\(\int \frac {x^2}{(c+a^2 c x^2)^2 \arctan (a x)^2} \, dx\) [552]

Optimal result

Integrand size = 22, antiderivative size = 43 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=-\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\text {Si}(2 \arctan (a x))}{a^3 c^2} \]

[Out]

-x^2/a/c^2/(a^2*x^2+1)/arctan(a*x)+Si(2*arctan(a*x))/a^3/c^2

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5062, 5090, 4491, 12, 3380} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {\text {Si}(2 \arctan (a x))}{a^3 c^2}-\frac {x^2}{a c^2 \left (a^2 x^2+1\right ) \arctan (a x)} \]

[In]

Int[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]

[Out]

-(x^2/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x])) + SinIntegral[2*ArcTan[a*x]]/(a^3*c^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rule 5062

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[
(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Dist[f*(m/(b*c*(p + 1))), Int[
(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e
, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]

Rule 5090

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[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])

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {2 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx}{a} \\ & = -\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {2 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\arctan (a x)\right )}{a^3 c^2} \\ & = -\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {2 \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\arctan (a x)\right )}{a^3 c^2} \\ & = -\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\arctan (a x)\right )}{a^3 c^2} \\ & = -\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\text {Si}(2 \arctan (a x))}{a^3 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {-\frac {a^2 x^2}{\left (1+a^2 x^2\right ) \arctan (a x)}+\text {Si}(2 \arctan (a x))}{a^3 c^2} \]

[In]

Integrate[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]

[Out]

(-((a^2*x^2)/((1 + a^2*x^2)*ArcTan[a*x])) + SinIntegral[2*ArcTan[a*x]])/(a^3*c^2)

Maple [A] (verified)

Time = 6.99 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {2 \,\operatorname {Si}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (2 \arctan \left (a x \right )\right )-1}{2 a^{3} c^{2} \arctan \left (a x \right )}\)	\(37\)
default	\(\frac {2 \,\operatorname {Si}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (2 \arctan \left (a x \right )\right )-1}{2 a^{3} c^{2} \arctan \left (a x \right )}\)	\(37\)

[In]

int(x^2/(a^2*c*x^2+c)^2/arctan(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/2/a^3/c^2*(2*Si(2*arctan(a*x))*arctan(a*x)+cos(2*arctan(a*x))-1)/arctan(a*x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.86 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=-\frac {2 \, a^{2} x^{2} - {\left (i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - {\left (-i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{2 \, {\left (a^{5} c^{2} x^{2} + a^{3} c^{2}\right )} \arctan \left (a x\right )} \]

[In]

integrate(x^2/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="fricas")

[Out]

-1/2*(2*a^2*x^2 - (I*a^2*x^2 + I)*arctan(a*x)*log_integral(-(a^2*x^2 + 2*I*a*x - 1)/(a^2*x^2 + 1)) - (-I*a^2*x
^2 - I)*arctan(a*x)*log_integral(-(a^2*x^2 - 2*I*a*x - 1)/(a^2*x^2 + 1)))/((a^5*c^2*x^2 + a^3*c^2)*arctan(a*x)
)

Sympy [F]

\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {\int \frac {x^{2}}{a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]

[In]

integrate(x**2/(a**2*c*x**2+c)**2/atan(a*x)**2,x)

[Out]

Integral(x**2/(a**4*x**4*atan(a*x)**2 + 2*a**2*x**2*atan(a*x)**2 + atan(a*x)**2), x)/c**2

Maxima [F]

\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^2/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="maxima")

[Out]

(4*(a^3*c^2*x^2 + a*c^2)*arctan(a*x)*integrate(1/2*x/((a^5*c^2*x^4 + 2*a^3*c^2*x^2 + a*c^2)*arctan(a*x)), x) -
x^2)/((a^3*c^2*x^2 + a*c^2)*arctan(a*x))

Giac [F]

\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^2/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

int(x^2/(atan(a*x)^2*(c + a^2*c*x^2)^2),x)

[Out]

int(x^2/(atan(a*x)^2*(c + a^2*c*x^2)^2), x)