Integrand size = 21, antiderivative size = 220 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))^2}{x^3} \, dx=-\frac {b c d (a+b \arctan (c x))}{x}-\frac {1}{2} c^2 d (a+b \arctan (c x))^2-\frac {d (a+b \arctan (c x))^2}{2 x^2}+2 e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+b^2 c^2 d \log (x)-\frac {1}{2} b^2 c^2 d \log \left (1+c^2 x^2\right )-i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 e \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]
-b*c*d*(a+b*arctan(c*x))/x-1/2*c^2*d*(a+b*arctan(c*x))^2-1/2*d*(a+b*arctan (c*x))^2/x^2-2*e*(a+b*arctan(c*x))^2*arctanh(-1+2/(1+I*c*x))+b^2*c^2*d*ln( x)-1/2*b^2*c^2*d*ln(c^2*x^2+1)-I*b*e*(a+b*arctan(c*x))*polylog(2,1-2/(1+I* c*x))+I*b*e*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-1/2*b^2*e*polylog( 3,1-2/(1+I*c*x))+1/2*b^2*e*polylog(3,-1+2/(1+I*c*x))
Time = 0.25 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))^2}{x^3} \, dx=-\frac {a^2 d}{2 x^2}-\frac {a b d (\arctan (c x)+c x (1+c x \arctan (c x)))}{x^2}+a^2 e \log (x)-\frac {b^2 d \left (2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2-2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )\right )}{2 x^2}+i a b e (\operatorname {PolyLog}(2,-i c x)-\operatorname {PolyLog}(2,i c x))+\frac {1}{24} b^2 e \left (-i \pi ^3+16 i \arctan (c x)^3+24 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-24 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+24 i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+24 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-12 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right ) \]
-1/2*(a^2*d)/x^2 - (a*b*d*(ArcTan[c*x] + c*x*(1 + c*x*ArcTan[c*x])))/x^2 + a^2*e*Log[x] - (b^2*d*(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]]))/(2*x^2) + I*a*b*e*(PolyLog[2, (-I )*c*x] - PolyLog[2, I*c*x]) + (b^2*e*((-I)*Pi^3 + (16*I)*ArcTan[c*x]^3 + 2 4*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)*ArcTan[c*x])] + 12*PolyLog [3, E^((-2*I)*ArcTan[c*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/24
Time = 0.63 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
\(\displaystyle \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))^2}{x^3} \, dx\) |
\(\Big \downarrow \) 5515 |
\(\displaystyle \int \left (\frac {d (a+b \arctan (c x))^2}{x^3}+\frac {e (a+b \arctan (c x))^2}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 e \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2-\frac {1}{2} c^2 d (a+b \arctan (c x))^2-\frac {d (a+b \arctan (c x))^2}{2 x^2}-\frac {b c d (a+b \arctan (c x))}{x}-i b e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))+i b e \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))-\frac {1}{2} b^2 c^2 d \log \left (c^2 x^2+1\right )+b^2 c^2 d \log (x)-\frac {1}{2} b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )+\frac {1}{2} b^2 e \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )\) |
-((b*c*d*(a + b*ArcTan[c*x]))/x) - (c^2*d*(a + b*ArcTan[c*x])^2)/2 - (d*(a + b*ArcTan[c*x])^2)/(2*x^2) + 2*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d*Log[x] - (b^2*c^2*d*Log[1 + c^2*x^2])/2 - I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*e*(a + b*ArcTan[c*x])* PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*e*PolyLog[3, -1 + 2/(1 + I*c*x)])/2
3.13.53.3.1 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 = 10.21 (sec) , antiderivative size = 1289, normalized size of antiderivative = 5.86
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1289\) |
default | \(\text {Expression too large to display}\) | \(1289\) |
parts | \(\text {Expression too large to display}\) | \(1318\) |
c^2*(a^2/c^2*e*ln(c*x)-1/2*a^2*d/c^2/x^2+b^2/c^2*(-1/2*I*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)/((1+I*c*x)^2/(c ^2*x^2+1)+1))^2*arctan(c*x)^2+arctan(c*x)^2*e*ln(c*x)+d*c^2*ln(1+(1+I*c*x) /(c^2*x^2+1)^(1/2))+d*c^2*ln((1+I*c*x)/(c^2*x^2+1)^(1/2)-1)+1/2*I*e*Pi*csg n(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)^2-1 /2*c*d*arctan(c*x)*(I*c*x-(c^2*x^2+1)^(1/2)+1)/x-2*I*e*arctan(c*x)*polylog (2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*e*Pi*arctan(c*x)^2-2*I*e*arctan(c*x )*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*e*arctan(c*x)*polylog(2,-(1+I*c *x)^2/(c^2*x^2+1))-1/2*I*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)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2-1/ 2*c*d*arctan(c*x)*(I*c*x+(c^2*x^2+1)^(1/2)+1)/x-1/2*I*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))*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+1/2*I*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))*csgn(((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+e*arctan(c*x)^2*ln (1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-e*ln((1+I*c*x)^2/(c^2*x^2+1)-1)*arctan(c*x )^2+e*arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*e*polylog(3,-(1+ I*c*x)^2/(c^2*x^2+1))+2*e*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*e*poly log(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*I*e*Pi*csgn(((1+I*c*x)^2/(c^2*x^...
\[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))^2}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
integral((a^2*e*x^2 + a^2*d + (b^2*e*x^2 + b^2*d)*arctan(c*x)^2 + 2*(a*b*e *x^2 + a*b*d)*arctan(c*x))/x^3, x)
\[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))^2}{x^3} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )}{x^{3}}\, dx \]
\[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))^2}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
-((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*a*b*d + a^2*e*log(x) - 1/2*a^ 2*d/x^2 - 1/96*(12*b^2*d*arctan(c*x)^2 - 3*b^2*d*log(c^2*x^2 + 1)^2 - (115 2*b^2*c^2*e*integrate(1/16*x^4*arctan(c*x)^2/(c^2*x^5 + x^3), x) + 3072*a* b*c^2*e*integrate(1/16*x^4*arctan(c*x)/(c^2*x^5 + x^3), x) + 1152*b^2*c^2* d*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^5 + x^3), x) + 96*b^2*c^2*d*inte grate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) - 192*b^2*c^2*d*inte grate(1/16*x^2*log(c^2*x^2 + 1)/(c^2*x^5 + x^3), x) + b^2*e*log(c^2*x^2 + 1)^3 + 384*b^2*c*d*integrate(1/16*x*arctan(c*x)/(c^2*x^5 + x^3), x) + 1152 *b^2*e*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^5 + x^3), x) + 96*b^2*e*int egrate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) + 3072*a*b*e*integr ate(1/16*x^2*arctan(c*x)/(c^2*x^5 + x^3), x) + 1152*b^2*d*integrate(1/16*a rctan(c*x)^2/(c^2*x^5 + x^3), x) + 96*b^2*d*integrate(1/16*log(c^2*x^2 + 1 )^2/(c^2*x^5 + x^3), x))*x^2)/x^2
Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))^2}{x^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right )}{x^3} \,d x \]