Integrand size = 20, antiderivative size = 156 \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a}{2 c^2 x}+\frac {a^3 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{4 c^2}-\frac {\arctan (a x)}{2 c^2 x^2}-\frac {a^2 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {i a^2 \arctan (a x)^2}{c^2}-\frac {2 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2} \]
-1/2*a/c^2/x+1/4*a^3*x/c^2/(a^2*x^2+1)-1/4*a^2*arctan(a*x)/c^2-1/2*arctan( a*x)/c^2/x^2-1/2*a^2*arctan(a*x)/c^2/(a^2*x^2+1)+I*a^2*arctan(a*x)^2/c^2-2 *a^2*arctan(a*x)*ln(2-2/(1-I*a*x))/c^2+I*a^2*polylog(2,-1+2/(1-I*a*x))/c^2
Time = 0.37 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60 \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2 \left (-\frac {4}{a x}+8 i \arctan (a x)^2+\arctan (a x) \left (-4-\frac {4}{a^2 x^2}-2 \cos (2 \arctan (a x))-16 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+8 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+\sin (2 \arctan (a x))\right )}{8 c^2} \]
(a^2*(-4/(a*x) + (8*I)*ArcTan[a*x]^2 + ArcTan[a*x]*(-4 - 4/(a^2*x^2) - 2*C os[2*ArcTan[a*x]] - 16*Log[1 - E^((2*I)*ArcTan[a*x])]) + (8*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + Sin[2*ArcTan[a*x]]))/(8*c^2)
Time = 1.43 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.49, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5501, 27, 5453, 5361, 264, 216, 5459, 5403, 2897, 5501, 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.
| \(\displaystyle \int \frac {\arctan (a x)}{x^3 \left (a^2 c x^2+c\right )^2} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{c x^3 \left (a^2 x^2+1\right )}dx}{c}-a^2 \int \frac {\arctan (a x)}{c^2 x \left (a^2 x^2+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{x^3 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{x^3}dx-a^2 \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{2 x^2}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx\right )+\frac {1}{2} a \left (a^2 \left (-\int \frac {1}{a^2 x^2+1}dx\right )-\frac {1}{x}\right )-\frac {\arctan (a x)}{2 x^2}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {-\left (a^2 \left (i \int \frac {\arctan (a x)}{x (a x+i)}dx-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {-\left (a^2 \left (i \left (i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx\right )}{c^2}+\frac {-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}-\frac {a^2 \left (a^2 \left (-\int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx\right )+i \int \frac {\arctan (a x)}{x (a x+i)}dx-\frac {1}{2} i \arctan (a x)^2\right )}{c^2}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}-\frac {a^2 \left (a^2 \left (-\int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx\right )+i \left (i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )}{c^2}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}-\frac {a^2 \left (a^2 \left (-\int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx\right )+i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )}{c^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}-\frac {a^2 \left (-\left (a^2 \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )\right )+i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )}{c^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}-\frac {a^2 \left (-\left (a^2 \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )\right )+i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )}{c^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )}{c^2}-\frac {a^2 \left (-\left (a^2 \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )\right )+i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )}{c^2}\) |
(-1/2*ArcTan[a*x]/x^2 + (a*(-x^(-1) - a*ArcTan[a*x]))/2 - a^2*((-1/2*I)*Ar cTan[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 - (a^2*((-1/2*I)*ArcTan[a*x]^2 - a^2*(-1/2*ArcTan [a*x]/(a^2*(1 + a^2*x^2)) + (x/(2*(1 + a^2*x^2)) + ArcTan[a*x]/(2*a))/(2*a )) + I*((-I)*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - PolyLog[2, -1 + 2/(1 - I *a*x)]/2)))/c^2
3.2.90.3.1 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2) ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
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 = 0.65 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.62
| method | result | size |
