Integrand size = 22, antiderivative size = 313 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\frac {c^2 x}{21 a^3}-\frac {c^2 x^3}{168 a}-\frac {1}{280} a c^2 x^5-\frac {c^2 \arctan (a x)}{21 a^4}-\frac {5 c^2 x^2 \arctan (a x)}{168 a^2}+\frac {1}{28} c^2 x^4 \arctan (a x)+\frac {1}{56} a^2 c^2 x^6 \arctan (a x)+\frac {2 i c^2 \arctan (a x)^2}{21 a^4}+\frac {c^2 x \arctan (a x)^2}{8 a^3}-\frac {c^2 x^3 \arctan (a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3+\frac {4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{21 a^4}+\frac {2 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{21 a^4} \]
1/21*c^2*x/a^3-1/168*c^2*x^3/a-1/280*a*c^2*x^5-1/21*c^2*arctan(a*x)/a^4-5/ 168*c^2*x^2*arctan(a*x)/a^2+1/28*c^2*x^4*arctan(a*x)+1/56*a^2*c^2*x^6*arct an(a*x)+2/21*I*c^2*arctan(a*x)^2/a^4+1/8*c^2*x*arctan(a*x)^2/a^3-1/24*c^2* x^3*arctan(a*x)^2/a-1/8*a*c^2*x^5*arctan(a*x)^2-3/56*a^3*c^2*x^7*arctan(a* x)^2-1/24*c^2*arctan(a*x)^3/a^4+1/4*c^2*x^4*arctan(a*x)^3+1/3*a^2*c^2*x^6* arctan(a*x)^3+1/8*a^4*c^2*x^8*arctan(a*x)^3+4/21*c^2*arctan(a*x)*ln(2/(1+I *a*x))/a^4+2/21*I*c^2*polylog(2,1-2/(1+I*a*x))/a^4
Time = 1.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.53 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\frac {c^2 \left (40 a x-5 a^3 x^3-3 a^5 x^5-5 \left (16 i-21 a x+7 a^3 x^3+21 a^5 x^5+9 a^7 x^7\right ) \arctan (a x)^2+35 \left (1+a^2 x^2\right )^3 \left (-1+3 a^2 x^2\right ) \arctan (a x)^3+5 \arctan (a x) \left (-8-5 a^2 x^2+6 a^4 x^4+3 a^6 x^6+32 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-80 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{840 a^4} \]
(c^2*(40*a*x - 5*a^3*x^3 - 3*a^5*x^5 - 5*(16*I - 21*a*x + 7*a^3*x^3 + 21*a ^5*x^5 + 9*a^7*x^7)*ArcTan[a*x]^2 + 35*(1 + a^2*x^2)^3*(-1 + 3*a^2*x^2)*Ar cTan[a*x]^3 + 5*ArcTan[a*x]*(-8 - 5*a^2*x^2 + 6*a^4*x^4 + 3*a^6*x^6 + 32*L og[1 + E^((2*I)*ArcTan[a*x])]) - (80*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] ))/(840*a^4)
Time = 2.23 (sec) , antiderivative size = 313, 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.
