6-6 a x \left (35+35 a^2 x^2+21 a^4 x^4+5 a^6 x^6\right ) \arctan (a x)+105 \left (1+a^2 x^2\right )^4 \arctan (a x)^2+48 \log \left (1+a^2 x^2\right )\right )}{840 a^2} \] Input:
Integrate[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]
Output:
(c^3*(57*a^2*x^2 + 24*a^4*x^4 + 5*a^6*x^6 - 6*a*x*(35 + 35*a^2*x^2 + 21*a^ 4*x^4 + 5*a^6*x^6)*ArcTan[a*x] + 105*(1 + a^2*x^2)^4*ArcTan[a*x]^2 + 48*Lo g[1 + a^2*x^2]))/(840*a^2)
Time = 0.65 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5465, 27, 5413, 5413, 5413, 5345, 240}
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.
| \(\displaystyle \int x \arctan (a x)^2 \left (a^2 c x^2+c\right )^3 \, dx\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^2}{8 a^2}-\frac {\int c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)dx}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^2}{8 a^2}-\frac {c^3 \int \left (a^2 x^2+1\right )^3 \arctan (a x)dx}{4 a}\) |
| \(\Big \downarrow \) 5413 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^2}{8 a^2}-\frac {c^3 \left (\frac {6}{7} \int \left (a^2 x^2+1\right )^2 \arctan (a x)dx+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^3}{42 a}\right )}{4 a}\) |
| \(\Big \downarrow \) 5413 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^2}{8 a^2}-\frac {c^3 \left (\frac {6}{7} \left (\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^3}{42 a}\right )}{4 a}\) |
| \(\Big \downarrow \) 5413 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^2}{8 a^2}-\frac {c^3 \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \arctan (a x)dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^3}{42 a}\right )}{4 a}\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^2}{8 a^2}-\frac {c^3 \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^3}{42 a}\right )}{4 a}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^2}{8 a^2}-\frac {c^3 \left (\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)+\frac {6}{7} \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )-\frac {\left (a^2 x^2+1\right )^3}{42 a}\right )}{4 a}\) |
Input:
Int[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]
Output:
(c^3*(1 + a^2*x^2)^4*ArcTan[a*x]^2)/(8*a^2) - (c^3*(-1/42*(1 + a^2*x^2)^3/ a + (x*(1 + a^2*x^2)^3*ArcTan[a*x])/7 + (6*(-1/20*(1 + a^2*x^2)^2/a + (x*( 1 + a^2*x^2)^2*ArcTan[a*x])/5 + (4*(-1/6*(1 + a^2*x^2)/a + (x*(1 + a^2*x^2 )*ArcTan[a*x])/3 + (2*(x*ArcTan[a*x] - Log[1 + a^2*x^2]/(2*a)))/3))/5))/7) )/(4*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2) ^q*((a + b*ArcTan[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Time = 1.15 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.84
| method | result | size |
| parts | \(\frac {c^{3} \arctan \left (a x \right )^{2} a^{6} x^{8}}{8}+\frac {c^{3} \arctan \left (a x \right )^{2} a^{4} x^{6}}{2}+\frac {3 c^{3} \arctan \left (a x \right )^{2} a^{2} x^{4}}{4}+\frac {c^{3} \arctan \left (a x \right )^{2} x^{2}}{2}+\frac {c^{3} \arctan \left (a x \right )^{2}}{8 a^{2}}-\frac {c^{3} \left (\frac {\arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 \arctan \left (a x \right ) a^{5} x^{5}}{5}+\arctan \left (a x \right ) x^{3} a^{3}+\arctan \left (a x \right ) a x -\frac {a^{6} x^{6}}{42}-\frac {4 a^{4} x^{4}}{35}-\frac {19 a^{2} x^{2}}{70}-\frac {8 \ln \left (a^{2} x^{2}+1\right )}{35}\right )}{4 a^{2}}\) | \(168\) |
| derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}}{2}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}}{4}+\frac {a^{2} c^{3} x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c^{3} \arctan \left (a x \right )^{2}}{8}-\frac {c^{3} \left (\frac {\arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 \arctan \left (a x \right ) a^{5} x^{5}}{5}+\arctan \left (a x \right ) x^{3} a^{3}+\arctan \left (a x \right ) a x -\frac {a^{6} x^{6}}{42}-\frac {4 a^{4} x^{4}}{35}-\frac {19 a^{2} x^{2}}{70}-\frac {8 \ln \left (a^{2} x^{2}+1\right )}{35}\right )}{4}}{a^{2}}\) | \(169\) |
| default | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}}{2}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}}{4}+\frac {a^{2} c^{3} x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c^{3} \arctan \left (a x \right )^{2}}{8}-\frac {c^{3} \left (\frac {\arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 \arctan \left (a x \right ) a^{5} x^{5}}{5}+\arctan \left (a x \right ) x^{3} a^{3}+\arctan \left (a x \right ) a x -\frac {a^{6} x^{6}}{42}-\frac {4 a^{4} x^{4}}{35}-\frac {19 a^{2} x^{2}}{70}-\frac {8 \ln \left (a^{2} x^{2}+1\right )}{35}\right )}{4}}{a^{2}}\) | \(169\) |
| parallelrisch | \(\frac {105 c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}-30 c^{3} \arctan \left (a x \right ) a^{7} x^{7}+420 a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}+5 a^{6} c^{3} x^{6}-126 a^{5} c^{3} x^{5} \arctan \left (a x \right )+630 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}+24 a^{4} c^{3} x^{4}-210 a^{3} c^{3} x^{3} \arctan \left (a x \right )+420 a^{2} c^{3} x^{2} \arctan \left (a x \right )^{2}+57 a^{2} c^{3} x^{2}-210 a \,c^{3} x \arctan \left (a x \right )+105 c^{3} \arctan \left (a x \right )^{2}+48 c^{3} \ln \left (a^{2} x^{2}+1\right )}{840 a^{2}}\) | \(190\) |
| risch | \(-\frac {c^{3} \left (a^{2} x^{2}+1\right )^{4} \ln \left (i a x +1\right )^{2}}{32 a^{2}}+\frac {c^{3} \left (35 a^{8} x^{8} \ln \left (-i a x +1\right )+10 i a^{7} x^{7}+140 a^{6} x^{6} \ln \left (-i a x +1\right )+42 i a^{5} x^{5}+210 x^{4} \ln \left (-i a x +1\right ) a^{4}+70 i a^{3} x^{3}+140 a^{2} x^{2} \ln \left (-i a x +1\right )+70 i a x +35 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{560 a^{2}}-\frac {c^{3} a^{6} x^{8} \ln \left (-i a x +1\right )^{2}}{32}-\frac {i c^{3} a^{5} x^{7} \ln \left (-i a x +1\right )}{56}-\frac {c^{3} a^{4} x^{6} \ln \left (-i a x +1\right )^{2}}{8}-\frac {3 i c^{3} a^{3} x^{5} \ln \left (-i a x +1\right )}{40}+\frac {c^{3} a^{4} x^{6}}{168}-\frac {3 c^{3} a^{2} x^{4} \ln \left (-i a x +1\right )^{2}}{16}-\frac {i c^{3} a \,x^{3} \ln \left (-i a x +1\right )}{8}+\frac {c^{3} x^{4} a^{2}}{35}-\frac {c^{3} x^{2} \ln \left (-i a x +1\right )^{2}}{8}-\frac {i c^{3} x \ln \left (-i a x +1\right )}{8 a}+\frac {19 c^{3} x^{2}}{280}-\frac {c^{3} \ln \left (-i a x +1\right )^{2}}{32 a^{2}}+\frac {2 c^{3} \ln \left (-a^{2} x^{2}-1\right )}{35 a^{2}}\) | \(378\) |
Input:
int(x*(a^2*c*x^2+c)^3*arctan(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/8*c^3*arctan(a*x)^2*a^6*x^8+1/2*c^3*arctan(a*x)^2*a^4*x^6+3/4*c^3*arctan (a*x)^2*a^2*x^4+1/2*c^3*arctan(a*x)^2*x^2+1/8*c^3*arctan(a*x)^2/a^2-1/4*c^ 3/a^2*(1/7*arctan(a*x)*a^7*x^7+3/5*arctan(a*x)*a^5*x^5+arctan(a*x)*x^3*a^3 +arctan(a*x)*a*x-1/42*a^6*x^6-4/35*a^4*x^4-19/70*a^2*x^2-8/35*ln(a^2*x^2+1 ))
