Integrand size = 22, antiderivative size = 225 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=-\frac {c^2 x}{210 a^2}+\frac {17 c^2 x^3}{630}+\frac {1}{105} a^2 c^2 x^5+\frac {c^2 \arctan (a x)}{210 a^3}-\frac {8 c^2 x^2 \arctan (a x)}{105 a}-\frac {9}{70} a c^2 x^4 \arctan (a x)-\frac {1}{21} a^3 c^2 x^6 \arctan (a x)-\frac {8 i c^2 \arctan (a x)^2}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^2+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2-\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{105 a^3} \] Output:
-1/210*c^2*x/a^2+17/630*c^2*x^3+1/105*a^2*c^2*x^5+1/210*c^2*arctan(a*x)/a^ 3-8/105*c^2*x^2*arctan(a*x)/a-9/70*a*c^2*x^4*arctan(a*x)-1/21*a^3*c^2*x^6* arctan(a*x)-8/105*I*c^2*arctan(a*x)^2/a^3+1/3*c^2*x^3*arctan(a*x)^2+2/5*a^ 2*c^2*x^5*arctan(a*x)^2+1/7*a^4*c^2*x^7*arctan(a*x)^2-16/105*c^2*arctan(a* x)*ln(2/(1+I*a*x))/a^3-8/105*I*c^2*polylog(2,1-2/(1+I*a*x))/a^3
Time = 0.93 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.59 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {c^2 \left (a x \left (-3+17 a^2 x^2+6 a^4 x^4\right )+6 \left (8 i+35 a^3 x^3+42 a^5 x^5+15 a^7 x^7\right ) \arctan (a x)^2-3 \arctan (a x) \left (-1+16 a^2 x^2+27 a^4 x^4+10 a^6 x^6+32 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+48 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{630 a^3} \] Input:
Integrate[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]
Output:
(c^2*(a*x*(-3 + 17*a^2*x^2 + 6*a^4*x^4) + 6*(8*I + 35*a^3*x^3 + 42*a^5*x^5 + 15*a^7*x^7)*ArcTan[a*x]^2 - 3*ArcTan[a*x]*(-1 + 16*a^2*x^2 + 27*a^4*x^4 + 10*a^6*x^6 + 32*Log[1 + E^((2*I)*ArcTan[a*x])]) + (48*I)*PolyLog[2, -E^ ((2*I)*ArcTan[a*x])]))/(630*a^3)
Time = 1.00 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5483, 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.
| \(\displaystyle \int x^2 \arctan (a x)^2 \left (a^2 c x^2+c\right )^2 \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^4 c^2 x^6 \arctan (a x)^2+2 a^2 c^2 x^4 \arctan (a x)^2+c^2 x^2 \arctan (a x)^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2-\frac {1}{21} a^3 c^2 x^6 \arctan (a x)-\frac {8 i c^2 \arctan (a x)^2}{105 a^3}+\frac {c^2 \arctan (a x)}{210 a^3}-\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{105 a^3}+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{105} a^2 c^2 x^5-\frac {c^2 x}{210 a^2}-\frac {9}{70} a c^2 x^4 \arctan (a x)+\frac {1}{3} c^2 x^3 \arctan (a x)^2-\frac {8 c^2 x^2 \arctan (a x)}{105 a}+\frac {17 c^2 x^3}{630}\) |
Input:
Int[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]
Output:
-1/210*(c^2*x)/a^2 + (17*c^2*x^3)/630 + (a^2*c^2*x^5)/105 + (c^2*ArcTan[a* x])/(210*a^3) - (8*c^2*x^2*ArcTan[a*x])/(105*a) - (9*a*c^2*x^4*ArcTan[a*x] )/70 - (a^3*c^2*x^6*ArcTan[a*x])/21 - (((8*I)/105)*c^2*ArcTan[a*x]^2)/a^3 + (c^2*x^3*ArcTan[a*x]^2)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^2)/5 + (a^4*c^2*x ^7*ArcTan[a*x]^2)/7 - (16*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(105*a^3) - (((8*I)/105)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2* d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])
Time = 2.07 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.19
| method | result | size |