| \(\displaystyle \int x^3 \arctan (a x)^3 \left (a^2 c x^2+c\right )^2 \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^4 c^2 x^7 \arctan (a x)^3+2 a^2 c^2 x^5 \arctan (a x)^3+c^2 x^3 \arctan (a x)^3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3-\frac {c^2 \arctan (a x)^3}{24 a^4}+\frac {2 i c^2 \arctan (a x)^2}{21 a^4}-\frac {c^2 \arctan (a x)}{21 a^4}+\frac {4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{21 a^4}+\frac {2 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{21 a^4}-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2+\frac {c^2 x \arctan (a x)^2}{8 a^3}+\frac {c^2 x}{21 a^3}+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{56} a^2 c^2 x^6 \arctan (a x)-\frac {5 c^2 x^2 \arctan (a x)}{168 a^2}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{28} c^2 x^4 \arctan (a x)-\frac {c^2 x^3 \arctan (a x)^2}{24 a}-\frac {1}{280} a c^2 x^5-\frac {c^2 x^3}{168 a}\) |
(c^2*x)/(21*a^3) - (c^2*x^3)/(168*a) - (a*c^2*x^5)/280 - (c^2*ArcTan[a*x]) /(21*a^4) - (5*c^2*x^2*ArcTan[a*x])/(168*a^2) + (c^2*x^4*ArcTan[a*x])/28 + (a^2*c^2*x^6*ArcTan[a*x])/56 + (((2*I)/21)*c^2*ArcTan[a*x]^2)/a^4 + (c^2* x*ArcTan[a*x]^2)/(8*a^3) - (c^2*x^3*ArcTan[a*x]^2)/(24*a) - (a*c^2*x^5*Arc Tan[a*x]^2)/8 - (3*a^3*c^2*x^7*ArcTan[a*x]^2)/56 - (c^2*ArcTan[a*x]^3)/(24 *a^4) + (c^2*x^4*ArcTan[a*x]^3)/4 + (a^2*c^2*x^6*ArcTan[a*x]^3)/3 + (a^4*c ^2*x^8*ArcTan[a*x]^3)/8 + (4*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(21*a^4) + (((2*I)/21)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4
3.4.71.3.1 Defintions of rubi rules used
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 = 3.50 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.06
| method | result | size |
| derivativedivides | \(\frac {\frac {c^{2} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {c^{2} \arctan \left (a x \right )^{3} a^{6} x^{6}}{3}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c^{2} \arctan \left (a x \right )^{3}}{24}-\frac {c^{2} \left (\frac {3 \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+a^{5} \arctan \left (a x \right )^{2} x^{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x -\frac {a^{6} \arctan \left (a x \right ) x^{6}}{7}-\frac {2 \arctan \left (a x \right ) a^{4} x^{4}}{7}+\frac {5 a^{2} \arctan \left (a x \right ) x^{2}}{21}+\frac {16 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{21}+\frac {a^{5} x^{5}}{35}+\frac {a^{3} x^{3}}{21}-\frac {8 a x}{21}+\frac {8 \arctan \left (a x \right )}{21}+\frac {8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{21}-\frac {8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{21}\right )}{8}}{a^{4}}\) | \(331\) |
| default | \(\frac {\frac {c^{2} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {c^{2} \arctan \left (a x \right )^{3} a^{6} x^{6}}{3}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c^{2} \arctan \left (a x \right )^{3}}{24}-\frac {c^{2} \left (\frac {3 \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+a^{5} \arctan \left (a x \right )^{2} x^{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x -\frac {a^{6} \arctan \left (a x \right ) x^{6}}{7}-\frac {2 \arctan \left (a x \right ) a^{4} x^{4}}{7}+\frac {5 a^{2} \arctan \left (a x \right ) x^{2}}{21}+\frac {16 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{21}+\frac {a^{5} x^{5}}{35}+\frac {a^{3} x^{3}}{21}-\frac {8 a x}{21}+\frac {8 \arctan \left (a x \right )}{21}+\frac {8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{21}-\frac {8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{21}\right )}{8}}{a^{4}}\) | \(331\) |
