Integrand size = 22, antiderivative size = 235 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x} \, dx=\frac {1}{12} a^2 c^2 x^2-\frac {3}{2} a c^2 x \arctan (a x)-\frac {1}{6} a^3 c^2 x^3 \arctan (a x)+\frac {3}{4} c^2 \arctan (a x)^2+a^2 c^2 x^2 \arctan (a x)^2+\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^2+2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {2}{3} c^2 \log \left (1+a^2 x^2\right )-i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \] Output:
1/12*a^2*c^2*x^2-3/2*a*c^2*x*arctan(a*x)-1/6*a^3*c^2*x^3*arctan(a*x)+3/4*c ^2*arctan(a*x)^2+a^2*c^2*x^2*arctan(a*x)^2+1/4*a^4*c^2*x^4*arctan(a*x)^2-2 *c^2*arctan(a*x)^2*arctanh(-1+2/(1+I*a*x))+2/3*c^2*ln(a^2*x^2+1)-I*c^2*arc tan(a*x)*polylog(2,1-2/(1+I*a*x))+I*c^2*arctan(a*x)*polylog(2,-1+2/(1+I*a* x))-1/2*c^2*polylog(3,1-2/(1+I*a*x))+1/2*c^2*polylog(3,-1+2/(1+I*a*x))
Time = 0.22 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x} \, dx=\frac {1}{24} c^2 \left (2-i \pi ^3+2 a^2 x^2-36 a x \arctan (a x)-4 a^3 x^3 \arctan (a x)+18 \arctan (a x)^2+24 a^2 x^2 \arctan (a x)^2+6 a^4 x^4 \arctan (a x)^2+16 i \arctan (a x)^3+24 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )-24 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+16 \log \left (1+a^2 x^2\right )+24 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+24 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-12 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right ) \] Input:
Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x,x]
Output:
(c^2*(2 - I*Pi^3 + 2*a^2*x^2 - 36*a*x*ArcTan[a*x] - 4*a^3*x^3*ArcTan[a*x] + 18*ArcTan[a*x]^2 + 24*a^2*x^2*ArcTan[a*x]^2 + 6*a^4*x^4*ArcTan[a*x]^2 + (16*I)*ArcTan[a*x]^3 + 24*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 24*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 16*Log[1 + a^2*x^2] + (2 4*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]*P olyLog[2, -E^((2*I)*ArcTan[a*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/24
Time = 0.77 (sec) , antiderivative size = 235, 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.
| \(\displaystyle \int \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^2}{x} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^4 c^2 x^3 \arctan (a x)^2+2 a^2 c^2 x \arctan (a x)^2+\frac {c^2 \arctan (a x)^2}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} a^4 c^2 x^4 \arctan (a x)^2-\frac {1}{6} a^3 c^2 x^3 \arctan (a x)+a^2 c^2 x^2 \arctan (a x)^2+\frac {1}{12} a^2 c^2 x^2+\frac {2}{3} c^2 \log \left (a^2 x^2+1\right )+2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-\frac {3}{2} a c^2 x \arctan (a x)+\frac {3}{4} c^2 \arctan (a x)^2-\frac {1}{2} c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {1}{2} c^2 \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )\) |
Input:
Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x,x]
Output:
(a^2*c^2*x^2)/12 - (3*a*c^2*x*ArcTan[a*x])/2 - (a^3*c^2*x^3*ArcTan[a*x])/6 + (3*c^2*ArcTan[a*x]^2)/4 + a^2*c^2*x^2*ArcTan[a*x]^2 + (a^4*c^2*x^4*ArcT an[a*x]^2)/4 + 2*c^2*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + (2*c^2*Log [1 + a^2*x^2])/3 - I*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*c^2 *ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (c^2*PolyLog[3, 1 - 2/(1 + I *a*x)])/2 + (c^2*PolyLog[3, -1 + 2/(1 + I*a*x)])/2
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 = 25.81 (sec) , antiderivative size = 1185, normalized size of antiderivative = 5.04
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1185\) |
| default | \(\text {Expression too large to display}\) | \(1185\) |
| parts | \(\text {Expression too large to display}\) | \(1696\) |
Input:
int((a^2*c*x^2+c)^2*arctan(a*x)^2/x,x,method=_RETURNVERBOSE)
Output:
1/4*a^4*c^2*x^4*arctan(a*x)^2+a^2*c^2*x^2*arctan(a*x)^2+c^2*arctan(a*x)^2* ln(a*x)-1/2*c^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))*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*arctan(a*x)^2+arctan(a*x)*(a*x-I)*(a*x+I)^2-a rctan(a*x)*(a*x-I)^2*(a*x+I)+I*arctan(a*x)*(a*x-I)^2+1/3*arctan(a*x)*(a*x- I)^3-4*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*polylog(3,-(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)/((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*a rctan(a*x)^2+2*arctan(a*x)*(a*x-I)+1/3*I*(a*x+I)-2*I*arctan(a*x)*(a*x-I)*( a*x+I)+4*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I*arctan(a *x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+polylog(3,-(1+I*a*x)^2/(a^2*x^2+1) )+8/3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)-1/6*(a*x+I)^2-I*Pi*arctan(a*x)^2+4*I*a rctan(a*x)*polylog(2,-(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+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-I*P i*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+2*arct an(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-2*arctan(a*x)^2*ln(1-(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*...
