Integrand size = 19, antiderivative size = 169 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {15 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac {15 \arctan (a x)}{64 a c^3}+\frac {\arctan (a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \arctan (a x)}{8 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{8 a c^3} \]
-1/32*x/c^3/(a^2*x^2+1)^2-15/64*x/c^3/(a^2*x^2+1)-15/64*arctan(a*x)/a/c^3+ 1/8*arctan(a*x)/a/c^3/(a^2*x^2+1)^2+3/8*arctan(a*x)/a/c^3/(a^2*x^2+1)+1/4* x*arctan(a*x)^2/c^3/(a^2*x^2+1)^2+3/8*x*arctan(a*x)^2/c^3/(a^2*x^2+1)+1/8* arctan(a*x)^3/a/c^3
Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.58 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-a x \left (17+15 a^2 x^2\right )+\left (17-6 a^2 x^2-15 a^4 x^4\right ) \arctan (a x)+8 a x \left (5+3 a^2 x^2\right ) \arctan (a x)^2+8 \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{64 a c^3 \left (1+a^2 x^2\right )^2} \]
(-(a*x*(17 + 15*a^2*x^2)) + (17 - 6*a^2*x^2 - 15*a^4*x^4)*ArcTan[a*x] + 8* a*x*(5 + 3*a^2*x^2)*ArcTan[a*x]^2 + 8*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/(64*a *c^3*(1 + a^2*x^2)^2)
Time = 0.51 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.30, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {5435, 27, 215, 215, 218, 5427, 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.
\(\displaystyle \int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^3} \, dx\) |
\(\Big \downarrow \) 5435 |
\(\displaystyle \frac {3 \int \frac {\arctan (a x)^2}{c^2 \left (a^2 x^2+1\right )^2}dx}{4 c}-\frac {1}{8} \int \frac {1}{\left (a^2 c x^2+c\right )^3}dx+\frac {x \arctan (a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{4 c^3}-\frac {1}{8} \int \frac {1}{\left (a^2 c x^2+c\right )^3}dx+\frac {x \arctan (a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {3 \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{4 c^3}+\frac {1}{8} \left (-\frac {3 \int \frac {1}{\left (a^2 c x^2+c\right )^2}dx}{4 c}-\frac {x}{4 c^3 \left (a^2 x^2+1\right )^2}\right )+\frac {x \arctan (a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {3 \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{4 c^3}+\frac {1}{8} \left (-\frac {3 \left (\frac {\int \frac {1}{a^2 c x^2+c}dx}{2 c}+\frac {x}{2 c^2 \left (a^2 x^2+1\right )}\right )}{4 c}-\frac {x}{4 c^3 \left (a^2 x^2+1\right )^2}\right )+\frac {x \arctan (a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {3 \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{4 c^3}+\frac {x \arctan (a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3 \left (\frac {x}{2 c^2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a c^2}\right )}{4 c}-\frac {x}{4 c^3 \left (a^2 x^2+1\right )^2}\right )\) |
\(\Big \downarrow \) 5427 |
\(\displaystyle \frac {3 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{4 c^3}+\frac {x \arctan (a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3 \left (\frac {x}{2 c^2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a c^2}\right )}{4 c}-\frac {x}{4 c^3 \left (a^2 x^2+1\right )^2}\right )\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle \frac {3 \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{4 c^3}+\frac {x \arctan (a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3 \left (\frac {x}{2 c^2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a c^2}\right )}{4 c}-\frac {x}{4 c^3 \left (a^2 x^2+1\right )^2}\right )\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {3 \left (-a \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{4 c^3}+\frac {x \arctan (a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3 \left (\frac {x}{2 c^2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a c^2}\right )}{4 c}-\frac {x}{4 c^3 \left (a^2 x^2+1\right )^2}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {x \arctan (a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{4 c^3}+\frac {1}{8} \left (-\frac {3 \left (\frac {x}{2 c^2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a c^2}\right )}{4 c}-\frac {x}{4 c^3 \left (a^2 x^2+1\right )^2}\right )\) |
ArcTan[a*x]/(8*a*c^3*(1 + a^2*x^2)^2) + (x*ArcTan[a*x]^2)/(4*c^3*(1 + a^2* x^2)^2) + (-1/4*x/(c^3*(1 + a^2*x^2)^2) - (3*(x/(2*c^2*(1 + a^2*x^2)) + Ar cTan[a*x]/(2*a*c^2)))/(4*c))/8 + (3*((x*ArcTan[a*x]^2)/(2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a) - a*(-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))))/(4*c^3)
3.4.2.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sym bol] :> Simp[x*((a + b*ArcTan[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b *ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2) Int[x*((a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S ymbol] :> Simp[b*p*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d* (q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d* (q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int[(d + e *x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] & & EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
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]
Time = 0.67 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.73
method | result | size |
