Integrand size = 14, antiderivative size = 82 \[ \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx=-i c (a+b \arctan (c x))^2-\frac {(a+b \arctan (c x))^2}{x}+2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right ) \]
-I*c*(a+b*arctan(c*x))^2-(a+b*arctan(c*x))^2/x+2*b*c*(a+b*arctan(c*x))*ln( 2-2/(1-I*c*x))-I*b^2*c*polylog(2,-1+2/(1-I*c*x))
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx=\frac {b^2 (-1-i c x) \arctan (c x)^2+2 b \arctan (c x) \left (-a+b c x \log \left (1-e^{2 i \arctan (c x)}\right )\right )-a \left (a-2 b c x \log (c x)+b c x \log \left (1+c^2 x^2\right )\right )-i b^2 c x \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )}{x} \]
(b^2*(-1 - I*c*x)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(-a + b*c*x*Log[1 - E^(( 2*I)*ArcTan[c*x])]) - a*(a - 2*b*c*x*Log[c*x] + b*c*x*Log[1 + c^2*x^2]) - I*b^2*c*x*PolyLog[2, E^((2*I)*ArcTan[c*x])])/x
Time = 0.43 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5361, 5459, 5403, 2897}
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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle 2 b c \int \frac {a+b \arctan (c x)}{x \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x))^2}{x}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {(a+b \arctan (c x))^2}{x}+2 b c \left (i \int \frac {a+b \arctan (c x)}{x (c x+i)}dx-\frac {i (a+b \arctan (c x))^2}{2 b}\right )\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {(a+b \arctan (c x))^2}{x}+2 b c \left (i \left (i b c \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{c^2 x^2+1}dx-i \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))\right )-\frac {i (a+b \arctan (c x))^2}{2 b}\right )\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle -\frac {(a+b \arctan (c x))^2}{x}+2 b c \left (i \left (-i \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )\right )-\frac {i (a+b \arctan (c x))^2}{2 b}\right )\) |
-((a + b*ArcTan[c*x])^2/x) + 2*b*c*(((-1/2*I)*(a + b*ArcTan[c*x])^2)/b + I *((-I)*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - (b*PolyLog[2, -1 + 2/( 1 - I*c*x)])/2))
3.1.20.3.1 Defintions of rubi rules used
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si mp[b*c*(p/d) Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2* d^2 + e^2, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1))), x] + Si mp[I/d Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (78 ) = 156\).
Time = 2.35 (sec) , antiderivative size = 270, normalized size of antiderivative = 3.29
| method | result | size |
| parts | \(-\frac {a^{2}}{x}+b^{2} c \left (-\frac {\arctan \left (c x \right )^{2}}{c x}+2 \ln \left (c x \right ) \arctan \left (c x \right )-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+i \ln \left (c x \right ) \ln \left (i c x +1\right )-i \ln \left (c x \right ) \ln \left (-i c x +1\right )+i \operatorname {dilog}\left (i c x +1\right )-i \operatorname {dilog}\left (-i c x +1\right )-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )+2 a b c \left (-\frac {\arctan \left (c x \right )}{c x}+\ln \left (c x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )\) | \(270\) |
| derivativedivides | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\arctan \left (c x \right )^{2}}{c x}+2 \ln \left (c x \right ) \arctan \left (c x \right )-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+i \ln \left (c x \right ) \ln \left (i c x +1\right )-i \ln \left (c x \right ) \ln \left (-i c x +1\right )+i \operatorname {dilog}\left (i c x +1\right )-i \operatorname {dilog}\left (-i c x +1\right )-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )+2 a b \left (-\frac {\arctan \left (c x \right )}{c x}+\ln \left (c x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )\right )\) | \(273\) |
| default | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\arctan \left (c x \right )^{2}}{c x}+2 \ln \left (c x \right ) \arctan \left (c x \right )-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+i \ln \left (c x \right ) \ln \left (i c x +1\right )-i \ln \left (c x \right ) \ln \left (-i c x +1\right )+i \operatorname {dilog}\left (i c x +1\right )-i \operatorname {dilog}\left (-i c x +1\right )-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )+2 a b \left (-\frac {\arctan \left (c x \right )}{c x}+\ln \left (c x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )\right )\) | \(273\) |
-a^2/x+b^2*c*(-1/c/x*arctan(c*x)^2+2*ln(c*x)*arctan(c*x)-arctan(c*x)*ln(c^ 2*x^2+1)+I*ln(c*x)*ln(1+I*c*x)-I*ln(c*x)*ln(1-I*c*x)+I*dilog(1+I*c*x)-I*di log(1-I*c*x)-1/2*I*(ln(c*x-I)*ln(c^2*x^2+1)-1/2*ln(c*x-I)^2-dilog(-1/2*I*( c*x+I))-ln(c*x-I)*ln(-1/2*I*(c*x+I)))+1/2*I*(ln(c*x+I)*ln(c^2*x^2+1)-1/2*l n(c*x+I)^2-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))))+2*a*b*c*(-1/ c/x*arctan(c*x)+ln(c*x)-1/2*ln(c^2*x^2+1))
\[ \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
\[ \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
\[ \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
-(c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arctan(c*x)/x)*a*b + 1/16*(4*(c*arct an(c*x)^3 + 4*c^2*integrate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^4 + x^2), x ) - 16*c^2*integrate(1/16*x^2*log(c^2*x^2 + 1)/(c^2*x^4 + x^2), x) + 32*c* integrate(1/16*x*arctan(c*x)/(c^2*x^4 + x^2), x) + 48*integrate(1/16*arcta n(c*x)^2/(c^2*x^4 + x^2), x) + 4*integrate(1/16*log(c^2*x^2 + 1)^2/(c^2*x^ 4 + x^2), x))*x - 4*arctan(c*x)^2 + log(c^2*x^2 + 1)^2)*b^2/x - a^2/x
\[ \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x^2} \,d x \]