Integrand size = 22, antiderivative size = 250 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^4} \, dx=-\frac {a^2 c^3}{3 x}+\frac {1}{3} a^4 c^3 x-\frac {2}{3} a^3 c^3 \arctan (a x)-\frac {a c^3 \arctan (a x)}{3 x^2}-\frac {1}{3} a^5 c^3 x^2 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^2}{x}+3 a^4 c^3 x \arctan (a x)^2+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^2+\frac {16}{3} a^3 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+\frac {16}{3} a^3 c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {8}{3} i a^3 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {8}{3} i a^3 c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right ) \]
-1/3*a^2*c^3/x+1/3*a^4*c^3*x-2/3*a^3*c^3*arctan(a*x)-1/3*a*c^3*arctan(a*x) /x^2-1/3*a^5*c^3*x^2*arctan(a*x)-1/3*c^3*arctan(a*x)^2/x^3-3*a^2*c^3*arcta n(a*x)^2/x+3*a^4*c^3*x*arctan(a*x)^2+1/3*a^6*c^3*x^3*arctan(a*x)^2+16/3*a^ 3*c^3*arctan(a*x)*ln(2/(1+I*a*x))+16/3*a^3*c^3*arctan(a*x)*ln(2-2/(1-I*a*x ))-8/3*I*a^3*c^3*polylog(2,-1+2/(1-I*a*x))+8/3*I*a^3*c^3*polylog(2,1-2/(1+ I*a*x))
Time = 0.52 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^4} \, dx=\frac {c^3 \left (-a^2 x^2+a^4 x^4-a x \arctan (a x)-2 a^3 x^3 \arctan (a x)-a^5 x^5 \arctan (a x)-\arctan (a x)^2-9 a^2 x^2 \arctan (a x)^2-16 i a^3 x^3 \arctan (a x)^2+9 a^4 x^4 \arctan (a x)^2+a^6 x^6 \arctan (a x)^2+16 a^3 x^3 \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )+16 a^3 x^3 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-8 i a^3 x^3 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-8 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right )}{3 x^3} \]
(c^3*(-(a^2*x^2) + a^4*x^4 - a*x*ArcTan[a*x] - 2*a^3*x^3*ArcTan[a*x] - a^5 *x^5*ArcTan[a*x] - ArcTan[a*x]^2 - 9*a^2*x^2*ArcTan[a*x]^2 - (16*I)*a^3*x^ 3*ArcTan[a*x]^2 + 9*a^4*x^4*ArcTan[a*x]^2 + a^6*x^6*ArcTan[a*x]^2 + 16*a^3 *x^3*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 16*a^3*x^3*ArcTan[a*x]*L og[1 + E^((2*I)*ArcTan[a*x])] - (8*I)*a^3*x^3*PolyLog[2, -E^((2*I)*ArcTan[ a*x])] - (8*I)*a^3*x^3*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(3*x^3)
Time = 0.79 (sec) , antiderivative size = 250, 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.
| \(\displaystyle \int \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^3}{x^4} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^2 \arctan (a x)^2+3 a^4 c^3 \arctan (a x)^2+\frac {3 a^2 c^3 \arctan (a x)^2}{x^2}+\frac {c^3 \arctan (a x)^2}{x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} a^6 c^3 x^3 \arctan (a x)^2-\frac {1}{3} a^5 c^3 x^2 \arctan (a x)+3 a^4 c^3 x \arctan (a x)^2+\frac {1}{3} a^4 c^3 x-\frac {2}{3} a^3 c^3 \arctan (a x)+\frac {16}{3} a^3 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+\frac {16}{3} a^3 c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {8}{3} i a^3 c^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+\frac {8}{3} i a^3 c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )-\frac {3 a^2 c^3 \arctan (a x)^2}{x}-\frac {a^2 c^3}{3 x}-\frac {c^3 \arctan (a x)^2}{3 x^3}-\frac {a c^3 \arctan (a x)}{3 x^2}\) |
-1/3*(a^2*c^3)/x + (a^4*c^3*x)/3 - (2*a^3*c^3*ArcTan[a*x])/3 - (a*c^3*ArcT an[a*x])/(3*x^2) - (a^5*c^3*x^2*ArcTan[a*x])/3 - (c^3*ArcTan[a*x]^2)/(3*x^ 3) - (3*a^2*c^3*ArcTan[a*x]^2)/x + 3*a^4*c^3*x*ArcTan[a*x]^2 + (a^6*c^3*x^ 3*ArcTan[a*x]^2)/3 + (16*a^3*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/3 + (16*a ^3*c^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/3 - ((8*I)/3)*a^3*c^3*PolyLog[2 , -1 + 2/(1 - I*a*x)] + ((8*I)/3)*a^3*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)]
3.3.81.3.1 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 = 1.43 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.29
| method | result | size |
| derivativedivides | \(a^{3} \left (\frac {a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}}{3}+3 a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {c^{3} \arctan \left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {3 c^{3} \arctan \left (a x \right )^{2}}{a x}-\frac {2 c^{3} \left (\frac {a^{2} \arctan \left (a x \right ) x^{2}}{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}-8 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a x}{2}+\arctan \left (a x \right )+\frac {1}{2 a x}-4 i \ln \left (a x \right ) \ln \left (i a x +1\right )+4 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-4 i \operatorname {dilog}\left (i a x +1\right )+4 i \operatorname {dilog}\left (-i a x +1\right )+4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{3}\right )\) | \(322\) |
