Integrand size = 23, antiderivative size = 320 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx=-\frac {a b e^2 x}{c}-\frac {b^2 e^2 x \arctan (c x)}{c}-\frac {b c d^2 (a+b \arctan (c x))}{x}-\frac {1}{2} c^2 d^2 (a+b \arctan (c x))^2+\frac {e^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d^2 (a+b \arctan (c x))^2}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))^2+4 d e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+b^2 c^2 d^2 \log (x)-\frac {1}{2} b^2 c^2 d^2 \log \left (1+c^2 x^2\right )+\frac {b^2 e^2 \log \left (1+c^2 x^2\right )}{2 c^2}-2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-b^2 d e \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+b^2 d e \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]
-a*b*e^2*x/c-b^2*e^2*x*arctan(c*x)/c-b*c*d^2*(a+b*arctan(c*x))/x-1/2*c^2*d ^2*(a+b*arctan(c*x))^2+1/2*e^2*(a+b*arctan(c*x))^2/c^2-1/2*d^2*(a+b*arctan (c*x))^2/x^2+1/2*e^2*x^2*(a+b*arctan(c*x))^2-4*d*e*(a+b*arctan(c*x))^2*arc tanh(-1+2/(1+I*c*x))+b^2*c^2*d^2*ln(x)-1/2*b^2*c^2*d^2*ln(c^2*x^2+1)+1/2*b ^2*e^2*ln(c^2*x^2+1)/c^2-2*I*b*d*e*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c* x))+2*I*b*d*e*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-b^2*d*e*polylog( 3,1-2/(1+I*c*x))+b^2*d*e*polylog(3,-1+2/(1+I*c*x))
Time = 0.41 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.15 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx=\frac {1}{2} \left (-\frac {a^2 d^2}{x^2}+a^2 e^2 x^2+\frac {2 a b e^2 \left (-c x+\left (1+c^2 x^2\right ) \arctan (c x)\right )}{c^2}-\frac {2 a b d^2 (\arctan (c x)+c x (1+c x \arctan (c x)))}{x^2}+4 a^2 d e \log (x)-\frac {b^2 d^2 \left (2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2-2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )\right )}{x^2}+\frac {b^2 e^2 \left (-2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2+\log \left (1+c^2 x^2\right )\right )}{c^2}+4 i a b d e (\operatorname {PolyLog}(2,-i c x)-\operatorname {PolyLog}(2,i c x))+\frac {1}{6} b^2 d e \left (-i \pi ^3+16 i \arctan (c x)^3+24 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-24 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+24 i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+24 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-12 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )\right ) \]
(-((a^2*d^2)/x^2) + a^2*e^2*x^2 + (2*a*b*e^2*(-(c*x) + (1 + c^2*x^2)*ArcTa n[c*x]))/c^2 - (2*a*b*d^2*(ArcTan[c*x] + c*x*(1 + c*x*ArcTan[c*x])))/x^2 + 4*a^2*d*e*Log[x] - (b^2*d^2*(2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x ]^2 - 2*c^2*x^2*Log[(c*x)/Sqrt[1 + c^2*x^2]]))/x^2 + (b^2*e^2*(-2*c*x*ArcT an[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]))/c^2 + (4*I)*a*b *d*e*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + (b^2*d*e*((-I)*Pi^3 + (1 6*I)*ArcTan[c*x]^3 + 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24 *ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog [2, E^((-2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcT an[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 12*PolyLog[3, -E^((2*I )*ArcTan[c*x])]))/6)/2
Time = 0.81 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5515, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 5515 |
| \(\displaystyle \int \left (\frac {d^2 (a+b \arctan (c x))^2}{x^3}+\frac {2 d e (a+b \arctan (c x))^2}{x}+e^2 x (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 d e \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 (a+b \arctan (c x))^2+\frac {e^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d^2 (a+b \arctan (c x))^2}{2 x^2}-\frac {b c d^2 (a+b \arctan (c x))}{x}-2 i b d e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))+2 i b d e \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))^2-\frac {a b e^2 x}{c}-\frac {b^2 e^2 x \arctan (c x)}{c}-\frac {1}{2} b^2 c^2 d^2 \log \left (c^2 x^2+1\right )+b^2 c^2 d^2 \log (x)+\frac {b^2 e^2 \log \left (c^2 x^2+1\right )}{2 c^2}-b^2 d e \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )+b^2 d e \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )\) |
