Integrand size = 20, antiderivative size = 41 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=-\frac {x}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{a^2 c^2} \]
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {-\frac {a x}{\left (1+a^2 x^2\right ) \arctan (a x)}+\operatorname {CosIntegral}(2 \arctan (a x))}{a^2 c^2} \]
Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(41)=82\).
Time = 0.78 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5503, 27, 5439, 3042, 3793, 2009, 5505, 3042, 3793, 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.
\(\displaystyle \int \frac {x}{\arctan (a x)^2 \left (a^2 c x^2+c\right )^2} \, dx\) |
\(\Big \downarrow \) 5503 |
\(\displaystyle \frac {\int \frac {1}{c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{a}-a \int \frac {x^2}{c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)}dx-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{a c^2}-\frac {a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)}\) |
\(\Big \downarrow \) 5439 |
\(\displaystyle -\frac {a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{c^2}+\frac {\int \frac {1}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{c^2}+\frac {\int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^2}{\arctan (a x)}d\arctan (a x)}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{c^2}+\frac {\int \left (\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}+\frac {1}{2 \arctan (a x)}\right )d\arctan (a x)}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{c^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)}\) |
\(\Big \downarrow \) 5505 |
\(\displaystyle -\frac {\int \frac {a^2 x^2}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2 c^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {\sin (\arctan (a x))^2}{\arctan (a x)}d\arctan (a x)}{a^2 c^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {\int \left (\frac {1}{2 \arctan (a x)}-\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}\right )d\arctan (a x)}{a^2 c^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {1}{2} \log (\arctan (a x))-\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))}{a^2 c^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)}\) |
-(x/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x])) - (-1/2*CosIntegral[2*ArcTan[a*x]] + Log[ArcTan[a*x]]/2)/(a^2*c^2) + (CosIntegral[2*ArcTan[a*x]]/2 + Log[ArcT an[a*x]]/2)/(a^2*c^2)
3.6.53.3.1 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[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] + (-Simp[c*((m + 2*q + 2)/(b*(p + 1))) Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 1)) Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x]) /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] & & LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1) Subst[Int[(a + b*x)^p*(Sin[x]^m/ Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p }, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q ] || GtQ[d, 0])
Time = 8.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {2 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\sin \left (2 \arctan \left (a x \right )\right )}{2 a^{2} c^{2} \arctan \left (a x \right )}\) | \(38\) |
default | \(\frac {2 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\sin \left (2 \arctan \left (a x \right )\right )}{2 a^{2} c^{2} \arctan \left (a x \right )}\) | \(38\) |
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.80 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) + {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2 \, a x}{2 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )} \arctan \left (a x\right )} \]
1/2*((a^2*x^2 + 1)*arctan(a*x)*log_integral(-(a^2*x^2 + 2*I*a*x - 1)/(a^2* x^2 + 1)) + (a^2*x^2 + 1)*arctan(a*x)*log_integral(-(a^2*x^2 - 2*I*a*x - 1 )/(a^2*x^2 + 1)) - 2*a*x)/((a^4*c^2*x^2 + a^2*c^2)*arctan(a*x))
\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {\int \frac {x}{a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]
\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]
-((a^3*c^2*x^2 + a*c^2)*arctan(a*x)*integrate((a^2*x^2 - 1)/((a^5*c^2*x^4 + 2*a^3*c^2*x^2 + a*c^2)*arctan(a*x)), x) + x)/((a^3*c^2*x^2 + a*c^2)*arct an(a*x))
\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]
Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]