Integrand size = 22, antiderivative size = 354 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^2} \, dx=-\frac {1}{20} a^3 c^3 x^2+\frac {21}{10} a^2 c^3 x \arctan (a x)+\frac {1}{10} a^4 c^3 x^3 \arctan (a x)-\frac {21}{20} a c^3 \arctan (a x)^2-\frac {6}{5} a^3 c^3 x^2 \arctan (a x)^2-\frac {3}{20} a^5 c^3 x^4 \arctan (a x)^2+\frac {6}{5} i a c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{x}+3 a^2 c^3 x \arctan (a x)^3+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3+\frac {33}{5} a c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {33}{5} i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {33}{10} a c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \]
-1/20*a^3*c^3*x^2+21/10*a^2*c^3*x*arctan(a*x)+1/10*a^4*c^3*x^3*arctan(a*x) -21/20*a*c^3*arctan(a*x)^2-6/5*a^3*c^3*x^2*arctan(a*x)^2-3/20*a^5*c^3*x^4* arctan(a*x)^2-3*I*a*c^3*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))-c^3*arctan(a *x)^3/x+3*a^2*c^3*x*arctan(a*x)^3+a^4*c^3*x^3*arctan(a*x)^3+1/5*a^6*c^3*x^ 5*arctan(a*x)^3+33/5*a*c^3*arctan(a*x)^2*ln(2/(1+I*a*x))-a*c^3*ln(a^2*x^2+ 1)+3*a*c^3*arctan(a*x)^2*ln(2-2/(1-I*a*x))+33/5*I*a*c^3*arctan(a*x)*polylo g(2,1-2/(1+I*a*x))+6/5*I*a*c^3*arctan(a*x)^3+3/2*a*c^3*polylog(3,-1+2/(1-I *a*x))+33/10*a*c^3*polylog(3,1-2/(1+I*a*x))
Time = 0.60 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.84 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^2} \, dx=\frac {c^3 \left (-2 a x-5 i a \pi ^3 x-2 a^3 x^3+84 a^2 x^2 \arctan (a x)+4 a^4 x^4 \arctan (a x)-42 a x \arctan (a x)^2-48 a^3 x^3 \arctan (a x)^2-6 a^5 x^5 \arctan (a x)^2-40 \arctan (a x)^3-48 i a x \arctan (a x)^3+120 a^2 x^2 \arctan (a x)^3+40 a^4 x^4 \arctan (a x)^3+8 a^6 x^6 \arctan (a x)^3+120 a x \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+264 a x \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-40 a x \log \left (1+a^2 x^2\right )+120 i a x \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-264 i a x \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+60 a x \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+132 a x \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{40 x} \]
(c^3*(-2*a*x - (5*I)*a*Pi^3*x - 2*a^3*x^3 + 84*a^2*x^2*ArcTan[a*x] + 4*a^4 *x^4*ArcTan[a*x] - 42*a*x*ArcTan[a*x]^2 - 48*a^3*x^3*ArcTan[a*x]^2 - 6*a^5 *x^5*ArcTan[a*x]^2 - 40*ArcTan[a*x]^3 - (48*I)*a*x*ArcTan[a*x]^3 + 120*a^2 *x^2*ArcTan[a*x]^3 + 40*a^4*x^4*ArcTan[a*x]^3 + 8*a^6*x^6*ArcTan[a*x]^3 + 120*a*x*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + 264*a*x*ArcTan[a*x ]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 40*a*x*Log[1 + a^2*x^2] + (120*I)*a*x *ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (264*I)*a*x*ArcTan[a*x]* PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 60*a*x*PolyLog[3, E^((-2*I)*ArcTan[a* x])] + 132*a*x*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(40*x)
Time = 1.43 (sec) , antiderivative size = 354, 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.