Time = 0.13 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.78 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {5 \, a^{6} c^{3} x^{6} + 24 \, a^{4} c^{3} x^{4} + 57 \, a^{2} c^{3} x^{2} + 48 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) + 105 \, {\left (a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{2} - 6 \, {\left (5 \, a^{7} c^{3} x^{7} + 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} + 35 \, a c^{3} x\right )} \arctan \left (a x\right )}{840 \, a^{2}} \] Input:
integrate(x*(a^2*c*x^2+c)^3*arctan(a*x)^2,x, algorithm="fricas")
Output:
1/840*(5*a^6*c^3*x^6 + 24*a^4*c^3*x^4 + 57*a^2*c^3*x^2 + 48*c^3*log(a^2*x^ 2 + 1) + 105*(a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 + 4*a^2*c^3*x^2 + c^3)*arctan(a*x)^2 - 6*(5*a^7*c^3*x^7 + 21*a^5*c^3*x^5 + 35*a^3*c^3*x^3 + 35*a*c^3*x)*arctan(a*x))/a^2
Time = 0.55 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.04 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\begin {cases} \frac {a^{6} c^{3} x^{8} \operatorname {atan}^{2}{\left (a x \right )}}{8} - \frac {a^{5} c^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{28} + \frac {a^{4} c^{3} x^{6} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {a^{4} c^{3} x^{6}}{168} - \frac {3 a^{3} c^{3} x^{5} \operatorname {atan}{\left (a x \right )}}{20} + \frac {3 a^{2} c^{3} x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{4} + \frac {a^{2} c^{3} x^{4}}{35} - \frac {a c^{3} x^{3} \operatorname {atan}{\left (a x \right )}}{4} + \frac {c^{3} x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {19 c^{3} x^{2}}{280} - \frac {c^{3} x \operatorname {atan}{\left (a x \right )}}{4 a} + \frac {2 c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{35 a^{2}} + \frac {c^{3} \operatorname {atan}^{2}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x*(a**2*c*x**2+c)**3*atan(a*x)**2,x)
Output:
Piecewise((a**6*c**3*x**8*atan(a*x)**2/8 - a**5*c**3*x**7*atan(a*x)/28 + a **4*c**3*x**6*atan(a*x)**2/2 + a**4*c**3*x**6/168 - 3*a**3*c**3*x**5*atan( a*x)/20 + 3*a**2*c**3*x**4*atan(a*x)**2/4 + a**2*c**3*x**4/35 - a*c**3*x** 3*atan(a*x)/4 + c**3*x**2*atan(a*x)**2/2 + 19*c**3*x**2/280 - c**3*x*atan( a*x)/(4*a) + 2*c**3*log(x**2 + a**(-2))/(35*a**2) + c**3*atan(a*x)**2/(8*a **2), Ne(a, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.66 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {{\left (a^{2} c x^{2} + c\right )}^{4} \arctan \left (a x\right )^{2}}{8 \, a^{2} c} + \frac {{\left (5 \, a^{4} c^{4} x^{6} + 24 \, a^{2} c^{4} x^{4} + 57 \, c^{4} x^{2} + \frac {48 \, c^{4} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a - 6 \, {\left (5 \, a^{6} c^{4} x^{7} + 21 \, a^{4} c^{4} x^{5} + 35 \, a^{2} c^{4} x^{3} + 35 \, c^{4} x\right )} \arctan \left (a x\right )}{840 \, a c} \] Input:
integrate(x*(a^2*c*x^2+c)^3*arctan(a*x)^2,x, algorithm="maxima")
Output:
1/8*(a^2*c*x^2 + c)^4*arctan(a*x)^2/(a^2*c) + 1/840*((5*a^4*c^4*x^6 + 24*a ^2*c^4*x^4 + 57*c^4*x^2 + 48*c^4*log(a^2*x^2 + 1)/a^2)*a - 6*(5*a^6*c^4*x^ 7 + 21*a^4*c^4*x^5 + 35*a^2*c^4*x^3 + 35*c^4*x)*arctan(a*x))/(a*c)
