Integrand size = 22, antiderivative size = 299 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^3} \, dx=\frac {1}{12} a^4 c^3 x^2-\frac {a c^3 \arctan (a x)}{x}-\frac {5}{2} a^3 c^3 x \arctan (a x)-\frac {1}{6} a^5 c^3 x^3 \arctan (a x)+\frac {3}{4} a^2 c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^2+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^2+6 a^2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+a^2 c^3 \log (x)+\frac {2}{3} a^2 c^3 \log \left (1+a^2 x^2\right )-3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \]
1/12*a^4*c^3*x^2-a*c^3*arctan(a*x)/x-5/2*a^3*c^3*x*arctan(a*x)-1/6*a^5*c^3 *x^3*arctan(a*x)+3/4*a^2*c^3*arctan(a*x)^2-1/2*c^3*arctan(a*x)^2/x^2+3/2*a ^4*c^3*x^2*arctan(a*x)^2+1/4*a^6*c^3*x^4*arctan(a*x)^2-6*a^2*c^3*arctan(a* x)^2*arctanh(-1+2/(1+I*a*x))+a^2*c^3*ln(x)+2/3*a^2*c^3*ln(a^2*x^2+1)-3*I*a ^2*c^3*arctan(a*x)*polylog(2,1-2/(1+I*a*x))+3*I*a^2*c^3*arctan(a*x)*polylo g(2,-1+2/(1+I*a*x))-3/2*a^2*c^3*polylog(3,1-2/(1+I*a*x))+3/2*a^2*c^3*polyl og(3,-1+2/(1+I*a*x))
Time = 0.32 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^3} \, dx=\frac {c^3 \left (2 a^2 x^2-3 i a^2 \pi ^3 x^2+2 a^4 x^4-24 a x \arctan (a x)-60 a^3 x^3 \arctan (a x)-4 a^5 x^5 \arctan (a x)-12 \arctan (a x)^2+18 a^2 x^2 \arctan (a x)^2+36 a^4 x^4 \arctan (a x)^2+6 a^6 x^6 \arctan (a x)^2+48 i a^2 x^2 \arctan (a x)^3+72 a^2 x^2 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )-72 a^2 x^2 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+24 a^2 x^2 \log \left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+28 a^2 x^2 \log \left (1+a^2 x^2\right )+72 i a^2 x^2 \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+72 i a^2 x^2 \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+36 a^2 x^2 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-36 a^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{24 x^2} \]
(c^3*(2*a^2*x^2 - (3*I)*a^2*Pi^3*x^2 + 2*a^4*x^4 - 24*a*x*ArcTan[a*x] - 60 *a^3*x^3*ArcTan[a*x] - 4*a^5*x^5*ArcTan[a*x] - 12*ArcTan[a*x]^2 + 18*a^2*x ^2*ArcTan[a*x]^2 + 36*a^4*x^4*ArcTan[a*x]^2 + 6*a^6*x^6*ArcTan[a*x]^2 + (4 8*I)*a^2*x^2*ArcTan[a*x]^3 + 72*a^2*x^2*ArcTan[a*x]^2*Log[1 - E^((-2*I)*Ar cTan[a*x])] - 72*a^2*x^2*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 24 *a^2*x^2*Log[(a*x)/Sqrt[1 + a^2*x^2]] + 28*a^2*x^2*Log[1 + a^2*x^2] + (72* I)*a^2*x^2*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (72*I)*a^2*x^2 *ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 36*a^2*x^2*PolyLog[3, E^ ((-2*I)*ArcTan[a*x])] - 36*a^2*x^2*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(2 4*x^2)
Time = 0.82 (sec) , antiderivative size = 299, 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.
