Integrand size = 22, antiderivative size = 284 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=a^2 c^2 x \arctan (a x)-\frac {1}{2} a c^2 \arctan (a x)^2-\frac {1}{2} a^3 c^2 x^2 \arctan (a x)^2+\frac {2}{3} i a c^2 \arctan (a x)^3-\frac {c^2 \arctan (a x)^3}{x}+2 a^2 c^2 x \arctan (a x)^3+\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^3+5 a c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {1}{2} a c^2 \log \left (1+a^2 x^2\right )+3 a c^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+5 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {5}{2} a c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \]
a^2*c^2*x*arctan(a*x)-1/2*a*c^2*arctan(a*x)^2-1/2*a^3*c^2*x^2*arctan(a*x)^ 2+2/3*I*a*c^2*arctan(a*x)^3-c^2*arctan(a*x)^3/x+2*a^2*c^2*x*arctan(a*x)^3+ 1/3*a^4*c^2*x^3*arctan(a*x)^3+5*a*c^2*arctan(a*x)^2*ln(2/(1+I*a*x))-1/2*a* c^2*ln(a^2*x^2+1)+3*a*c^2*arctan(a*x)^2*ln(2-2/(1-I*a*x))-3*I*a*c^2*arctan (a*x)*polylog(2,-1+2/(1-I*a*x))+5*I*a*c^2*arctan(a*x)*polylog(2,1-2/(1+I*a *x))+3/2*a*c^2*polylog(3,-1+2/(1-I*a*x))+5/2*a*c^2*polylog(3,1-2/(1+I*a*x) )
Time = 0.32 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=\frac {c^2 \left (-3 i a \pi ^3 x+24 a^2 x^2 \arctan (a x)-12 a x \arctan (a x)^2-12 a^3 x^3 \arctan (a x)^2-24 \arctan (a x)^3-16 i a x \arctan (a x)^3+48 a^2 x^2 \arctan (a x)^3+8 a^4 x^4 \arctan (a x)^3+72 a x \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+120 a x \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-12 a x \log \left (1+a^2 x^2\right )+72 i a x \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-120 i a x \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+36 a x \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+60 a x \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{24 x} \]
(c^2*((-3*I)*a*Pi^3*x + 24*a^2*x^2*ArcTan[a*x] - 12*a*x*ArcTan[a*x]^2 - 12 *a^3*x^3*ArcTan[a*x]^2 - 24*ArcTan[a*x]^3 - (16*I)*a*x*ArcTan[a*x]^3 + 48* a^2*x^2*ArcTan[a*x]^3 + 8*a^4*x^4*ArcTan[a*x]^3 + 72*a*x*ArcTan[a*x]^2*Log [1 - E^((-2*I)*ArcTan[a*x])] + 120*a*x*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcT an[a*x])] - 12*a*x*Log[1 + a^2*x^2] + (72*I)*a*x*ArcTan[a*x]*PolyLog[2, E^ ((-2*I)*ArcTan[a*x])] - (120*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTa n[a*x])] + 36*a*x*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 60*a*x*PolyLog[3, - E^((2*I)*ArcTan[a*x])]))/(24*x)
Time = 0.92 (sec) , antiderivative size = 284, 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.
