Integrand size = 22, antiderivative size = 336 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=-\frac {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+a^3 c^3 \log (x)+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-a^3 c^3 \log \left (1+a^2 x^2\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \]
-a^2*c^3*arctan(a*x)/x+a^4*c^3*x*arctan(a*x)-a^3*c^3*arctan(a*x)^2-1/2*a*c ^3*arctan(a*x)^2/x^2-1/2*a^5*c^3*x^2*arctan(a*x)^2-1/3*c^3*arctan(a*x)^3/x ^3-3*a^2*c^3*arctan(a*x)^3/x+3*a^4*c^3*x*arctan(a*x)^3+1/3*a^6*c^3*x^3*arc tan(a*x)^3+a^3*c^3*ln(x)+8*a^3*c^3*arctan(a*x)^2*ln(2/(1+I*a*x))-a^3*c^3*l n(a^2*x^2+1)+8*a^3*c^3*arctan(a*x)^2*ln(2-2/(1-I*a*x))+8*I*a^3*c^3*arctan( a*x)*polylog(2,1-2/(1+I*a*x))-8*I*a^3*c^3*arctan(a*x)*polylog(2,-1+2/(1-I* a*x))+4*a^3*c^3*polylog(3,-1+2/(1-I*a*x))+4*a^3*c^3*polylog(3,1-2/(1+I*a*x ))
Time = 0.66 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\frac {c^3 \left (-2 i a^3 \pi ^3 x^3-6 a^2 x^2 \arctan (a x)+6 a^4 x^4 \arctan (a x)-3 a x \arctan (a x)^2-6 a^3 x^3 \arctan (a x)^2-3 a^5 x^5 \arctan (a x)^2-2 \arctan (a x)^3-18 a^2 x^2 \arctan (a x)^3+18 a^4 x^4 \arctan (a x)^3+2 a^6 x^6 \arctan (a x)^3+48 a^3 x^3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+48 a^3 x^3 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+6 a^3 x^3 \log (a x)-6 a^3 x^3 \log \left (1+a^2 x^2\right )+48 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-48 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+24 a^3 x^3 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+24 a^3 x^3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{6 x^3} \]
(c^3*((-2*I)*a^3*Pi^3*x^3 - 6*a^2*x^2*ArcTan[a*x] + 6*a^4*x^4*ArcTan[a*x] - 3*a*x*ArcTan[a*x]^2 - 6*a^3*x^3*ArcTan[a*x]^2 - 3*a^5*x^5*ArcTan[a*x]^2 - 2*ArcTan[a*x]^3 - 18*a^2*x^2*ArcTan[a*x]^3 + 18*a^4*x^4*ArcTan[a*x]^3 + 2*a^6*x^6*ArcTan[a*x]^3 + 48*a^3*x^3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTa n[a*x])] + 48*a^3*x^3*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 6*a^3 *x^3*Log[a*x] - 6*a^3*x^3*Log[1 + a^2*x^2] + (48*I)*a^3*x^3*ArcTan[a*x]*Po lyLog[2, E^((-2*I)*ArcTan[a*x])] - (48*I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, - E^((2*I)*ArcTan[a*x])] + 24*a^3*x^3*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 2 4*a^3*x^3*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(6*x^3)
Time = 1.30 (sec) , antiderivative size = 336, 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.
| \(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^3}{x^4} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^2 \arctan (a x)^3+3 a^4 c^3 \arctan (a x)^3+\frac {3 a^2 c^3 \arctan (a x)^3}{x^2}+\frac {c^3 \arctan (a x)^3}{x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2+3 a^4 c^3 x \arctan (a x)^3+a^4 c^3 x \arctan (a x)-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )-a^3 c^3 \arctan (a x)^2+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+a^3 c^3 \log (x)-\frac {3 a^2 c^3 \arctan (a x)^3}{x}-\frac {a^2 c^3 \arctan (a x)}{x}-a^3 c^3 \log \left (a^2 x^2+1\right )-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {a c^3 \arctan (a x)^2}{2 x^2}\) |
-((a^2*c^3*ArcTan[a*x])/x) + a^4*c^3*x*ArcTan[a*x] - a^3*c^3*ArcTan[a*x]^2 - (a*c^3*ArcTan[a*x]^2)/(2*x^2) - (a^5*c^3*x^2*ArcTan[a*x]^2)/2 - (c^3*Ar cTan[a*x]^3)/(3*x^3) - (3*a^2*c^3*ArcTan[a*x]^3)/x + 3*a^4*c^3*x*ArcTan[a* x]^3 + (a^6*c^3*x^3*ArcTan[a*x]^3)/3 + a^3*c^3*Log[x] + 8*a^3*c^3*ArcTan[a *x]^2*Log[2/(1 + I*a*x)] - a^3*c^3*Log[1 + a^2*x^2] + 8*a^3*c^3*ArcTan[a*x ]^2*Log[2 - 2/(1 - I*a*x)] - (8*I)*a^3*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/( 1 - I*a*x)] + (8*I)*a^3*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + 4* a^3*c^3*PolyLog[3, -1 + 2/(1 - I*a*x)] + 4*a^3*c^3*PolyLog[3, 1 - 2/(1 + I *a*x)]
3.4.86.3.1 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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 138.35 (sec) , antiderivative size = 1948, normalized size of antiderivative = 5.80
