Integrand size = 22, antiderivative size = 311 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^4} \, dx=-\frac {a^2 c^2 \arctan (a x)}{x}-\frac {1}{2} a^3 c^2 \arctan (a x)^2-\frac {a c^2 \arctan (a x)^2}{2 x^2}-\frac {2}{3} i a^3 c^2 \arctan (a x)^3-\frac {c^2 \arctan (a x)^3}{3 x^3}-\frac {2 a^2 c^2 \arctan (a x)^3}{x}+a^4 c^2 x \arctan (a x)^3+a^3 c^2 \log (x)+3 a^3 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {1}{2} a^3 c^2 \log \left (1+a^2 x^2\right )+5 a^3 c^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-5 i a^3 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+3 i a^3 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {5}{2} a^3 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {3}{2} a^3 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \]
-a^2*c^2*arctan(a*x)/x-1/2*a^3*c^2*arctan(a*x)^2-1/2*a*c^2*arctan(a*x)^2/x ^2-2/3*I*a^3*c^2*arctan(a*x)^3-1/3*c^2*arctan(a*x)^3/x^3-2*a^2*c^2*arctan( a*x)^3/x+a^4*c^2*x*arctan(a*x)^3+a^3*c^2*ln(x)+3*a^3*c^2*arctan(a*x)^2*ln( 2/(1+I*a*x))-1/2*a^3*c^2*ln(a^2*x^2+1)+5*a^3*c^2*arctan(a*x)^2*ln(2-2/(1-I *a*x))-5*I*a^3*c^2*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))+3*I*a^3*c^2*arcta n(a*x)*polylog(2,1-2/(1+I*a*x))+5/2*a^3*c^2*polylog(3,-1+2/(1-I*a*x))+3/2* a^3*c^2*polylog(3,1-2/(1+I*a*x))
Time = 0.48 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^4} \, dx=\frac {c^2 \left (-5 i a^3 \pi ^3 x^3-24 a^2 x^2 \arctan (a x)-12 a x \arctan (a x)^2-12 a^3 x^3 \arctan (a x)^2-8 \arctan (a x)^3-48 a^2 x^2 \arctan (a x)^3+16 i a^3 x^3 \arctan (a x)^3+24 a^4 x^4 \arctan (a x)^3+120 a^3 x^3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+72 a^3 x^3 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+24 a^3 x^3 \log (a x)-12 a^3 x^3 \log \left (1+a^2 x^2\right )+120 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-72 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+60 a^3 x^3 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+36 a^3 x^3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{24 x^3} \]
(c^2*((-5*I)*a^3*Pi^3*x^3 - 24*a^2*x^2*ArcTan[a*x] - 12*a*x*ArcTan[a*x]^2 - 12*a^3*x^3*ArcTan[a*x]^2 - 8*ArcTan[a*x]^3 - 48*a^2*x^2*ArcTan[a*x]^3 + (16*I)*a^3*x^3*ArcTan[a*x]^3 + 24*a^4*x^4*ArcTan[a*x]^3 + 120*a^3*x^3*ArcT an[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + 72*a^3*x^3*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 24*a^3*x^3*Log[a*x] - 12*a^3*x^3*Log[1 + a^2*x ^2] + (120*I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (72 *I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 60*a^3*x^3*Po lyLog[3, E^((-2*I)*ArcTan[a*x])] + 36*a^3*x^3*PolyLog[3, -E^((2*I)*ArcTan[ a*x])]))/(24*x^3)
Time = 0.97 (sec) , antiderivative size = 311, 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.