| \(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^3}{x^2} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^4 \arctan (a x)^3+3 a^4 c^3 x^2 \arctan (a x)^3+3 a^2 c^3 \arctan (a x)^3+\frac {c^3 \arctan (a x)^3}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3-\frac {3}{20} a^5 c^3 x^4 \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{10} a^4 c^3 x^3 \arctan (a x)-\frac {6}{5} a^3 c^3 x^2 \arctan (a x)^2-\frac {1}{20} a^3 c^3 x^2+3 a^2 c^3 x \arctan (a x)^3+\frac {21}{10} a^2 c^3 x \arctan (a x)-a c^3 \log \left (a^2 x^2+1\right )-3 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+\frac {33}{5} i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {6}{5} i a c^3 \arctan (a x)^3-\frac {21}{20} a c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{x}+\frac {33}{5} a c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} a c^3 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+\frac {33}{10} a c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )\) |
-1/20*(a^3*c^3*x^2) + (21*a^2*c^3*x*ArcTan[a*x])/10 + (a^4*c^3*x^3*ArcTan[ a*x])/10 - (21*a*c^3*ArcTan[a*x]^2)/20 - (6*a^3*c^3*x^2*ArcTan[a*x]^2)/5 - (3*a^5*c^3*x^4*ArcTan[a*x]^2)/20 + ((6*I)/5)*a*c^3*ArcTan[a*x]^3 - (c^3*A rcTan[a*x]^3)/x + 3*a^2*c^3*x*ArcTan[a*x]^3 + a^4*c^3*x^3*ArcTan[a*x]^3 + (a^6*c^3*x^5*ArcTan[a*x]^3)/5 + (33*a*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] )/5 - a*c^3*Log[1 + a^2*x^2] + 3*a*c^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x) ] - (3*I)*a*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((33*I)/5)*a* c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*a*c^3*PolyLog[3, -1 + 2 /(1 - I*a*x)])/2 + (33*a*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/10
3.4.84.3.1 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 = 163.40 (sec) , antiderivative size = 1894, normalized size of antiderivative = 5.35
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1894\) |
| default | \(\text {Expression too large to display}\) | \(1894\) |
| parts | \(\text {Expression too large to display}\) | \(1897\) |
a*(1/5*c^3*arctan(a*x)^3*a^5*x^5+c^3*arctan(a*x)^3*a^3*x^3+3*c^3*arctan(a* x)^3*a*x-c^3*arctan(a*x)^3/a/x-3/5*c^3*(1/4*a^4*arctan(a*x)^2*x^4+2*x^2*ar ctan(a*x)^2*a^2+8*arctan(a*x)^2*ln(a^2*x^2+1)-5*arctan(a*x)^2*ln(a*x)-16*a rctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+10*I*arctan(a*x)*polylog(2,(1 +I*a*x)/(a^2*x^2+1)^(1/2))-10/3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+5*arctan(a*x )^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-5*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1) ^(1/2)+1)-1/12*I*(I-64*arctan(a*x)^3+I*a^2*x^2-40*arctan(a*x)-48*csgn(I*(1 +I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*arctan(a*x)^2 *Pi+48*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*arctan(a*x)^2*Pi+48*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*arctan(a*x)^2*Pi-96*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*((1+ I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2*Pi+48*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*arctan(a*x)^2*Pi+96*csgn(I* (1+I*a*x)^2/(a^2*x^2+1))^2*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x) ^2*Pi-48*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)*arctan( a*x)^2*Pi+30*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(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)) *arctan(a*x)^2*Pi+30*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2...
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^2} \, dx=c^{3} \left (\int 3 a^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx + \int 3 a^{4} x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**3*(Integral(3*a**2*atan(a*x)**3, x) + Integral(atan(a*x)**3/x**2, x) + Integral(3*a**4*x**2*atan(a*x)**3, x) + Integral(a**6*x**4*atan(a*x)**3, x ))
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \]
1/320*(8*(a^6*c^3*x^6 + 5*a^4*c^3*x^4 + 15*a^2*c^3*x^2 - 5*c^3)*arctan(a*x )^3 - 6*(a^6*c^3*x^6 + 5*a^4*c^3*x^4 + 15*a^2*c^3*x^2 - 5*c^3)*arctan(a*x) *log(a^2*x^2 + 1)^2 + 5*(8960*a^8*c^3*integrate(1/160*x^8*arctan(a*x)^3/(a ^2*x^4 + x^2), x) + 960*a^8*c^3*integrate(1/160*x^8*arctan(a*x)*log(a^2*x^ 2 + 1)^2/(a^2*x^4 + x^2), x) + 768*a^8*c^3*integrate(1/160*x^8*arctan(a*x) *log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 768*a^7*c^3*integrate(1/160*x^7*ar ctan(a*x)^2/(a^2*x^4 + x^2), x) + 192*a^7*c^3*integrate(1/160*x^7*log(a^2* x^2 + 1)^2/(a^2*x^4 + x^2), x) + 35840*a^6*c^3*integrate(1/160*x^6*arctan( a*x)^3/(a^2*x^4 + x^2), x) + 3840*a^6*c^3*integrate(1/160*x^6*arctan(a*x)* log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 3840*a^6*c^3*integrate(1/160*x^6* arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 3840*a^5*c^3*integrate( 1/160*x^5*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 960*a^5*c^3*integrate(1/160* x^5*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 56*a*c^3*arctan(a*x)^4 + 5376 0*a^4*c^3*integrate(1/160*x^4*arctan(a*x)^3/(a^2*x^4 + x^2), x) + 5760*a^4 *c^3*integrate(1/160*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x ) + 11520*a^4*c^3*integrate(1/160*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^ 4 + x^2), x) - 11520*a^3*c^3*integrate(1/160*x^3*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 3*a*c^3*log(a^2*x^2 + 1)^3 + 3840*a^2*c^3*integrate(1/160*x^2*a rctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 3840*a^2*c^3*integrate (1/160*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + 3840*a*c^...
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3}{x^2} \,d x \]