-((a*b*e^2*x)/c) - (b^2*e^2*x*ArcTan[c*x])/c - (b*c*d^2*(a + b*ArcTan[c*x] ))/x - (c^2*d^2*(a + b*ArcTan[c*x])^2)/2 + (e^2*(a + b*ArcTan[c*x])^2)/(2* c^2) - (d^2*(a + b*ArcTan[c*x])^2)/(2*x^2) + (e^2*x^2*(a + b*ArcTan[c*x])^ 2)/2 + 4*d*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d^ 2*Log[x] - (b^2*c^2*d^2*Log[1 + c^2*x^2])/2 + (b^2*e^2*Log[1 + c^2*x^2])/( 2*c^2) - (2*I)*b*d*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + ( 2*I)*b*d*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - b^2*d*e*Po lyLog[3, 1 - 2/(1 + I*c*x)] + b^2*d*e*PolyLog[3, -1 + 2/(1 + I*c*x)]
3.13.60.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*ArcTan[c*x] )^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d , e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 15.75 (sec) , antiderivative size = 1500, normalized size of antiderivative = 4.69
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1500\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1521\) |
| default | \(\text {Expression too large to display}\) | \(1521\) |
I*b^2*d*e*Pi*arctan(c*x)^2+2*I*b^2*d*e*arctan(c*x)*polylog(2,-(1+I*c*x)^2/ (c^2*x^2+1))-4*I*b^2*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))*arctan(c*x)*d *e-4*I*b^2*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))*arctan(c*x)*d*e-b^2*e^2* x*arctan(c*x)/c+2*a*b*c^2*(1/2*arctan(c*x)/c^2*e^2*x^2+2*arctan(c*x)/c^2*d *e*ln(c*x)-1/2*arctan(c*x)*d^2/c^2/x^2-1/2/c^4*(c*x*e^2+c^3*d^2/x+(c^4*d^2 -e^2)*arctan(c*x)+4*c^2*d*e*(-1/2*I*ln(c*x)*ln(1+I*c*x)+1/2*I*ln(c*x)*ln(1 -I*c*x)-1/2*I*dilog(1+I*c*x)+1/2*I*dilog(1-I*c*x))))+I*b^2*d*e*Pi*csgn(I*( (1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1 +I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2+4*b^2* polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))*d*e-b^2*d*e*polylog(3,-(1+I*c*x)^2/ (c^2*x^2+1))+4*b^2*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))*d*e+1/2*b^2*arc tan(c*x)^2*e^2*x^2-1/2*b^2*arctan(c*x)^2*d^2/x^2-b^2/c^2*e^2*ln((1+I*c*x)^ 2/(c^2*x^2+1)+1)+1/2*b^2/c^2*e^2*arctan(c*x)^2+b^2*c^2*ln(1+(1+I*c*x)/(c^2 *x^2+1)^(1/2))*d^2+b^2*c^2*ln((1+I*c*x)/(c^2*x^2+1)^(1/2)-1)*d^2-1/2*b^2*c ^2*arctan(c*x)^2*d^2-I*b^2*d*e*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c *x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+I*b^2*d*e*Pi*csgn(((1+I*c*x)^2/(c^2* x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2+I*b^2*d*e*Pi*csgn(I *((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2+ I*b^2*d*e*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1)) *csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan...
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
integral((a^2*e^2*x^4 + 2*a^2*d*e*x^2 + a^2*d^2 + (b^2*e^2*x^4 + 2*b^2*d*e *x^2 + b^2*d^2)*arctan(c*x)^2 + 2*(a*b*e^2*x^4 + 2*a*b*d*e*x^2 + a*b*d^2)* arctan(c*x))/x^3, x)
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
1/2*a^2*e^2*x^2 - ((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*a*b*d^2 + 2* a^2*d*e*log(x) - 1/2*a^2*d^2/x^2 + 1/96*((1152*b^2*c^2*e^2*integrate(1/16* x^6*arctan(c*x)^2/(c^2*x^5 + x^3), x) + 96*b^2*c^2*e^2*integrate(1/16*x^6* log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) + 3072*a*b*c^2*e^2*integrate(1/16*x ^6*arctan(c*x)/(c^2*x^5 + x^3), x) + 192*b^2*c^2*e^2*integrate(1/16*x^6*lo g(c^2*x^2 + 1)/(c^2*x^5 + x^3), x) + 2304*b^2*c^2*d*e*integrate(1/16*x^4*a rctan(c*x)^2/(c^2*x^5 + x^3), x) + 6144*a*b*c^2*d*e*integrate(1/16*x^4*arc tan(c*x)/(c^2*x^5 + x^3), x) + 1152*b^2*c^2*d^2*integrate(1/16*x^2*arctan( c*x)^2/(c^2*x^5 + x^3), x) + 96*b^2*c^2*d^2*integrate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) - 192*b^2*c^2*d^2*integrate(1/16*x^2*log(c^2*x ^2 + 1)/(c^2*x^5 + x^3), x) + 2*b^2*d*e*log(c^2*x^2 + 1)^3 - 384*b^2*c*e^2 *integrate(1/16*x^5*arctan(c*x)/(c^2*x^5 + x^3), x) + 384*b^2*c*d^2*integr ate(1/16*x*arctan(c*x)/(c^2*x^5 + x^3), x) + 1152*b^2*e^2*integrate(1/16*x ^4*arctan(c*x)^2/(c^2*x^5 + x^3), x) + 3072*a*b*e^2*integrate(1/16*x^4*arc tan(c*x)/(c^2*x^5 + x^3), x) + 2304*b^2*d*e*integrate(1/16*x^2*arctan(c*x) ^2/(c^2*x^5 + x^3), x) + 192*b^2*d*e*integrate(1/16*x^2*log(c^2*x^2 + 1)^2 /(c^2*x^5 + x^3), x) + 6144*a*b*d*e*integrate(1/16*x^2*arctan(c*x)/(c^2*x^ 5 + x^3), x) + 1152*b^2*d^2*integrate(1/16*arctan(c*x)^2/(c^2*x^5 + x^3), x) + 96*b^2*d^2*integrate(1/16*log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) + b^ 2*e^2*log(c^2*x^2 + 1)^3/c^2)*x^2 + 12*(b^2*e^2*x^4 - b^2*d^2)*arctan(c...
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2}{x^3} \,d x \]