Integrand size = 22, antiderivative size = 67 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}-\frac {1}{a^3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\text {Si}(4 \arctan (a x))}{2 a^3 c^3} \]
1/a^3/c^3/(a^2*x^2+1)^2/arctan(a*x)-1/a^3/c^3/(a^2*x^2+1)/arctan(a*x)+1/2* Si(4*arctan(a*x))/a^3/c^3
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {-2 a^2 x^2+\left (1+a^2 x^2\right )^2 \arctan (a x) \text {Si}(4 \arctan (a x))}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)} \]
(-2*a^2*x^2 + (1 + a^2*x^2)^2*ArcTan[a*x]*SinIntegral[4*ArcTan[a*x]])/(2*a ^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])
Time = 0.71 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.52, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5499, 27, 5437, 5505, 4906, 27, 2009, 3042, 3780}
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.
\(\displaystyle \int \frac {x^2}{\arctan (a x)^2 \left (a^2 c x^2+c\right )^3} \, dx\) |
\(\Big \downarrow \) 5499 |
\(\displaystyle \frac {\int \frac {1}{c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx}{a^2 c}-\frac {\int \frac {1}{c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx}{a^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)^2}dx}{a^2 c^3}-\frac {\int \frac {1}{\left (a^2 x^2+1\right )^3 \arctan (a x)^2}dx}{a^2 c^3}\) |
\(\Big \downarrow \) 5437 |
\(\displaystyle \frac {-2 a \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-4 a \int \frac {x}{\left (a^2 x^2+1\right )^3 \arctan (a x)}dx-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
\(\Big \downarrow \) 5505 |
\(\displaystyle \frac {-\frac {2 \int \frac {a x}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {4 \int \frac {a x}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {-\frac {2 \int \frac {\sin (2 \arctan (a x))}{2 \arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {4 \int \left (\frac {\sin (2 \arctan (a x))}{4 \arctan (a x)}+\frac {\sin (4 \arctan (a x))}{8 \arctan (a x)}\right )d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {\int \frac {\sin (2 \arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {4 \int \left (\frac {\sin (2 \arctan (a x))}{4 \arctan (a x)}+\frac {\sin (4 \arctan (a x))}{8 \arctan (a x)}\right )d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}}{a^2 c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\int \frac {\sin (2 \arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}-\frac {4 \left (\frac {1}{4} \text {Si}(2 \arctan (a x))+\frac {1}{8} \text {Si}(4 \arctan (a x))\right )}{a}}{a^2 c^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\int \frac {\sin (2 \arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a}-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}}{a^2 c^3}-\frac {-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}-\frac {4 \left (\frac {1}{4} \text {Si}(2 \arctan (a x))+\frac {1}{8} \text {Si}(4 \arctan (a x))\right )}{a}}{a^2 c^3}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {-\frac {1}{a \left (a^2 x^2+1\right ) \arctan (a x)}-\frac {\text {Si}(2 \arctan (a x))}{a}}{a^2 c^3}-\frac {-\frac {1}{a \left (a^2 x^2+1\right )^2 \arctan (a x)}-\frac {4 \left (\frac {1}{4} \text {Si}(2 \arctan (a x))+\frac {1}{8} \text {Si}(4 \arctan (a x))\right )}{a}}{a^2 c^3}\) |
(-(1/(a*(1 + a^2*x^2)*ArcTan[a*x])) - SinIntegral[2*ArcTan[a*x]]/a)/(a^2*c ^3) - (-(1/(a*(1 + a^2*x^2)^2*ArcTan[a*x])) - (4*(SinIntegral[2*ArcTan[a*x ]]/4 + SinIntegral[4*ArcTan[a*x]]/8))/a)/(a^2*c^3)
3.6.60.3.1 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[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
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_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_.)*(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[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 = 8.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.55
method | result | size |
derivativedivides | \(\frac {4 \,\operatorname {Si}\left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (4 \arctan \left (a x \right )\right )-1}{8 a^{3} c^{3} \arctan \left (a x \right )}\) | \(37\) |
default | \(\frac {4 \,\operatorname {Si}\left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (4 \arctan \left (a x \right )\right )-1}{8 a^{3} c^{3} \arctan \left (a x \right )}\) | \(37\) |
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.93 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=-\frac {4 \, a^{2} x^{2} - {\left (i \, a^{4} x^{4} + 2 i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - {\left (-i \, a^{4} x^{4} - 2 i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right )}{4 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )} \arctan \left (a x\right )} \]
-1/4*(4*a^2*x^2 - (I*a^4*x^4 + 2*I*a^2*x^2 + I)*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) ) - (-I*a^4*x^4 - 2*I*a^2*x^2 - I)*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^7*c^3* x^4 + 2*a^5*c^3*x^2 + a^3*c^3)*arctan(a*x))
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {\int \frac {x^{2}}{a^{6} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \]
Integral(x**2/(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
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}} \,d x } \]
-((a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)*arctan(a*x)*integrate(2*(a^2*x^3 - x)/((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 ) + x^2)/((a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)*arctan(a*x))
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]