Integrand size = 22, antiderivative size = 86 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}-\frac {x}{a^3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{2 a^4 c^3}-\frac {\operatorname {CosIntegral}(4 \arctan (a x))}{2 a^4 c^3} \]
x/a^3/c^3/(a^2*x^2+1)^2/arctan(a*x)-x/a^3/c^3/(a^2*x^2+1)/arctan(a*x)+1/2* Ci(2*arctan(a*x))/a^4/c^3-1/2*Ci(4*arctan(a*x))/a^4/c^3
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {-2 a^3 x^3+\left (1+a^2 x^2\right )^2 \arctan (a x) \operatorname {CosIntegral}(2 \arctan (a x))-\left (1+a^2 x^2\right )^2 \arctan (a x) \operatorname {CosIntegral}(4 \arctan (a x))}{2 a^4 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)} \]
(-2*a^3*x^3 + (1 + a^2*x^2)^2*ArcTan[a*x]*CosIntegral[2*ArcTan[a*x]] - (1 + a^2*x^2)^2*ArcTan[a*x]*CosIntegral[4*ArcTan[a*x]])/(2*a^4*c^3*(1 + a^2*x ^2)^2*ArcTan[a*x])
Leaf count is larger than twice the leaf count of optimal. \(177\) vs. \(2(86)=172\).
Time = 1.53 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5499, 27, 5503, 5439, 3042, 3793, 2009, 5505, 3042, 3793, 2009, 4906, 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.
| \(\displaystyle \int \frac {x^3}{\arctan (a x)^2 \left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5499 |
| \(\displaystyle \frac {\int \frac {x}{c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx}{a^2 c}-\frac {\int \frac {x}{c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx}{a^2 c^3}-\frac {\int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx}{a^2 c^3}\) |
| \(\Big \downarrow \) 5503 |
| \(\displaystyle \frac {\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx}{a}-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {\frac {\int \frac {1}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx}{a}-3 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
| \(\Big \downarrow \) 5439 |
| \(\displaystyle \frac {-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \frac {1}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-3 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx+\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^2}{\arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-3 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx+\frac {\int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^4}{\arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\int \left (\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}+\frac {1}{2 \arctan (a x)}\right )d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-3 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx+\frac {\int \left (\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}+\frac {\cos (4 \arctan (a x))}{8 \arctan (a x)}+\frac {3}{8 \arctan (a x)}\right )d\arctan (a x)}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-a \int \frac {x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-3 a \int \frac {x^2}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
| \(\Big \downarrow \) 5505 |
| \(\displaystyle \frac {-\frac {\int \frac {a^2 x^2}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {3 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sin (\arctan (a x))^2}{\arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {3 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {-\frac {\int \left (\frac {1}{2 \arctan (a x)}-\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}\right )d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {3 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \log (\arctan (a x))-\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {3 \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \log (\arctan (a x))-\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {3 \int \left (\frac {1}{8 \arctan (a x)}-\frac {\cos (4 \arctan (a x))}{8 \arctan (a x)}\right )d\arctan (a x)}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \log (\arctan (a x))-\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{2} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {3 \left (\frac {1}{8} \log (\arctan (a x))-\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))\right )}{a^2}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2}-\frac {x}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
-((-(x/(a*(1 + a^2*x^2)^2*ArcTan[a*x])) - (3*(-1/8*CosIntegral[4*ArcTan[a* x]] + Log[ArcTan[a*x]]/8))/a^2 + (CosIntegral[2*ArcTan[a*x]]/2 + CosIntegr al[4*ArcTan[a*x]]/8 + (3*Log[ArcTan[a*x]])/8)/a^2)/(a^2*c^3)) + (-(x/(a*(1 + a^2*x^2)*ArcTan[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)/(a ^2*c^3)
3.6.59.3.1 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[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
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[1/e Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar cTan[c*x])^p, x], x] - Simp[d/e 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] && IGtQ[m, 1] && NeQ[p, -1]
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 = 9.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(-\frac {4 \,\operatorname {Ci}\left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\sin \left (4 \arctan \left (a x \right )\right )+2 \sin \left (2 \arctan \left (a x \right )\right )}{8 a^{4} c^{3} \arctan \left (a x \right )}\) | \(60\) |
| default | \(-\frac {4 \,\operatorname {Ci}\left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\sin \left (4 \arctan \left (a x \right )\right )+2 \sin \left (2 \arctan \left (a x \right )\right )}{8 a^{4} c^{3} \arctan \left (a x \right )}\) | \(60\) |
-1/8/a^4/c^3*(4*Ci(4*arctan(a*x))*arctan(a*x)-4*Ci(2*arctan(a*x))*arctan(a *x)-sin(4*arctan(a*x))+2*sin(2*arctan(a*x)))/arctan(a*x)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 292, normalized size of antiderivative = 3.40 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=-\frac {4 \, a^{3} x^{3} + {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{4 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )} \arctan \left (a x\right )} \]
-1/4*(4*a^3*x^3 + (a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)*log_integral((a^4* x^4 + 4*I*a^3*x^3 - 6*a^2*x^2 - 4*I*a*x + 1)/(a^4*x^4 + 2*a^2*x^2 + 1)) + (a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)*log_integral((a^4*x^4 - 4*I*a^3*x^3 - 6*a^2*x^2 + 4*I*a*x + 1)/(a^4*x^4 + 2*a^2*x^2 + 1)) - (a^4*x^4 + 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^4*x^4 + 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^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3)*arctan(a*x))
\[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {\int \frac {x^{3}}{a^{6} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \]
Integral(x**3/(a**6*x**6*atan(a*x)**2 + 3*a**4*x**4*atan(a*x)**2 + 3*a**2* x**2*atan(a*x)**2 + atan(a*x)**2), x)/c**3
\[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}} \,d x } \]
-(x^3 + (a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)*arctan(a*x)*integrate((a^2*x ^4 - 3*x^2)/((a^7*c^3*x^6 + 3*a^5*c^3*x^4 + 3*a^3*c^3*x^2 + a*c^3)*arctan( a*x)), x))/((a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)*arctan(a*x))
\[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]