| default | \(a^{3} \left (\frac {a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}}{3}+3 a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {c^{3} \arctan \left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {3 c^{3} \arctan \left (a x \right )^{2}}{a x}-\frac {2 c^{3} \left (\frac {a^{2} \arctan \left (a x \right ) x^{2}}{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}-8 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a x}{2}+\arctan \left (a x \right )+\frac {1}{2 a x}-4 i \ln \left (a x \right ) \ln \left (i a x +1\right )+4 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-4 i \operatorname {dilog}\left (i a x +1\right )+4 i \operatorname {dilog}\left (-i a x +1\right )+4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{3}\right )\) | \(322\) |
| parts | \(\frac {a^{6} c^{3} x^{3} \arctan \left (a x \right )^{2}}{3}+3 a^{4} c^{3} x \arctan \left (a x \right )^{2}-\frac {3 a^{2} c^{3} \arctan \left (a x \right )^{2}}{x}-\frac {c^{3} \arctan \left (a x \right )^{2}}{3 x^{3}}-\frac {2 c^{3} \left (\frac {a^{5} \arctan \left (a x \right ) x^{2}}{2}+8 a^{3} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {a \arctan \left (a x \right )}{2 x^{2}}-8 a^{3} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a^{3} \left (a x -2 \arctan \left (a x \right )-\frac {1}{a x}+8 i \ln \left (a x \right ) \ln \left (i a x +1\right )-8 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+8 i \operatorname {dilog}\left (i a x +1\right )-8 i \operatorname {dilog}\left (-i a x +1\right )-8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{2}\right )}{3}\) | \(328\) |
a^3*(1/3*a^3*c^3*x^3*arctan(a*x)^2+3*a*c^3*x*arctan(a*x)^2-1/3*c^3*arctan( a*x)^2/a^3/x^3-3*c^3*arctan(a*x)^2/a/x-2/3*c^3*(1/2*a^2*arctan(a*x)*x^2+8* arctan(a*x)*ln(a^2*x^2+1)+1/2*arctan(a*x)/a^2/x^2-8*arctan(a*x)*ln(a*x)-1/ 2*a*x+arctan(a*x)+1/2/a/x-4*I*ln(a*x)*ln(1+I*a*x)+4*I*ln(a*x)*ln(1-I*a*x)- 4*I*dilog(1+I*a*x)+4*I*dilog(1-I*a*x)+4*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)-4*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)))
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x^{4}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^4} \, dx=c^{3} \left (\int 3 a^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{4}}\, dx + \int \frac {3 a^{2} \operatorname {atan}^{2}{\left (a x \right )}}{x^{2}}\, dx + \int a^{6} x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
c**3*(Integral(3*a**4*atan(a*x)**2, x) + Integral(atan(a*x)**2/x**4, x) + Integral(3*a**2*atan(a*x)**2/x**2, x) + Integral(a**6*x**2*atan(a*x)**2, x ))
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x^{4}} \,d x } \]
1/48*(24*(72*a^8*c^3*integrate(1/48*x^8*arctan(a*x)^2/(a^2*x^6 + x^4), x) + 6*a^8*c^3*integrate(1/48*x^8*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 8* a^8*c^3*integrate(1/48*x^8*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) - 16*a^7*c ^3*integrate(1/48*x^7*arctan(a*x)/(a^2*x^6 + x^4), x) + 288*a^6*c^3*integr ate(1/48*x^6*arctan(a*x)^2/(a^2*x^6 + x^4), x) + 24*a^6*c^3*integrate(1/48 *x^6*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 72*a^6*c^3*integrate(1/48*x^ 6*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 3*a^3*c^3*arctan(a*x)^3 - 144*a^5 *c^3*integrate(1/48*x^5*arctan(a*x)/(a^2*x^6 + x^4), x) + 36*a^4*c^3*integ rate(1/48*x^4*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) - 72*a^4*c^3*integrat e(1/48*x^4*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 144*a^3*c^3*integrate(1/ 48*x^3*arctan(a*x)/(a^2*x^6 + x^4), x) + 288*a^2*c^3*integrate(1/48*x^2*ar ctan(a*x)^2/(a^2*x^6 + x^4), x) + 24*a^2*c^3*integrate(1/48*x^2*log(a^2*x^ 2 + 1)^2/(a^2*x^6 + x^4), x) - 8*a^2*c^3*integrate(1/48*x^2*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 16*a*c^3*integrate(1/48*x*arctan(a*x)/(a^2*x^6 + x^4), x) + 72*c^3*integrate(1/48*arctan(a*x)^2/(a^2*x^6 + x^4), x) + 6*c^3 *integrate(1/48*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x))*x^3 + 4*(a^6*c^3*x ^6 + 9*a^4*c^3*x^4 - 9*a^2*c^3*x^2 - c^3)*arctan(a*x)^2 - (a^6*c^3*x^6 + 9 *a^4*c^3*x^4 - 9*a^2*c^3*x^2 - c^3)*log(a^2*x^2 + 1)^2)/x^3
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3}{x^4} \,d x \]