Time = 0.13 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.12 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {{\left (a^{2} c x^{2} + c\right )}^{4} \arctan \left (a x\right )^{2}}{8 \, a^{2} c} - \frac {5 \, {\left (12 \, x^{7} \arctan \left (a x\right ) - a {\left (\frac {2 \, a^{4} x^{6} - 3 \, a^{2} x^{4} + 6 \, x^{2}}{a^{6}} - \frac {6 \, \log \left (a^{2} x^{2} + 1\right )}{a^{8}}\right )}\right )} a^{6} c^{3} + 63 \, {\left (4 \, x^{5} \arctan \left (a x\right ) - a {\left (\frac {a^{2} x^{4} - 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )}\right )} a^{4} c^{3} + 210 \, {\left (2 \, x^{3} \arctan \left (a x\right ) - a {\left (\frac {x^{2}}{a^{2}} - \frac {\log \left (a^{2} x^{2} + 1\right )}{a^{4}}\right )}\right )} a^{2} c^{3} + \frac {210 \, {\left (2 \, a x \arctan \left (a x\right ) - \log \left (a^{2} x^{2} + 1\right )\right )} c^{3}}{a}}{1680 \, a} \] Input:
integrate(x*(a^2*c*x^2+c)^3*arctan(a*x)^2,x, algorithm="giac")
Output:
1/8*(a^2*c*x^2 + c)^4*arctan(a*x)^2/(a^2*c) - 1/1680*(5*(12*x^7*arctan(a*x ) - a*((2*a^4*x^6 - 3*a^2*x^4 + 6*x^2)/a^6 - 6*log(a^2*x^2 + 1)/a^8))*a^6* c^3 + 63*(4*x^5*arctan(a*x) - a*((a^2*x^4 - 2*x^2)/a^4 + 2*log(a^2*x^2 + 1 )/a^6))*a^4*c^3 + 210*(2*x^3*arctan(a*x) - a*(x^2/a^2 - log(a^2*x^2 + 1)/a ^4))*a^2*c^3 + 210*(2*a*x*arctan(a*x) - log(a^2*x^2 + 1))*c^3/a)/a
Time = 0.71 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.78 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx={\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {c^3}{8\,a^2}+\frac {c^3\,x^2}{2}+\frac {3\,a^2\,c^3\,x^4}{4}+\frac {a^4\,c^3\,x^6}{2}+\frac {a^6\,c^3\,x^8}{8}\right )+\frac {19\,c^3\,x^2}{280}-a^2\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {c^3\,x}{4\,a^3}+\frac {3\,a\,c^3\,x^5}{20}+\frac {c^3\,x^3}{4\,a}+\frac {a^3\,c^3\,x^7}{28}\right )+\frac {2\,c^3\,\ln \left (a^2\,x^2+1\right )}{35\,a^2}+\frac {a^2\,c^3\,x^4}{35}+\frac {a^4\,c^3\,x^6}{168} \] Input:
int(x*atan(a*x)^2*(c + a^2*c*x^2)^3,x)
Output:
atan(a*x)^2*(c^3/(8*a^2) + (c^3*x^2)/2 + (3*a^2*c^3*x^4)/4 + (a^4*c^3*x^6) /2 + (a^6*c^3*x^8)/8) + (19*c^3*x^2)/280 - a^2*atan(a*x)*((c^3*x)/(4*a^3) + (3*a*c^3*x^5)/20 + (c^3*x^3)/(4*a) + (a^3*c^3*x^7)/28) + (2*c^3*log(a^2* x^2 + 1))/(35*a^2) + (a^2*c^3*x^4)/35 + (a^4*c^3*x^6)/168
Time = 0.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.76 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^{3} \left (105 \mathit {atan} \left (a x \right )^{2} a^{8} x^{8}+420 \mathit {atan} \left (a x \right )^{2} a^{6} x^{6}+630 \mathit {atan} \left (a x \right )^{2} a^{4} x^{4}+420 \mathit {atan} \left (a x \right )^{2} a^{2} x^{2}+105 \mathit {atan} \left (a x \right )^{2}-30 \mathit {atan} \left (a x \right ) a^{7} x^{7}-126 \mathit {atan} \left (a x \right ) a^{5} x^{5}-210 \mathit {atan} \left (a x \right ) a^{3} x^{3}-210 \mathit {atan} \left (a x \right ) a x +48 \,\mathrm {log}\left (a^{2} x^{2}+1\right )+5 a^{6} x^{6}+24 a^{4} x^{4}+57 a^{2} x^{2}\right )}{840 a^{2}} \] Input:
int(x*(a^2*c*x^2+c)^3*atan(a*x)^2,x)
Output:
(c**3*(105*atan(a*x)**2*a**8*x**8 + 420*atan(a*x)**2*a**6*x**6 + 630*atan( a*x)**2*a**4*x**4 + 420*atan(a*x)**2*a**2*x**2 + 105*atan(a*x)**2 - 30*ata n(a*x)*a**7*x**7 - 126*atan(a*x)*a**5*x**5 - 210*atan(a*x)*a**3*x**3 - 210 *atan(a*x)*a*x + 48*log(a**2*x**2 + 1) + 5*a**6*x**6 + 24*a**4*x**4 + 57*a **2*x**2))/(840*a**2)