| parts | \(\frac {a^{4} c^{2} x^{8} \arctan \left (a x \right )^{3}}{8}+\frac {a^{2} c^{2} x^{6} \arctan \left (a x \right )^{3}}{3}+\frac {c^{2} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c^{2} \left (\frac {3 a^{3} \arctan \left (a x \right )^{2} x^{7}}{7}+a \arctan \left (a x \right )^{2} x^{5}+\frac {\arctan \left (a x \right )^{2} x^{3}}{3 a}-\frac {\arctan \left (a x \right )^{2} x}{a^{3}}+\frac {\arctan \left (a x \right )^{3}}{a^{4}}-\frac {2 \left (\frac {3 a^{6} \arctan \left (a x \right ) x^{6}}{2}+3 \arctan \left (a x \right ) a^{4} x^{4}-\frac {5 a^{2} \arctan \left (a x \right ) x^{2}}{2}-8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {3 a^{5} x^{5}}{10}-\frac {a^{3} x^{3}}{2}+4 a x -4 \arctan \left (a x \right )-4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+7 \arctan \left (a x \right )^{3}\right )}{21 a^{4}}\right )}{8}\) | \(337\) |
1/a^4*(1/8*c^2*arctan(a*x)^3*a^8*x^8+1/3*c^2*arctan(a*x)^3*a^6*x^6+1/4*a^4 *c^2*x^4*arctan(a*x)^3-1/24*c^2*arctan(a*x)^3-1/8*c^2*(3/7*arctan(a*x)^2*a ^7*x^7+a^5*arctan(a*x)^2*x^5+1/3*a^3*arctan(a*x)^2*x^3-a*arctan(a*x)^2*x-1 /7*a^6*arctan(a*x)*x^6-2/7*arctan(a*x)*a^4*x^4+5/21*a^2*arctan(a*x)*x^2+16 /21*arctan(a*x)*ln(a^2*x^2+1)+1/35*a^5*x^5+1/21*a^3*x^3-8/21*a*x+8/21*arct an(a*x)+8/21*I*(ln(a*x-I)*ln(a^2*x^2+1)-dilog(-1/2*I*(I+a*x))-ln(a*x-I)*ln (-1/2*I*(I+a*x))-1/2*ln(a*x-I)^2)-8/21*I*(ln(I+a*x)*ln(a^2*x^2+1)-dilog(1/ 2*I*(a*x-I))-ln(I+a*x)*ln(1/2*I*(a*x-I))-1/2*ln(I+a*x)^2)))
\[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=c^{2} \left (\int x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{7} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**2*(Integral(x**3*atan(a*x)**3, x) + Integral(2*a**2*x**5*atan(a*x)**3, x) + Integral(a**4*x**7*atan(a*x)**3, x))
\[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
1/2688*(28*(129024*a^9*c^2*integrate(1/2688*x^9*arctan(a*x)^3/(a^5*x^2 + a ^3), x) - 24192*a^8*c^2*integrate(1/2688*x^8*arctan(a*x)^2/(a^5*x^2 + a^3) , x) - 6048*a^8*c^2*integrate(1/2688*x^8*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3 ), x) - 3456*a^8*c^2*integrate(1/2688*x^8*log(a^2*x^2 + 1)/(a^5*x^2 + a^3) , x) + 387072*a^7*c^2*integrate(1/2688*x^7*arctan(a*x)^3/(a^5*x^2 + a^3), x) + 6912*a^7*c^2*integrate(1/2688*x^7*arctan(a*x)/(a^5*x^2 + a^3), x) - 6 4512*a^6*c^2*integrate(1/2688*x^6*arctan(a*x)^2/(a^5*x^2 + a^3), x) - 1612 8*a^6*c^2*integrate(1/2688*x^6*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) - 80 64*a^6*c^2*integrate(1/2688*x^6*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 387 072*a^5*c^2*integrate(1/2688*x^5*arctan(a*x)^3/(a^5*x^2 + a^3), x) + 16128 *a^5*c^2*integrate(1/2688*x^5*arctan(a*x)/(a^5*x^2 + a^3), x) - 48384*a^4* c^2*integrate(1/2688*x^4*arctan(a*x)^2/(a^5*x^2 + a^3), x) - 12096*a^4*c^2 *integrate(1/2688*x^4*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) - 2688*a^4*c^ 2*integrate(1/2688*x^4*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 129024*a^3*c ^2*integrate(1/2688*x^3*arctan(a*x)^3/(a^5*x^2 + a^3), x) + 5376*a^3*c^2*i ntegrate(1/2688*x^3*arctan(a*x)/(a^5*x^2 + a^3), x) + 8064*a^2*c^2*integra te(1/2688*x^2*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) - 16128*a*c^2*integrate (1/2688*x*arctan(a*x)/(a^5*x^2 + a^3), x) + 2016*c^2*integrate(1/2688*log( a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) + c^2*arctan(a*x)^3/a^4)*a^4 + 56*(3*a^ 8*c^2*x^8 + 8*a^6*c^2*x^6 + 6*a^4*c^2*x^4 - c^2)*arctan(a*x)^3 - 4*(9*a...
\[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]