| parts | \(\frac {a^{4} c^{2} x^{7} \arctan \left (a x \right )^{2}}{7}+\frac {2 a^{2} c^{2} x^{5} \arctan \left (a x \right )^{2}}{5}+\frac {c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {2 c^{2} \left (\frac {5 a^{3} \arctan \left (a x \right ) x^{6}}{2}+\frac {27 a \arctan \left (a x \right ) x^{4}}{4}+\frac {4 \arctan \left (a x \right ) x^{2}}{a}-\frac {4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{a^{3}}-\frac {2 a^{5} x^{5}+\frac {17 a^{3} x^{3}}{3}-a x +\arctan \left (a x \right )+8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )-8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{4 a^{3}}\right )}{105}\) | \(267\) |
| derivativedivides | \(\frac {\frac {c^{2} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {2 c^{2} \arctan \left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {a^{3} c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {2 c^{2} \left (\frac {5 a^{6} \arctan \left (a x \right ) x^{6}}{2}+\frac {27 x^{4} \arctan \left (a x \right ) a^{4}}{4}+4 x^{2} a^{2} \arctan \left (a x \right )-4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{2}-\frac {17 a^{3} x^{3}}{12}+\frac {a x}{4}-\frac {\arctan \left (a x \right )}{4}-2 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+2 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )\right )}{105}}{a^{3}}\) | \(269\) |
| default | \(\frac {\frac {c^{2} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {2 c^{2} \arctan \left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {a^{3} c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {2 c^{2} \left (\frac {5 a^{6} \arctan \left (a x \right ) x^{6}}{2}+\frac {27 x^{4} \arctan \left (a x \right ) a^{4}}{4}+4 x^{2} a^{2} \arctan \left (a x \right )-4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{2}-\frac {17 a^{3} x^{3}}{12}+\frac {a x}{4}-\frac {\arctan \left (a x \right )}{4}-2 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+2 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )\right )}{105}}{a^{3}}\) | \(269\) |
| risch | \(-\frac {177151 i c^{2}}{2315250 a^{3}}-\frac {c^{2} \ln \left (i a x +1\right )^{2} x^{3}}{12}-\frac {c^{2} \ln \left (-i a x +1\right )^{2} x^{3}}{12}+\frac {17 x^{3} c^{2}}{630}-\frac {c^{2} x}{210 a^{2}}+\frac {a^{2} c^{2} x^{5}}{105}+\frac {c^{2} \arctan \left (a x \right )}{210 a^{3}}+\frac {c^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{6}-\frac {c^{2} a^{4} \ln \left (-i a x +1\right )^{2} x^{7}}{28}-\frac {c^{2} a^{2} \ln \left (-i a x +1\right )^{2} x^{5}}{10}-\frac {c^{2} a^{2} \ln \left (i a x +1\right )^{2} x^{5}}{10}-\frac {c^{2} a^{4} \ln \left (i a x +1\right )^{2} x^{7}}{28}-\frac {8 i c^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{105 a^{3}}+\frac {2 i c^{2} \ln \left (-i a x +1\right )^{2}}{105 a^{3}}-\frac {2 i c^{2} \ln \left (i a x +1\right )^{2}}{105 a^{3}}-\frac {9 i c^{2} a \ln \left (-i a x +1\right ) x^{4}}{140}-\frac {4 i c^{2} \ln \left (-i a x +1\right ) x^{2}}{105 a}+\frac {c^{2} a^{4} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{7}}{14}+\frac {c^{2} a^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{5}+\frac {8 i c^{2} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{105 a^{3}}-\frac {8 i c^{2} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{105 a^{3}}-\frac {4 i c^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{105 a^{3}}+\frac {9 i c^{2} a \ln \left (i a x +1\right ) x^{4}}{140}+\frac {i c^{2} a^{3} \ln \left (i a x +1\right ) x^{6}}{42}+\frac {4 i c^{2} \ln \left (i a x +1\right ) x^{2}}{105 a}-\frac {i c^{2} a^{3} \ln \left (-i a x +1\right ) x^{6}}{42}\) | \(495\) |
