Integrand size = 20, antiderivative size = 117 \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{4 c^2}+\frac {\arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^2}{2 c^2}+\frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {i \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2} \]
-1/4*a*x/c^2/(a^2*x^2+1)-1/4*arctan(a*x)/c^2+1/2*arctan(a*x)/c^2/(a^2*x^2+ 1)-1/2*I*arctan(a*x)^2/c^2+arctan(a*x)*ln(2-2/(1-I*a*x))/c^2-1/2*I*polylog (2,-1+2/(1-I*a*x))/c^2
Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.62 \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^2} \, dx=-\frac {4 i \arctan (a x)^2-2 \arctan (a x) \left (\cos (2 \arctan (a x))+4 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+4 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+\sin (2 \arctan (a x))}{8 c^2} \]
-1/8*((4*I)*ArcTan[a*x]^2 - 2*ArcTan[a*x]*(Cos[2*ArcTan[a*x]] + 4*Log[1 - E^((2*I)*ArcTan[a*x])]) + (4*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + Sin[2* ArcTan[a*x]])/c^2
Time = 0.60 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5501, 27, 5459, 5403, 2897, 5465, 215, 216}
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.
| \(\displaystyle \int \frac {\arctan (a x)}{x \left (a^2 c x^2+c\right )^2} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{c x \left (a^2 x^2+1\right )}dx}{c}-a^2 \int \frac {x \arctan (a x)}{c^2 \left (a^2 x^2+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {a^2 \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {i \int \frac {\arctan (a x)}{x (a x+i)}dx-\frac {1}{2} i \arctan (a x)^2}{c^2}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {a^2 \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {i \left (i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{2} i \arctan (a x)^2}{c^2}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle -\frac {a^2 \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2}{c^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle -\frac {a^2 \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2}{c^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle -\frac {a^2 \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2}{c^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {a^2 \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2}{c^2}\) |
-((a^2*(-1/2*ArcTan[a*x]/(a^2*(1 + a^2*x^2)) + (x/(2*(1 + a^2*x^2)) + ArcT an[a*x]/(2*a))/(2*a)))/c^2) + ((-1/2*I)*ArcTan[a*x]^2 + I*((-I)*ArcTan[a*x ]*Log[2 - 2/(1 - I*a*x)] - PolyLog[2, -1 + 2/(1 - I*a*x)]/2))/c^2
3.2.88.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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_)]*(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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 )^(q_), x_Symbol] :> Simp[1/d Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c *x])^p, x], x] - Simp[e/d Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.38 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.92
| method | result | size |
| parts | \(-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{2}}+\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) \ln \left (x \right )}{c^{2}}-\frac {a \left (\frac {x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )}{2 a}-\frac {i \ln \left (x \right ) \left (\ln \left (i a x +1\right )-\ln \left (-i a x +1\right )\right )}{a}-\frac {i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{a}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (a^{2} x^{2}+1\right )-a^{2} \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{a^{2} \underline {\hspace {1.25 ex}}\alpha }+2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )+2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 a^{2}}\right )}{2 c^{2}}\) | \(225\) |
| derivativedivides | \(\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{2}}+\frac {\arctan \left (a x \right ) \ln \left (a x \right )}{c^{2}}-\frac {\frac {a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )}{2}-i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x \right ) \ln \left (-i a x +1\right )-i \operatorname {dilog}\left (i a x +1\right )+i \operatorname {dilog}\left (-i a x +1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{2 c^{2}}\) | \(260\) |
| default | \(\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{2}}+\frac {\arctan \left (a x \right ) \ln \left (a x \right )}{c^{2}}-\frac {\frac {a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )}{2}-i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x \right ) \ln \left (-i a x +1\right )-i \operatorname {dilog}\left (i a x +1\right )+i \operatorname {dilog}\left (-i a x +1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{2 c^{2}}\) | \(260\) |
| risch | \(-\frac {i \ln \left (i a x +1\right )}{8 c^{2} \left (i a x +1\right )}-\frac {i \ln \left (-i a x +1\right )^{2}}{8 c^{2}}-\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c^{2}}+\frac {i \ln \left (i a x +1\right )^{2}}{8 c^{2}}+\frac {i \operatorname {dilog}\left (i a x +1\right )}{2 c^{2}}+\frac {i \ln \left (i a x +1\right )}{16 c^{2} \left (i a x -1\right )}+\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c^{2}}-\frac {i \ln \left (-i a x +1\right )}{16 c^{2} \left (-i a x -1\right )}-\frac {\arctan \left (a x \right )}{8 c^{2}}-\frac {\ln \left (-i a x +1\right ) a x}{16 c^{2} \left (-i a x -1\right )}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c^{2}}+\frac {i \ln \left (-i a x +1\right )}{8 c^{2} \left (-i a x +1\right )}-\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2 c^{2}}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c^{2}}-\frac {i}{8 c^{2} \left (i a x +1\right )}+\frac {i}{8 c^{2} \left (-i a x +1\right )}-\frac {\ln \left (i a x +1\right ) a x}{16 c^{2} \left (i a x -1\right )}\) | \(313\) |
-1/2/c^2*arctan(a*x)*ln(a^2*x^2+1)+1/2*arctan(a*x)/c^2/(a^2*x^2+1)+1/c^2*a rctan(a*x)*ln(x)-1/2/c^2*a*(1/2*x/(a^2*x^2+1)+1/2/a*arctan(a*x)-I*ln(x)*(l n(1+I*a*x)-ln(1-I*a*x))/a-I*(dilog(1+I*a*x)-dilog(1-I*a*x))/a-1/4/a^2*sum( 1/_alpha*(2*ln(x-_alpha)*ln(a^2*x^2+1)-a^2*(1/a^2/_alpha*ln(x-_alpha)^2+2* _alpha*ln(x-_alpha)*ln(1/2*(x+_alpha)/_alpha)+2*_alpha*dilog(1/2*(x+_alpha )/_alpha))),_alpha=RootOf(_Z^2*a^2+1)))
\[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x} \,d x } \]
Exception generated. \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: RecursionError} \]
\[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x} \,d x } \]
\[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )}{x\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]