| \(\displaystyle \int \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^3}{x^3} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^3 \arctan (a x)^2+3 a^4 c^3 x \arctan (a x)^2+\frac {3 a^2 c^3 \arctan (a x)^2}{x}+\frac {c^3 \arctan (a x)^2}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} a^6 c^3 x^4 \arctan (a x)^2-\frac {1}{6} a^5 c^3 x^3 \arctan (a x)+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^2+\frac {1}{12} a^4 c^3 x^2-\frac {5}{2} a^3 c^3 x \arctan (a x)+6 a^2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )+\frac {3}{4} a^2 c^3 \arctan (a x)^2-\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )+\frac {2}{3} a^2 c^3 \log \left (a^2 x^2+1\right )+a^2 c^3 \log (x)-\frac {c^3 \arctan (a x)^2}{2 x^2}-\frac {a c^3 \arctan (a x)}{x}\) |
(a^4*c^3*x^2)/12 - (a*c^3*ArcTan[a*x])/x - (5*a^3*c^3*x*ArcTan[a*x])/2 - ( a^5*c^3*x^3*ArcTan[a*x])/6 + (3*a^2*c^3*ArcTan[a*x]^2)/4 - (c^3*ArcTan[a*x ]^2)/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x]^2)/2 + (a^6*c^3*x^4*ArcTan[a*x]^ 2)/4 + 6*a^2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c^3*Log[x] + (2*a^2*c^3*Log[1 + a^2*x^2])/3 - (3*I)*a^2*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*I)*a^2*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x )] - (3*a^2*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*a^2*c^3*PolyLog[3, - 1 + 2/(1 + I*a*x)])/2
3.3.80.3.1 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 = 67.32 (sec) , antiderivative size = 1318, normalized size of antiderivative = 4.41
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1318\) |
| default | \(\text {Expression too large to display}\) | \(1318\) |
| parts | \(\text {Expression too large to display}\) | \(1754\) |
a^2*(1/4*a^4*c^3*x^4*arctan(a*x)^2+3/2*a^2*c^3*x^2*arctan(a*x)^2-1/2*c^3*a rctan(a*x)^2/a^2/x^2+3*c^3*arctan(a*x)^2*ln(a*x)-1/2*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)^2-3/2*ar ctan(a*x)^2+1/3*arctan(a*x)*(a*x-I)^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)+4*arctan(a*x)*(a*x-I)-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))-2*ln( (1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+3*polyl og(3,-(1+I*a*x)^2/(a^2*x^2+1))+14/3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+arctan(a *x)*(a*x-I)*(I+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))^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*arctan(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+arctan(a*x)*(I*a* x-(a^2*x^2+1)^(1/2)+1)/a/x-2*I*arctan(a*x)*(a*x-I)*(I+a*x)+arctan(a*x)*(I* a*x+(a^2*x^2+1)^(1/2)+1)/a/x+3*I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*cs gn(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)/((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*arctan(a* x)^2+I*arctan(a*x)*(a*x-I)^2-1/6*(I+a*x)^2-arctan(a*x)*(a*x-I)^2*(I+a*x...
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^3} \, dx=c^{3} \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {3 a^{2} \operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx + \int 3 a^{4} x \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{3} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
c**3*(Integral(atan(a*x)**2/x**3, x) + Integral(3*a**2*atan(a*x)**2/x, x) + Integral(3*a**4*x*atan(a*x)**2, x) + Integral(a**6*x**3*atan(a*x)**2, x) )
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
1/64*(4*(192*a^8*c^3*integrate(1/16*x^8*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 16*a^8*c^3*integrate(1/16*x^8*log(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x) + 1 6*a^8*c^3*integrate(1/16*x^8*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) - 32*a^7 *c^3*integrate(1/16*x^7*arctan(a*x)/(a^2*x^5 + x^3), x) + 768*a^6*c^3*inte grate(1/16*x^6*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 64*a^6*c^3*integrate(1/ 16*x^6*log(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x) + 96*a^6*c^3*integrate(1/16* x^6*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) - 192*a^5*c^3*integrate(1/16*x^5* arctan(a*x)/(a^2*x^5 + x^3), x) + 1152*a^4*c^3*integrate(1/16*x^4*arctan(a *x)^2/(a^2*x^5 + x^3), x) + a^2*c^3*log(a^2*x^2 + 1)^3 + 768*a^2*c^3*integ rate(1/16*x^2*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 64*a^2*c^3*integrate(1/1 6*x^2*log(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x) - 32*a^2*c^3*integrate(1/16*x ^2*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) + 64*a*c^3*integrate(1/16*x*arctan (a*x)/(a^2*x^5 + x^3), x) + 192*c^3*integrate(1/16*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 16*c^3*integrate(1/16*log(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x)) *x^2 + 4*(a^6*c^3*x^6 + 6*a^4*c^3*x^4 - 2*c^3)*arctan(a*x)^2 - (a^6*c^3*x^ 6 + 6*a^4*c^3*x^4 - 2*c^3)*log(a^2*x^2 + 1)^2)/x^2
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3}{x^3} \,d x \]