| parts | \(\frac {\arctan \left (a x \right ) a^{2} \ln \left (a^{2} x^{2}+1\right )}{c^{2}}-\frac {a^{2} \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{2 c^{2} x^{2}}-\frac {2 \arctan \left (a x \right ) a^{2} \ln \left (x \right )}{c^{2}}-\frac {a \left (-4 a^{2} \left (-\frac {i \ln \left (x \right ) \left (\ln \left (i a x +1\right )-\ln \left (-i a x +1\right )\right )}{2 a}-\frac {i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{2 a}\right )-\frac {a^{2} x}{2 \left (a^{2} x^{2}+1\right )}+\frac {a \arctan \left (a x \right )}{2}+\frac {1}{x}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (a^{2} x^{2}+1\right )-a^{2} \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{a^{2} \underline {\hspace {1.25 ex}}\alpha }+2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )+2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{2}\right )}{2 c^{2}}\) | \(253\) |
| derivativedivides | \(a^{2} \left (-\frac {\arctan \left (a x \right )}{2 c^{2} a^{2} x^{2}}-\frac {2 \arctan \left (a x \right ) \ln \left (a x \right )}{c^{2}}+\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{c^{2}}-\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {2 i \ln \left (a x \right ) \ln \left (i a x +1\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+2 i \operatorname {dilog}\left (i a x +1\right )-2 i \operatorname {dilog}\left (-i a x +1\right )-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\frac {1}{a x}-\frac {a x}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2}}{2 c^{2}}\right )\) | \(286\) |
| default | \(a^{2} \left (-\frac {\arctan \left (a x \right )}{2 c^{2} a^{2} x^{2}}-\frac {2 \arctan \left (a x \right ) \ln \left (a x \right )}{c^{2}}+\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{c^{2}}-\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {2 i \ln \left (a x \right ) \ln \left (i a x +1\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+2 i \operatorname {dilog}\left (i a x +1\right )-2 i \operatorname {dilog}\left (-i a x +1\right )-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\frac {1}{a x}-\frac {a x}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2}}{2 c^{2}}\right )\) | \(286\) |
| risch | \(-\frac {a}{2 c^{2} x}+\frac {a^{2} \arctan \left (a x \right )}{8 c^{2}}+\frac {i a^{2}}{8 c^{2} \left (i a x +1\right )}-\frac {i a^{2} \operatorname {dilog}\left (i a x +1\right )}{c^{2}}-\frac {i a^{2} \ln \left (i a x +1\right )^{2}}{4 c^{2}}+\frac {i a^{2} \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{2 c^{2}}-\frac {i a^{2} \ln \left (i a x \right )}{4 c^{2}}+\frac {i a^{2} \ln \left (i a x +1\right )}{4 c^{2}}+\frac {i \ln \left (i a x +1\right )}{4 c^{2} x^{2}}+\frac {i a^{2} \operatorname {dilog}\left (-i a x +1\right )}{c^{2}}-\frac {i a^{2}}{8 c^{2} \left (-i a x +1\right )}+\frac {i a^{2} \ln \left (-i a x +1\right )^{2}}{4 c^{2}}-\frac {i a^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{2 c^{2}}+\frac {i a^{2} \ln \left (-i a x \right )}{4 c^{2}}-\frac {i a^{2} \ln \left (-i a x +1\right )}{4 c^{2}}-\frac {i \ln \left (-i a x +1\right )}{4 c^{2} x^{2}}+\frac {a^{3} \ln \left (-i a x +1\right ) x}{16 c^{2} \left (-i a x -1\right )}-\frac {i a^{2} \ln \left (-i a x +1\right )}{8 c^{2} \left (-i a x +1\right )}+\frac {i a^{2} \ln \left (-i a x +1\right )}{16 c^{2} \left (-i a x -1\right )}+\frac {i a^{2} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{2 c^{2}}+\frac {a^{3} \ln \left (i a x +1\right ) x}{16 c^{2} \left (i a x -1\right )}+\frac {i a^{2} \ln \left (i a x +1\right )}{8 c^{2} \left (i a x +1\right )}-\frac {i a^{2} \ln \left (i a x +1\right )}{16 c^{2} \left (i a x -1\right )}-\frac {i a^{2} \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{2 c^{2}}\) | \(469\) |
1/c^2*arctan(a*x)*a^2*ln(a^2*x^2+1)-1/2*a^2*arctan(a*x)/c^2/(a^2*x^2+1)-1/ 2*arctan(a*x)/c^2/x^2-2/c^2*arctan(a*x)*a^2*ln(x)-1/2*a/c^2*(-4*a^2*(-1/2* I*ln(x)*(ln(1+I*a*x)-ln(1-I*a*x))/a-1/2*I*(dilog(1+I*a*x)-dilog(1-I*a*x))/ a)-1/2*a^2*x/(a^2*x^2+1)+1/2*a*arctan(a*x)+1/x+1/2*sum(1/_alpha*(2*ln(x-_a lpha)*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)))
\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}{\left (a x \right )}}{a^{4} x^{7} + 2 a^{2} x^{5} + x^{3}}\, dx}{c^{2}} \]
\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )}{x^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]