Integrand size = 22, antiderivative size = 207 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=-\frac {a c^2 \arctan (a x)}{x}-a^3 c^2 x \arctan (a x)-\frac {c^2 \arctan (a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^2+4 a^2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \log (x)-2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-a^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+a^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \]
-a*c^2*arctan(a*x)/x-a^3*c^2*x*arctan(a*x)-1/2*c^2*arctan(a*x)^2/x^2+1/2*a ^4*c^2*x^2*arctan(a*x)^2-4*a^2*c^2*arctan(a*x)^2*arctanh(-1+2/(1+I*a*x))+a ^2*c^2*ln(x)-2*I*a^2*c^2*arctan(a*x)*polylog(2,1-2/(1+I*a*x))+2*I*a^2*c^2* arctan(a*x)*polylog(2,-1+2/(1+I*a*x))-a^2*c^2*polylog(3,1-2/(1+I*a*x))+a^2 *c^2*polylog(3,-1+2/(1+I*a*x))
Time = 0.24 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=a^2 c^2 \left (-\frac {i \pi ^3}{12}-\frac {\arctan (a x)}{a x}-a x \arctan (a x)-\frac {\arctan (a x)^2}{2 a^2 x^2}+\frac {1}{2} a^2 x^2 \arctan (a x)^2+\frac {4}{3} i \arctan (a x)^3+2 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )-2 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+\log \left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+\frac {1}{2} \log \left (1+a^2 x^2\right )+2 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+2 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right ) \]
a^2*c^2*((-1/12*I)*Pi^3 - ArcTan[a*x]/(a*x) - a*x*ArcTan[a*x] - ArcTan[a*x ]^2/(2*a^2*x^2) + (a^2*x^2*ArcTan[a*x]^2)/2 + ((4*I)/3)*ArcTan[a*x]^3 + 2* ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 2*ArcTan[a*x]^2*Log[1 + E^ ((2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] + Log[1 + a^2*x^2]/2 + (2*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (2*I)*ArcTan[a*x]* PolyLog[2, -E^((2*I)*ArcTan[a*x])] + PolyLog[3, E^((-2*I)*ArcTan[a*x])] - PolyLog[3, -E^((2*I)*ArcTan[a*x])])
Time = 0.63 (sec) , antiderivative size = 207, 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.
| \(\displaystyle \int \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^4 c^2 x \arctan (a x)^2+\frac {2 a^2 c^2 \arctan (a x)^2}{x}+\frac {c^2 \arctan (a x)^2}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} a^4 c^2 x^2 \arctan (a x)^2-a^3 c^2 x \arctan (a x)+4 a^2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-a^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+a^2 c^2 \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )+a^2 c^2 \log (x)-\frac {c^2 \arctan (a x)^2}{2 x^2}-\frac {a c^2 \arctan (a x)}{x}\) |
-((a*c^2*ArcTan[a*x])/x) - a^3*c^2*x*ArcTan[a*x] - (c^2*ArcTan[a*x]^2)/(2* x^2) + (a^4*c^2*x^2*ArcTan[a*x]^2)/2 + 4*a^2*c^2*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c^2*Log[x] - (2*I)*a^2*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (2*I)*a^2*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x) ] - a^2*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)] + a^2*c^2*PolyLog[3, -1 + 2/(1 + I*a*x)]
3.3.72.3.1 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 = 58.06 (sec) , antiderivative size = 1184, normalized size of antiderivative = 5.72
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1184\) |
| default | \(\text {Expression too large to display}\) | \(1184\) |
| parts | \(\text {Expression too large to display}\) | \(1614\) |
a^2*(1/2*a^2*c^2*x^2*arctan(a*x)^2+2*c^2*arctan(a*x)^2*ln(a*x)-1/2*c^2*arc tan(a*x)^2/a^2/x^2-c^2*(2*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-2*ar ctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-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)^2-4*polylog(3,-(1+I*a *x)/(a^2*x^2+1)^(1/2))-2*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+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)^2-4*polylog(3,(1+I*a*x)/(a^2*x ^2+1)^(1/2))+1/2*arctan(a*x)*(I*a*x-(a^2*x^2+1)^(1/2)+1)/a/x+polylog(3,-(1 +I*a*x)^2/(a^2*x^2+1))+ln((1+I*a*x)^2/(a^2*x^2+1)+1)-ln((1+I*a*x)/(a^2*x^2 +1)^(1/2)+1)-ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-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+1/2*arctan(a*x)* (I*a*x+(a^2*x^2+1)^(1/2)+1)/a/x-2*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^ 2*x^2+1))+arctan(a*x)*(a*x-I)+4*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^ 2+1)^(1/2))+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-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))*arctan(a*x)^2+4*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-I *Pi*arctan(a*x)^2+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*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csgn(I*((1+I*...