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1948\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1949\) |
| default | \(\text {Expression too large to display}\) | \(1949\) |
1/3*a^6*c^3*x^3*arctan(a*x)^3+3*a^4*c^3*x*arctan(a*x)^3-3*a^2*c^3*arctan(a *x)^3/x-1/3*c^3*arctan(a*x)^3/x^3-c^3*(1/2*a^5*arctan(a*x)^2*x^2+1/2*a*arc tan(a*x)^2/x^2-8*a^3*arctan(a*x)^2*ln(a*x)+8*a^3*arctan(a*x)^2*ln(a^2*x^2+ 1)-a^3*(16*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-8*arctan(a*x)^2*l n((1+I*a*x)^2/(a^2*x^2+1)-1)+8*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2 )+1)-16*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+16*polylog(3 ,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+8*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^ (1/2))-16*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+16*polylog( 3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-8*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2 *x^2+1))+4*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-1/3*arctan(a*x)*(3-12*I*arc tan(a*x)*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^ 3*a*x-12*I*arctan(a*x)*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/ (a^2*x^2+1)+1))^3*a*x+12*I*arctan(a*x)*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1) /((1+I*a*x)^2/(a^2*x^2+1)+1))^2*a*x+12*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/(( 1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*Pi*arctan(a*x)*a*x-12*I*csgn(I*((1+I*a*x)^2 /(a^2*x^2+1)+1)^2)^3*Pi*arctan(a*x)*a*x+12*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1 ))^3*Pi*arctan(a*x)*a*x+6*I*a*x-3*a^2*x^2+3*x*arctan(a*x)*a-12*I*csgn(I*(1 +I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^ 2+1)+1)^2)^2*Pi*arctan(a*x)*a*x-12*I*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2) *csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*Pi*arc...
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=c^{3} \left (\int 3 a^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{4}}\, dx + \int \frac {3 a^{2} \operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx + \int a^{6} x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**3*(Integral(3*a**4*atan(a*x)**3, x) + Integral(atan(a*x)**3/x**4, x) + Integral(3*a**2*atan(a*x)**3/x**2, x) + Integral(a**6*x**2*atan(a*x)**3, x ))
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]
1/192*(3*(1792*a^8*c^3*integrate(1/32*x^8*arctan(a*x)^3/(a^2*x^6 + x^4), x ) + 192*a^8*c^3*integrate(1/32*x^8*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 256*a^8*c^3*integrate(1/32*x^8*arctan(a*x)*log(a^2*x^2 + 1)/ (a^2*x^6 + x^4), x) - 256*a^7*c^3*integrate(1/32*x^7*arctan(a*x)^2/(a^2*x^ 6 + x^4), x) + 64*a^7*c^3*integrate(1/32*x^7*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 84*a^3*c^3*arctan(a*x)^4 + 7168*a^6*c^3*integrate(1/32*x^6*arc tan(a*x)^3/(a^2*x^6 + x^4), x) + 768*a^6*c^3*integrate(1/32*x^6*arctan(a*x )*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 2304*a^6*c^3*integrate(1/32*x^6 *arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) - 2304*a^5*c^3*integrate (1/32*x^5*arctan(a*x)^2/(a^2*x^6 + x^4), x) + 3*a^3*c^3*log(a^2*x^2 + 1)^3 + 1152*a^4*c^3*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) - 2304*a^4*c^3*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1) /(a^2*x^6 + x^4), x) + 2304*a^3*c^3*integrate(1/32*x^3*arctan(a*x)^2/(a^2* x^6 + x^4), x) - 576*a^3*c^3*integrate(1/32*x^3*log(a^2*x^2 + 1)^2/(a^2*x^ 6 + x^4), x) + 7168*a^2*c^3*integrate(1/32*x^2*arctan(a*x)^3/(a^2*x^6 + x^ 4), x) + 768*a^2*c^3*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^ 2*x^6 + x^4), x) - 256*a^2*c^3*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 256*a*c^3*integrate(1/32*x*arctan(a*x)^2/(a^2*x ^6 + x^4), x) - 64*a*c^3*integrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^ 4), x) + 1792*c^3*integrate(1/32*arctan(a*x)^3/(a^2*x^6 + x^4), x) + 19...
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3}{x^4} \,d x \]