Integrand size = 19, antiderivative size = 65 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=-\frac {1}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{a c^2} \]
Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {-1+2 a x \arctan (a x)-2 \left (1+a^2 x^2\right ) \arctan (a x)^2 \operatorname {CosIntegral}(2 \arctan (a x))}{2 c^2 \left (a+a^3 x^2\right ) \arctan (a x)^2} \]
(-1 + 2*a*x*ArcTan[a*x] - 2*(1 + a^2*x^2)*ArcTan[a*x]^2*CosIntegral[2*ArcT an[a*x]])/(2*c^2*(a + a^3*x^2)*ArcTan[a*x]^2)
Time = 0.91 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.68, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {5437, 27, 5503, 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.
| \(\displaystyle \int \frac {1}{\arctan (a x)^3 \left (a^2 c x^2+c\right )^2} \, dx\) |
| \(\Big \downarrow \) 5437 |
| \(\displaystyle -a \int \frac {x}{c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx}{c^2}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
| \(\Big \downarrow \) 5503 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{a}-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
| \(\Big \downarrow \) 5439 |
| \(\displaystyle -\frac {a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \frac {1}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^2}{\arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \left (\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}+\frac {1}{2 \arctan (a x)}\right )d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
| \(\Big \downarrow \) 5505 |
| \(\displaystyle -\frac {a \left (-\frac {\int \frac {a^2 x^2}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (-\frac {\int \frac {\sin (\arctan (a x))^2}{\arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {a \left (-\frac {\int \left (\frac {1}{2 \arctan (a x)}-\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}\right )d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {1}{2} \log (\arctan (a x))-\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}\) |
-1/2*1/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) - (a*(-(x/(a*(1 + a^2*x^2)*ArcT an[a*x])) - (-1/2*CosIntegral[2*ArcTan[a*x]] + Log[ArcTan[a*x]]/2)/a^2 + ( CosIntegral[2*ArcTan[a*x]]/2 + Log[ArcTan[a*x]]/2)/a^2))/c^2
3.7.30.3.1 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_S ymbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Simp[2*c*((q + 1)/(b*(p + 1))) Int[x*(d + e*x^2)^q*(a + b*Arc Tan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && LtQ[p, -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 = 8.88 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80
| method | result | size |
| derivativedivides | \(-\frac {4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-2 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (2 \arctan \left (a x \right )\right )+1}{4 a \,c^{2} \arctan \left (a x \right )^{2}}\) | \(52\) |
| default | \(-\frac {4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-2 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (2 \arctan \left (a x \right )\right )+1}{4 a \,c^{2} \arctan \left (a x \right )^{2}}\) | \(52\) |
-1/4/a/c^2*(4*Ci(2*arctan(a*x))*arctan(a*x)^2-2*sin(2*arctan(a*x))*arctan( a*x)+cos(2*arctan(a*x))+1)/arctan(a*x)^2
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=-\frac {{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) + {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2 \, a x \arctan \left (a x\right ) + 1}{2 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\right )^{2}} \]
-1/2*((a^2*x^2 + 1)*arctan(a*x)^2*log_integral(-(a^2*x^2 + 2*I*a*x - 1)/(a ^2*x^2 + 1)) + (a^2*x^2 + 1)*arctan(a*x)^2*log_integral(-(a^2*x^2 - 2*I*a* x - 1)/(a^2*x^2 + 1)) - 2*a*x*arctan(a*x) + 1)/((a^3*c^2*x^2 + a*c^2)*arct an(a*x)^2)
\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {\int \frac {1}{a^{4} x^{4} \operatorname {atan}^{3}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \]
\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}} \,d x } \]
1/2*(2*(a^3*c^2*x^2 + a*c^2)*arctan(a*x)^2*integrate((a^2*x^2 - 1)/((a^4*c ^2*x^4 + 2*a^2*c^2*x^2 + c^2)*arctan(a*x)), x) + 2*a*x*arctan(a*x) - 1)/(( a^3*c^2*x^2 + a*c^2)*arctan(a*x)^2)
\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]