parallelrisch | \(\frac {8 a^{4} \arctan \left (a x \right )^{3} x^{4}-15 \arctan \left (a x \right ) a^{4} x^{4}+24 a^{3} \arctan \left (a x \right )^{2} x^{3}+16 \arctan \left (a x \right )^{3} x^{2} a^{2}-15 a^{3} x^{3}-6 a^{2} \arctan \left (a x \right ) x^{2}+40 a \arctan \left (a x \right )^{2} x +8 \arctan \left (a x \right )^{3}-17 a x +17 \arctan \left (a x \right )}{64 c^{3} \left (a^{2} x^{2}+1\right )^{2} a}\) | \(123\) |
derivativedivides | \(\frac {\frac {a x \arctan \left (a x \right )^{2}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 a x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {15}{8} a^{3} x^{3}+\frac {17}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {15 \arctan \left (a x \right )}{16}+\arctan \left (a x \right )^{3}}{4 c^{3}}}{a}\) | \(143\) |
default | \(\frac {\frac {a x \arctan \left (a x \right )^{2}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 a x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {15}{8} a^{3} x^{3}+\frac {17}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {15 \arctan \left (a x \right )}{16}+\arctan \left (a x \right )^{3}}{4 c^{3}}}{a}\) | \(143\) |
parts | \(\frac {x \arctan \left (a x \right )^{2}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{3}}{8 a \,c^{3}}-\frac {\frac {-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {15}{8} a^{3} x^{3}+\frac {17}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {15 \arctan \left (a x \right )}{16}}{a}+\frac {\arctan \left (a x \right )^{3}}{a}}{4 c^{3}}\) | \(149\) |
risch | \(\frac {i \ln \left (i a x +1\right )^{3}}{64 c^{3} a}-\frac {i \left (3 x^{4} \ln \left (-i a x +1\right ) a^{4}+6 a^{2} x^{2} \ln \left (-i a x +1\right )-6 i a^{3} x^{3}+3 \ln \left (-i a x +1\right )-10 i a x \right ) \ln \left (i a x +1\right )^{2}}{64 c^{3} \left (a^{2} x^{2}+1\right )^{2} a}+\frac {i \left (3 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}+6 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-12 i x^{3} \ln \left (-i a x +1\right ) a^{3}-12 a^{2} x^{2}+3 \ln \left (-i a x +1\right )^{2}-20 i a x \ln \left (-i a x +1\right )-16\right ) \ln \left (i a x +1\right )}{64 \left (a x +i\right )^{2} a \,c^{3} \left (a x -i\right )^{2}}-\frac {i \left (2 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}+15 \ln \left (i a x -1\right ) a^{4} x^{4}-15 \ln \left (-i a x -1\right ) a^{4} x^{4}+4 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}-12 i a^{3} x^{3} \ln \left (-i a x +1\right )^{2}+30 \ln \left (i a x -1\right ) a^{2} x^{2}-30 \ln \left (-i a x -1\right ) a^{2} x^{2}-24 a^{2} x^{2} \ln \left (-i a x +1\right )-30 i a^{3} x^{3}+2 \ln \left (-i a x +1\right )^{3}-20 i a x \ln \left (-i a x +1\right )^{2}+15 \ln \left (i a x -1\right )-15 \ln \left (-i a x -1\right )-32 \ln \left (-i a x +1\right )-34 i a x \right )}{128 \left (a x +i\right )^{2} a \,c^{3} \left (a x -i\right )^{2}}\) | \(461\) |
1/64*(8*a^4*arctan(a*x)^3*x^4-15*arctan(a*x)*a^4*x^4+24*a^3*arctan(a*x)^2* x^3+16*arctan(a*x)^3*x^2*a^2-15*a^3*x^3-6*a^2*arctan(a*x)*x^2+40*a*arctan( a*x)^2*x+8*arctan(a*x)^3-17*a*x+17*arctan(a*x))/c^3/(a^2*x^2+1)^2/a
Time = 0.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.67 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {15 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 8 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right )^{2} + 17 \, a x + {\left (15 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 17\right )} \arctan \left (a x\right )}{64 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
-1/64*(15*a^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^3 - 8*(3*a^3*x ^3 + 5*a*x)*arctan(a*x)^2 + 17*a*x + (15*a^4*x^4 + 6*a^2*x^2 - 17)*arctan( a*x))/(a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)
\[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Time = 0.34 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.37 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {1}{8} \, {\left (\frac {3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{2} + c^{3}} + \frac {3 \, \arctan \left (a x\right )}{a c^{3}}\right )} \arctan \left (a x\right )^{2} - \frac {{\left (15 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 17 \, a x + 15 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{64 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} + \frac {{\left (3 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )}{8 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \]
1/8*((3*a^2*x^3 + 5*x)/(a^4*c^3*x^4 + 2*a^2*c^3*x^2 + c^3) + 3*arctan(a*x) /(a*c^3))*arctan(a*x)^2 - 1/64*(15*a^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*a rctan(a*x)^3 + 17*a*x + 15*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x))*a^2/(a^7 *c^3*x^4 + 2*a^5*c^3*x^2 + a^3*c^3) + 1/8*(3*a^2*x^2 - 3*(a^4*x^4 + 2*a^2* x^2 + 1)*arctan(a*x)^2 + 4)*a*arctan(a*x)/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a ^2*c^3)
\[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
Time = 0.60 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.93 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {1}{2\,a^3\,c^3}+\frac {3\,x^2}{8\,a\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {15\,\mathrm {atan}\left (a\,x\right )}{64\,a\,c^3}-\frac {\frac {15\,a^2\,x^3}{8}+\frac {17\,x}{8}}{8\,a^4\,c^3\,x^4+16\,a^2\,c^3\,x^2+8\,c^3}+\frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {3\,x^3}{8\,c^3}+\frac {5\,x}{8\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}+\frac {{\mathrm {atan}\left (a\,x\right )}^3}{8\,a\,c^3} \]