| \(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^4 c^2 \arctan (a x)^3+\frac {2 a^2 c^2 \arctan (a x)^3}{x^2}+\frac {c^2 \arctan (a x)^3}{x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 c^2 x \arctan (a x)^3-5 i a^3 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+3 i a^3 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )-\frac {2}{3} i a^3 c^2 \arctan (a x)^3-\frac {1}{2} a^3 c^2 \arctan (a x)^2+3 a^3 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+5 a^3 c^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+\frac {5}{2} a^3 c^2 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+\frac {3}{2} a^3 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+a^3 c^2 \log (x)-\frac {2 a^2 c^2 \arctan (a x)^3}{x}-\frac {a^2 c^2 \arctan (a x)}{x}-\frac {1}{2} a^3 c^2 \log \left (a^2 x^2+1\right )-\frac {c^2 \arctan (a x)^3}{3 x^3}-\frac {a c^2 \arctan (a x)^2}{2 x^2}\) |
-((a^2*c^2*ArcTan[a*x])/x) - (a^3*c^2*ArcTan[a*x]^2)/2 - (a*c^2*ArcTan[a*x ]^2)/(2*x^2) - ((2*I)/3)*a^3*c^2*ArcTan[a*x]^3 - (c^2*ArcTan[a*x]^3)/(3*x^ 3) - (2*a^2*c^2*ArcTan[a*x]^3)/x + a^4*c^2*x*ArcTan[a*x]^3 + a^3*c^2*Log[x ] + 3*a^3*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] - (a^3*c^2*Log[1 + a^2*x^2] )/2 + 5*a^3*c^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (5*I)*a^3*c^2*ArcTa n[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (3*I)*a^3*c^2*ArcTan[a*x]*PolyLog[ 2, 1 - 2/(1 + I*a*x)] + (5*a^3*c^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (3* a^3*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/2
3.4.78.3.1 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 = 103.36 (sec) , antiderivative size = 1883, normalized size of antiderivative = 6.05
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1883\) |
| default | \(\text {Expression too large to display}\) | \(1883\) |
| parts | \(\text {Expression too large to display}\) | \(1884\) |
a^3*(c^2*arctan(a*x)^3*a*x-1/3*c^2*arctan(a*x)^3/a^3/x^3-2*c^2*arctan(a*x) ^3/a/x-c^2*(1/2*arctan(a*x)^2/a^2/x^2-5*arctan(a*x)^2*ln(a*x)+4*arctan(a*x )^2*ln(a^2*x^2+1)-8*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+1/6*arct an(a*x)*(12*I*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^ 2*x^2+1))*Pi*arctan(a*x)*a*x+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)^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)*Pi*arctan(a*x)*a*x+15*I*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)- 1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*Pi*a rctan(a*x)*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+6*I*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)+1)^2)*csgn (I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*Pi*arctan(a*x)*a*x+15*I*csgn(I*((1+I*a*x )^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2 +1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*Pi*arctan(a*x)*a*x-15*I*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*Pi*arctan(a*x)*a*x-1 5*I*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*Pi*arc tan(a*x)*a*x+15*I*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/ (a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^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*arctan(a*x)*a*x+16*I*arctan(a*x)^2*a*x-15*I*...
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^4} \, dx=c^{2} \left (\int a^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{4}}\, dx + \int \frac {2 a^{2} \operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx\right ) \]
c**2*(Integral(a**4*atan(a*x)**3, x) + Integral(atan(a*x)**3/x**4, x) + In tegral(2*a**2*atan(a*x)**3/x**2, x))
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]
1/192*(3*(42*a^3*c^2*arctan(a*x)^4 + 1792*a^6*c^2*integrate(1/32*x^6*arcta n(a*x)^3/(a^2*x^6 + x^4), x) + 192*a^6*c^2*integrate(1/32*x^6*arctan(a*x)* log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 768*a^6*c^2*integrate(1/32*x^6*ar ctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) - 768*a^5*c^2*integrate(1/3 2*x^5*arctan(a*x)^2/(a^2*x^6 + x^4), x) + a^3*c^2*log(a^2*x^2 + 1)^3 + 576 *a^4*c^2*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4) , x) - 1536*a^4*c^2*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x ^6 + x^4), x) + 1536*a^3*c^2*integrate(1/32*x^3*arctan(a*x)^2/(a^2*x^6 + x ^4), x) - 384*a^3*c^2*integrate(1/32*x^3*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4 ), x) + 5376*a^2*c^2*integrate(1/32*x^2*arctan(a*x)^3/(a^2*x^6 + x^4), x) + 576*a^2*c^2*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^2*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a ^2*x^6 + x^4), x) + 256*a*c^2*integrate(1/32*x*arctan(a*x)^2/(a^2*x^6 + x^ 4), x) - 64*a*c^2*integrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 1792*c^2*integrate(1/32*arctan(a*x)^3/(a^2*x^6 + x^4), x) + 192*c^2*inte grate(1/32*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x))*x^3 + 8*(3* a^4*c^2*x^4 - 6*a^2*c^2*x^2 - c^2)*arctan(a*x)^3 - 6*(3*a^4*c^2*x^4 - 6*a^ 2*c^2*x^2 - c^2)*arctan(a*x)*log(a^2*x^2 + 1)^2)/x^3
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2}{x^4} \,d x \]