Integrand size = 22, antiderivative size = 287 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=\frac {29}{180} a^2 c^3 x^2+\frac {1}{60} a^4 c^3 x^4-\frac {11}{6} a c^3 x \arctan (a x)-\frac {7}{18} a^3 c^3 x^3 \arctan (a x)-\frac {1}{15} a^5 c^3 x^5 \arctan (a x)+\frac {11}{12} c^3 \arctan (a x)^2+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {34}{45} c^3 \log \left (1+a^2 x^2\right )-i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \]
29/180*a^2*c^3*x^2+1/60*a^4*c^3*x^4-11/6*a*c^3*x*arctan(a*x)-7/18*a^3*c^3* x^3*arctan(a*x)-1/15*a^5*c^3*x^5*arctan(a*x)+11/12*c^3*arctan(a*x)^2+3/2*a ^2*c^3*x^2*arctan(a*x)^2+3/4*a^4*c^3*x^4*arctan(a*x)^2+1/6*a^6*c^3*x^6*arc tan(a*x)^2-2*c^3*arctan(a*x)^2*arctanh(-1+2/(1+I*a*x))+34/45*c^3*ln(a^2*x^ 2+1)-I*c^3*arctan(a*x)*polylog(2,1-2/(1+I*a*x))+I*c^3*arctan(a*x)*polylog( 2,-1+2/(1+I*a*x))-1/2*c^3*polylog(3,1-2/(1+I*a*x))+1/2*c^3*polylog(3,-1+2/ (1+I*a*x))
Time = 0.46 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=\frac {1}{360} c^3 \left (52-15 i \pi ^3+58 a^2 x^2+6 a^4 x^4-660 a x \arctan (a x)-140 a^3 x^3 \arctan (a x)-24 a^5 x^5 \arctan (a x)+330 \arctan (a x)^2+540 a^2 x^2 \arctan (a x)^2+270 a^4 x^4 \arctan (a x)^2+60 a^6 x^6 \arctan (a x)^2+240 i \arctan (a x)^3+360 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )-360 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+272 \log \left (1+a^2 x^2\right )+360 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+360 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+180 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-180 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right ) \]
(c^3*(52 - (15*I)*Pi^3 + 58*a^2*x^2 + 6*a^4*x^4 - 660*a*x*ArcTan[a*x] - 14 0*a^3*x^3*ArcTan[a*x] - 24*a^5*x^5*ArcTan[a*x] + 330*ArcTan[a*x]^2 + 540*a ^2*x^2*ArcTan[a*x]^2 + 270*a^4*x^4*ArcTan[a*x]^2 + 60*a^6*x^6*ArcTan[a*x]^ 2 + (240*I)*ArcTan[a*x]^3 + 360*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x ])] - 360*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 272*Log[1 + a^2*x ^2] + (360*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (360*I)*Arc Tan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 180*PolyLog[3, E^((-2*I)*Arc Tan[a*x])] - 180*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/360
Time = 0.93 (sec) , antiderivative size = 287, 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.
| \(\displaystyle \int \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^3}{x} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^5 \arctan (a x)^2+3 a^4 c^3 x^3 \arctan (a x)^2+3 a^2 c^3 x \arctan (a x)^2+\frac {c^3 \arctan (a x)^2}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2-\frac {1}{15} a^5 c^3 x^5 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{60} a^4 c^3 x^4-\frac {7}{18} a^3 c^3 x^3 \arctan (a x)+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {29}{180} a^2 c^3 x^2+\frac {34}{45} c^3 \log \left (a^2 x^2+1\right )+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-\frac {11}{6} a c^3 x \arctan (a x)+\frac {11}{12} c^3 \arctan (a x)^2-\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )\) |
(29*a^2*c^3*x^2)/180 + (a^4*c^3*x^4)/60 - (11*a*c^3*x*ArcTan[a*x])/6 - (7* a^3*c^3*x^3*ArcTan[a*x])/18 - (a^5*c^3*x^5*ArcTan[a*x])/15 + (11*c^3*ArcTa n[a*x]^2)/12 + (3*a^2*c^3*x^2*ArcTan[a*x]^2)/2 + (3*a^4*c^3*x^4*ArcTan[a*x ]^2)/4 + (a^6*c^3*x^6*ArcTan[a*x]^2)/6 + 2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2 /(1 + I*a*x)] + (34*c^3*Log[1 + a^2*x^2])/45 - I*c^3*ArcTan[a*x]*PolyLog[2 , 1 - 2/(1 + I*a*x)] + I*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (c^3*PolyLog[3, -1 + 2/(1 + I*a*x) ])/2