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=c^{2} \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {2 a^{2} \operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx + \int a^{4} x \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
c**2*(Integral(atan(a*x)**2/x**3, x) + Integral(2*a**2*atan(a*x)**2/x, x) + Integral(a**4*x*atan(a*x)**2, x))
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
-1/32*(2*a^4*c^2*x^4 - 4*a^4*c^2*x^2*integrate(4*x*arctan(a*x)^2 + x*log(a ^2*x^2 + 1)^2, x) - 8*a^3*c^2*x^2*integrate(-1/4*(12*(a^2*x^2 + 1)*a*x*arc tan(a*x)^2 - 3*(a^2*x^2 + 1)*a*x*log(a^2*x^2 + 1)^2 + 12*(a^2*x^2 + 1)*arc tan(a*x)*log(a^2*x^2 + 1) + (4*(a^2*x^2 + 1)^2*arctan(a*x)*cos(3*arctan(a* x))*log(a^2*x^2 + 1) - 12*(a^2*x^2 + 1)^(3/2)*arctan(a*x)*cos(2*arctan(a*x ))*log(a^2*x^2 + 1) - 4*sqrt(a^2*x^2 + 1)*arctan(a*x)*log(a^2*x^2 + 1) + ( 4*(a^2*x^2 + 1)^2*arctan(a*x)^2 - (a^2*x^2 + 1)^2*log(a^2*x^2 + 1)^2)*sin( 3*arctan(a*x)) - 3*(4*(a^2*x^2 + 1)^(3/2)*arctan(a*x)^2 - (a^2*x^2 + 1)^(3 /2)*log(a^2*x^2 + 1)^2)*sin(2*arctan(a*x)))*sqrt(a^2*x^2 + 1))/((a^2*x^2 + 1)^4*cos(3*arctan(a*x))^2 + (a^2*x^2 + 1)^4*sin(3*arctan(a*x))^2 - 6*(a^2 *x^2 + 1)^(7/2)*sin(3*arctan(a*x))*sin(2*arctan(a*x)) + 9*(a^2*x^2 + 1)^3* cos(2*arctan(a*x))^2 + 9*(a^2*x^2 + 1)^3*sin(2*arctan(a*x))^2 + a^2*x^2 + 6*(a^2*x^2 + 1)^2*cos(2*arctan(a*x)) + 9*(a^2*x^2 + 1)^2 - 2*(3*(a^2*x^2 + 1)^(7/2)*cos(2*arctan(a*x)) + (a^2*x^2 + 1)^(5/2))*cos(3*arctan(a*x)) + 6 *((a^2*x^2 + 1)^2*a*x*sin(3*arctan(a*x)) - 3*(a^2*x^2 + 1)^(3/2)*a*x*sin(2 *arctan(a*x)) + (a^2*x^2 + 1)^2*cos(3*arctan(a*x)) - 3*(a^2*x^2 + 1)^(3/2) *cos(2*arctan(a*x)) - sqrt(a^2*x^2 + 1))*sqrt(a^2*x^2 + 1) + 1), x) - 8*a^ 3*c^2*x^2*integrate(1/4*(4*(a^2*x^2 + 1)*arctan(a*x)*log(a^2*x^2 + 1) - (4 *(a^2*x^2 + 1)*a*x*arctan(a*x)^2 - (a^2*x^2 + 1)*a*x*log(a^2*x^2 + 1)^2 + 4*(a^2*x^2 + 1)*arctan(a*x)*log(a^2*x^2 + 1))*cos(2*arctan(a*x)) - (4*(...
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2}{x^3} \,d x \]