Input:
int(x^2*(a^2*c*x^2+c)^2*arctan(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/7*a^4*c^2*x^7*arctan(a*x)^2+2/5*a^2*c^2*x^5*arctan(a*x)^2+1/3*c^2*x^3*ar ctan(a*x)^2-2/105*c^2*(5/2*a^3*arctan(a*x)*x^6+27/4*a*arctan(a*x)*x^4+4/a* arctan(a*x)*x^2-4/a^3*arctan(a*x)*ln(a^2*x^2+1)-1/4/a^3*(2*a^5*x^5+17/3*a^ 3*x^3-a*x+arctan(a*x)+8*I*(ln(a*x-I)*ln(a^2*x^2+1)-1/2*ln(a*x-I)^2-dilog(- 1/2*I*(a*x+I))-ln(a*x-I)*ln(-1/2*I*(a*x+I)))-8*I*(ln(a*x+I)*ln(a^2*x^2+1)- 1/2*ln(a*x+I)^2-dilog(1/2*I*(a*x-I))-ln(a*x+I)*ln(1/2*I*(a*x-I)))))
\[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{2} \,d x } \] Input:
integrate(x^2*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="fricas")
Output:
integral((a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2)*arctan(a*x)^2, x)
\[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=c^{2} \left (\int x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 2 a^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{4} x^{6} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \] Input:
integrate(x**2*(a**2*c*x**2+c)**2*atan(a*x)**2,x)
Output:
c**2*(Integral(x**2*atan(a*x)**2, x) + Integral(2*a**2*x**4*atan(a*x)**2, x) + Integral(a**4*x**6*atan(a*x)**2, x))
\[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{2} \,d x } \] Input:
integrate(x^2*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="maxima")
Output:
1/420*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 + 35*c^2*x^3)*arctan(a*x)^2 - 1/168 0*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 + 35*c^2*x^3)*log(a^2*x^2 + 1)^2 + inte grate(1/1680*(1260*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x^4 + c^2*x^2) *arctan(a*x)^2 + 105*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x^4 + c^2*x^ 2)*log(a^2*x^2 + 1)^2 - 8*(15*a^5*c^2*x^7 + 42*a^3*c^2*x^5 + 35*a*c^2*x^3) *arctan(a*x) + 4*(15*a^6*c^2*x^8 + 42*a^4*c^2*x^6 + 35*a^2*c^2*x^4)*log(a^ 2*x^2 + 1))/(a^2*x^2 + 1), x)
\[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{2} \,d x } \] Input:
integrate(x^2*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)^2*x^2*arctan(a*x)^2, x)
Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \] Input:
int(x^2*atan(a*x)^2*(c + a^2*c*x^2)^2,x)
Output:
int(x^2*atan(a*x)^2*(c + a^2*c*x^2)^2, x)
\[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {c^{2} \left (90 \mathit {atan} \left (a x \right )^{2} a^{7} x^{7}+252 \mathit {atan} \left (a x \right )^{2} a^{5} x^{5}+210 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}+48 \mathit {atan} \left (a x \right )^{2} a x -30 \mathit {atan} \left (a x \right ) a^{6} x^{6}-81 \mathit {atan} \left (a x \right ) a^{4} x^{4}-48 \mathit {atan} \left (a x \right ) a^{2} x^{2}+3 \mathit {atan} \left (a x \right )-48 \left (\int \mathit {atan} \left (a x \right )^{2}d x \right ) a +6 a^{5} x^{5}+17 a^{3} x^{3}-3 a x \right )}{630 a^{3}} \] Input:
int(x^2*(a^2*c*x^2+c)^2*atan(a*x)^2,x)
Output:
(c**2*(90*atan(a*x)**2*a**7*x**7 + 252*atan(a*x)**2*a**5*x**5 + 210*atan(a *x)**2*a**3*x**3 + 48*atan(a*x)**2*a*x - 30*atan(a*x)*a**6*x**6 - 81*atan( a*x)*a**4*x**4 - 48*atan(a*x)*a**2*x**2 + 3*atan(a*x) - 48*int(atan(a*x)** 2,x)*a + 6*a**5*x**5 + 17*a**3*x**3 - 3*a*x))/(630*a**3)