3.3.78.3.1 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 = 68.15 (sec) , antiderivative size = 1405, normalized size of antiderivative = 4.90
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1405\) |
| default | \(\text {Expression too large to display}\) | \(1405\) |
| parts | \(\text {Expression too large to display}\) | \(1948\) |
1/6*a^6*c^3*x^6*arctan(a*x)^2+3/4*a^4*c^3*x^4*arctan(a*x)^2+3/2*a^2*c^3*x^ 2*arctan(a*x)^2+c^3*arctan(a*x)^2*ln(a*x)-1/6*c^3*(-3*I*Pi*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))*csgn(I*((1+I*a*x) ^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2+2*arctan(a*x) *(a*x-I)*(I+a*x)^4+3*I*arctan(a*x)*(a*x-I)^2+4*arctan(a*x)*(a*x-I)^3*(I+a* x)^2-11/2*arctan(a*x)^2-5/3*arctan(a*x)*(a*x-I)^3+23/15*I*(I+a*x)+2/5*I*(I +a*x)^3+6*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-6*arctan(a*x)^2*ln(1 -(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/ 2)+1)+6*arctan(a*x)*(a*x-I)+2/5*arctan(a*x)*(a*x-I)^5-12*polylog(3,-(1+I*a *x)/(a^2*x^2+1)^(1/2))-12*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/10*(I+a *x)^4-4*arctan(a*x)*(a*x-I)^2*(I+a*x)^3+3*polylog(3,-(1+I*a*x)^2/(a^2*x^2+ 1))+136/15*ln((1+I*a*x)^2/(a^2*x^2+1)+1)-2*arctan(a*x)*(a*x-I)^4*(I+a*x)-5 *arctan(a*x)*(a*x-I)*(I+a*x)^2-8*I*arctan(a*x)*(a*x-I)*(I+a*x)^3+3*I*Pi*cs gn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^ 2-3*I*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 rctan(a*x)^2-3*I*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*arctan(a*x)^2-8*I*arctan(a*x)*(a*x-I)^3*(I+a*x)-6*I*arctan(a*x )*(a*x-I)*(I+a*x)+12*I*arctan(a*x)*(a*x-I)^2*(I+a*x)^2+3*I*Pi*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*arctan(a*x)^2+3*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)...
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=c^{3} \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
c**3*(Integral(atan(a*x)**2/x, x) + Integral(3*a**2*x*atan(a*x)**2, x) + I ntegral(3*a**4*x**3*atan(a*x)**2, x) + Integral(a**6*x**5*atan(a*x)**2, x) )
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x} \,d x } \]
36*a^8*c^3*integrate(1/48*x^8*arctan(a*x)^2/(a^2*x^3 + x), x) + 3*a^8*c^3* integrate(1/48*x^8*log(a^2*x^2 + 1)^2/(a^2*x^3 + x), x) + 2*a^8*c^3*integr ate(1/48*x^8*log(a^2*x^2 + 1)/(a^2*x^3 + x), x) - 4*a^7*c^3*integrate(1/48 *x^7*arctan(a*x)/(a^2*x^3 + x), x) + 144*a^6*c^3*integrate(1/48*x^6*arctan (a*x)^2/(a^2*x^3 + x), x) + 12*a^6*c^3*integrate(1/48*x^6*log(a^2*x^2 + 1) ^2/(a^2*x^3 + x), x) + 9*a^6*c^3*integrate(1/48*x^6*log(a^2*x^2 + 1)/(a^2* x^3 + x), x) - 18*a^5*c^3*integrate(1/48*x^5*arctan(a*x)/(a^2*x^3 + x), x) + 216*a^4*c^3*integrate(1/48*x^4*arctan(a*x)^2/(a^2*x^3 + x), x) + 18*a^4 *c^3*integrate(1/48*x^4*log(a^2*x^2 + 1)^2/(a^2*x^3 + x), x) + 18*a^4*c^3* integrate(1/48*x^4*log(a^2*x^2 + 1)/(a^2*x^3 + x), x) - 36*a^3*c^3*integra te(1/48*x^3*arctan(a*x)/(a^2*x^3 + x), x) + 144*a^2*c^3*integrate(1/48*x^2 *arctan(a*x)^2/(a^2*x^3 + x), x) + 1/24*c^3*log(a^2*x^2 + 1)^3 + 36*c^3*in tegrate(1/48*arctan(a*x)^2/(a^2*x^3 + x), x) + 3*c^3*integrate(1/48*log(a^ 2*x^2 + 1)^2/(a^2*x^3 + x), x) + 1/48*(2*a^6*c^3*x^6 + 9*a^4*c^3*x^4 + 18* a^2*c^3*x^2)*arctan(a*x)^2 - 1/192*(2*a^6*c^3*x^6 + 9*a^4*c^3*x^4 + 18*a^2 *c^3*x^2)*log(a^2*x^2 + 1)^2
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3}{x} \,d x \]