\(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^2}{x^2} \, dx\) |
\(\Big \downarrow \) 5483 |
\(\displaystyle \int \left (a^4 c^2 x^2 \arctan (a x)^3+2 a^2 c^2 \arctan (a x)^3+\frac {c^2 \arctan (a x)^3}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} a^4 c^2 x^3 \arctan (a x)^3-\frac {1}{2} a^3 c^2 x^2 \arctan (a x)^2+2 a^2 c^2 x \arctan (a x)^3+a^2 c^2 x \arctan (a x)-\frac {1}{2} a c^2 \log \left (a^2 x^2+1\right )-3 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+5 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {2}{3} i a c^2 \arctan (a x)^3-\frac {1}{2} a c^2 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{x}+5 a c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} a c^2 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+\frac {5}{2} a c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )\) |
a^2*c^2*x*ArcTan[a*x] - (a*c^2*ArcTan[a*x]^2)/2 - (a^3*c^2*x^2*ArcTan[a*x] ^2)/2 + ((2*I)/3)*a*c^2*ArcTan[a*x]^3 - (c^2*ArcTan[a*x]^3)/x + 2*a^2*c^2* x*ArcTan[a*x]^3 + (a^4*c^2*x^3*ArcTan[a*x]^3)/3 + 5*a*c^2*ArcTan[a*x]^2*Lo g[2/(1 + I*a*x)] - (a*c^2*Log[1 + a^2*x^2])/2 + 3*a*c^2*ArcTan[a*x]^2*Log[ 2 - 2/(1 - I*a*x)] - (3*I)*a*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x) ] + (5*I)*a*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*a*c^2*PolyL og[3, -1 + 2/(1 - I*a*x)])/2 + (5*a*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/2
3.4.76.3.1 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 = 76.25 (sec) , antiderivative size = 1793, normalized size of antiderivative = 6.31
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1793\) |
default | \(\text {Expression too large to display}\) | \(1793\) |
parts | \(\text {Expression too large to display}\) | \(1794\) |
a*(1/3*c^2*arctan(a*x)^3*a^3*x^3+2*c^2*arctan(a*x)^3*a*x-c^2*arctan(a*x)^3 /a/x-c^2*(1/2*x^2*arctan(a*x)^2*a^2-3*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*I*arct an(a*x)*(-9*arctan(a*x)*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))*Pi+9*arctan(a*x)*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-12*arctan(a*x) *Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+24*arctan(a*x)*Pi*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))-12*arctan( a*x)*Pi*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))^2-12*arctan(a*x)*Pi*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*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+12*arctan(a*x)*Pi* csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*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)+12*arctan(a*x)*P 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)^3-12*arcta n(a*x)*Pi*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* csgn(I*(1+I*a*x)^2/(a^2*x^2+1))+12*arctan(a*x)*Pi*csgn(I*(1+I*a*x)^2/(a^2* x^2+1))^3-24*arctan(a*x)*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+ I*a*x)^2/(a^2*x^2+1))^2+12*arctan(a*x)*Pi*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))+9*arctan(a*x)*csgn(I/((1+I*a*x)^2...
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=c^{2} \left (\int 2 a^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx + \int a^{4} x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**2*(Integral(2*a**2*atan(a*x)**3, x) + Integral(atan(a*x)**3/x**2, x) + Integral(a**4*x**2*atan(a*x)**3, x))
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \]
1/96*(4*(a^4*c^2*x^4 + 6*a^2*c^2*x^2 - 3*c^2)*arctan(a*x)^3 - 3*(a^4*c^2*x ^4 + 6*a^2*c^2*x^2 - 3*c^2)*arctan(a*x)*log(a^2*x^2 + 1)^2 + 3*(896*a^6*c^ 2*integrate(1/32*x^6*arctan(a*x)^3/(a^2*x^4 + x^2), x) + 96*a^6*c^2*integr ate(1/32*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 128*a^6* c^2*integrate(1/32*x^6*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 128*a^5*c^2*integrate(1/32*x^5*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 32*a^5* c^2*integrate(1/32*x^5*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 21*a*c^2*a rctan(a*x)^4 + 2688*a^4*c^2*integrate(1/32*x^4*arctan(a*x)^3/(a^2*x^4 + x^ 2), x) + 288*a^4*c^2*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^ 2*x^4 + x^2), x) + 768*a^4*c^2*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 768*a^3*c^2*integrate(1/32*x^3*arctan(a*x)^2/(a ^2*x^4 + x^2), x) + a*c^2*log(a^2*x^2 + 1)^3 + 288*a^2*c^2*integrate(1/32* x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 384*a^2*c^2*integ rate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + 384*a*c^2 *integrate(1/32*x*arctan(a*x)^2/(a^2*x^4 + x^2), x) - 96*a*c^2*integrate(1 /32*x*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 896*c^2*integrate(1/32*arct an(a*x)^3/(a^2*x^4 + x^2), x) + 96*c^2*integrate(1/32*arctan(a*x)*log(a^2* x^2 + 1)^2/(a^2*x^4 + x^2), x))*x)/x
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2}{x^2} \,d x \]