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{x} \,d x } \] Input:
integrate((a^2*c*x^2+c)^2*arctan(a*x)^2/x,x, algorithm="fricas")
Output:
integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*arctan(a*x)^2/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x} \, dx=c^{2} \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx + \int 2 a^{2} x \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{4} x^{3} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \] Input:
integrate((a**2*c*x**2+c)**2*atan(a*x)**2/x,x)
Output:
c**2*(Integral(atan(a*x)**2/x, x) + Integral(2*a**2*x*atan(a*x)**2, x) + I ntegral(a**4*x**3*atan(a*x)**2, x))
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{x} \,d x } \] Input:
integrate((a^2*c*x^2+c)^2*arctan(a*x)^2/x,x, algorithm="maxima")
Output:
12*a^6*c^2*integrate(1/16*x^6*arctan(a*x)^2/(a^2*x^3 + x), x) + a^6*c^2*in tegrate(1/16*x^6*log(a^2*x^2 + 1)^2/(a^2*x^3 + x), x) + a^6*c^2*integrate( 1/16*x^6*log(a^2*x^2 + 1)/(a^2*x^3 + x), x) - 2*a^5*c^2*integrate(1/16*x^5 *arctan(a*x)/(a^2*x^3 + x), x) + 36*a^4*c^2*integrate(1/16*x^4*arctan(a*x) ^2/(a^2*x^3 + x), x) + 3*a^4*c^2*integrate(1/16*x^4*log(a^2*x^2 + 1)^2/(a^ 2*x^3 + x), x) + 4*a^4*c^2*integrate(1/16*x^4*log(a^2*x^2 + 1)/(a^2*x^3 + x), x) - 8*a^3*c^2*integrate(1/16*x^3*arctan(a*x)/(a^2*x^3 + x), x) + 36*a ^2*c^2*integrate(1/16*x^2*arctan(a*x)^2/(a^2*x^3 + x), x) + 1/32*c^2*log(a ^2*x^2 + 1)^3 + 1/16*(a^4*c^2*x^4 + 4*a^2*c^2*x^2)*arctan(a*x)^2 + 12*c^2* integrate(1/16*arctan(a*x)^2/(a^2*x^3 + x), x) + c^2*integrate(1/16*log(a^ 2*x^2 + 1)^2/(a^2*x^3 + x), x) - 1/64*(a^4*c^2*x^4 + 4*a^2*c^2*x^2)*log(a^ 2*x^2 + 1)^2
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{x} \,d x } \] Input:
integrate((a^2*c*x^2+c)^2*arctan(a*x)^2/x,x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)^2*arctan(a*x)^2/x, x)
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2}{x} \,d x \] Input:
int((atan(a*x)^2*(c + a^2*c*x^2)^2)/x,x)
Output:
int((atan(a*x)^2*(c + a^2*c*x^2)^2)/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x} \, dx=\frac {c^{2} \left (3 \mathit {atan} \left (a x \right )^{2} a^{4} x^{4}+12 \mathit {atan} \left (a x \right )^{2} a^{2} x^{2}+9 \mathit {atan} \left (a x \right )^{2}-2 \mathit {atan} \left (a x \right ) a^{3} x^{3}-18 \mathit {atan} \left (a x \right ) a x +12 \left (\int \frac {\mathit {atan} \left (a x \right )^{2}}{x}d x \right )+8 \,\mathrm {log}\left (a^{2} x^{2}+1\right )+a^{2} x^{2}\right )}{12} \] Input:
int((a^2*c*x^2+c)^2*atan(a*x)^2/x,x)
Output:
(c**2*(3*atan(a*x)**2*a**4*x**4 + 12*atan(a*x)**2*a**2*x**2 + 9*atan(a*x)* *2 - 2*atan(a*x)*a**3*x**3 - 18*atan(a*x)*a*x + 12*int(atan(a*x)**2/x,x) + 8*log(a**2*x**2 + 1) + a**2*x**2))/12