\(\int (\frac {x^3}{(1+a^2 x^2) \arctan (a x)^3}-\frac {3 x^2}{2 a \arctan (a x)^2}) \, dx\) [643]

Optimal result

Integrand size = 38, antiderivative size = 16 \[ \int \left (\frac {x^3}{\left (1+a^2 x^2\right ) \arctan (a x)^3}-\frac {3 x^2}{2 a \arctan (a x)^2}\right ) \, dx=-\frac {x^3}{2 a \arctan (a x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {5046} \[ \int \left (\frac {x^3}{\left (1+a^2 x^2\right ) \arctan (a x)^3}-\frac {3 x^2}{2 a \arctan (a x)^2}\right ) \, dx=-\frac {x^3}{2 a \arctan (a x)^2} \]

[In]

Int[x^3/((1 + a^2*x^2)*ArcTan[a*x]^3) - (3*x^2)/(2*a*ArcTan[a*x]^2),x]

[Out]

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

Rule 5046

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

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (\frac {x^3}{\left (1+a^2 x^2\right ) \arctan (a x)^3}-\frac {3 x^2}{2 a \arctan (a x)^2}\right ) \, dx=-\frac {x^3}{2 a \arctan (a x)^2} \]

[In]

Integrate[x^3/((1 + a^2*x^2)*ArcTan[a*x]^3) - (3*x^2)/(2*a*ArcTan[a*x]^2),x]

[Out]

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

Maple [A] (verified)

Time = 114.78 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
parallelrisch	\(-\frac {x^{3}}{2 a \arctan \left (a x \right )^{2}}\)	\(15\)
risch	\(\frac {2 x^{3}}{a \left (\ln \left (-i a x +1\right )-\ln \left (i a x +1\right )\right )^{2}}\)	\(30\)
default	\(-\frac {x^{3}}{2 a \arctan \left (a x \right )^{2}}-\frac {x^{2}}{2 a^{2} \arctan \left (a x \right )^{3}}-\frac {1}{2 a^{4} \arctan \left (a x \right )^{3}}-\frac {3 x}{2 a^{3} \arctan \left (a x \right )^{4}}-\frac {3 i}{2 a^{4} \arctan \left (a x \right )^{4}}+\frac {\left (-i \arctan \left (a x \right )+x \arctan \left (a x \right ) a +3\right ) \left (a x +i\right )}{2 \arctan \left (a x \right )^{4} a^{4}}\)	\(98\)
parts	\(-\frac {x^{3}}{2 a \arctan \left (a x \right )^{2}}-\frac {x^{2}}{2 a^{2} \arctan \left (a x \right )^{3}}-\frac {1}{2 a^{4} \arctan \left (a x \right )^{3}}-\frac {3 x}{2 a^{3} \arctan \left (a x \right )^{4}}-\frac {3 i}{2 a^{4} \arctan \left (a x \right )^{4}}+\frac {\left (-i \arctan \left (a x \right )+x \arctan \left (a x \right ) a +3\right ) \left (a x +i\right )}{2 \arctan \left (a x \right )^{4} a^{4}}\)	\(98\)

[In]

int(x^3/(a^2*x^2+1)/arctan(a*x)^3-3/2*x^2/a/arctan(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (\frac {x^3}{\left (1+a^2 x^2\right ) \arctan (a x)^3}-\frac {3 x^2}{2 a \arctan (a x)^2}\right ) \, dx=-\frac {x^{3}}{2 \, a \arctan \left (a x\right )^{2}} \]

[In]

integrate(x^3/(a^2*x^2+1)/arctan(a*x)^3-3/2*x^2/a/arctan(a*x)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \left (\frac {x^3}{\left (1+a^2 x^2\right ) \arctan (a x)^3}-\frac {3 x^2}{2 a \arctan (a x)^2}\right ) \, dx=- \frac {\int \left (- \frac {2 a x^{3}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\right )\, dx + \int \frac {3 x^{2} \operatorname {atan}{\left (a x \right )}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x^{4} \operatorname {atan}{\left (a x \right )}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{2 a} \]

[In]

integrate(x**3/(a**2*x**2+1)/atan(a*x)**3-3/2*x**2/a/atan(a*x)**2,x)

[Out]

-(Integral(-2*a*x**3/(a**2*x**2*atan(a*x)**3 + atan(a*x)**3), x) + Integral(3*x**2*atan(a*x)/(a**2*x**2*atan(a
*x)**3 + atan(a*x)**3), x) + Integral(3*a**2*x**4*atan(a*x)/(a**2*x**2*atan(a*x)**3 + atan(a*x)**3), x))/(2*a)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (\frac {x^3}{\left (1+a^2 x^2\right ) \arctan (a x)^3}-\frac {3 x^2}{2 a \arctan (a x)^2}\right ) \, dx=-\frac {x^{3}}{2 \, a \arctan \left (a x\right )^{2}} \]

[In]

integrate(x^3/(a^2*x^2+1)/arctan(a*x)^3-3/2*x^2/a/arctan(a*x)^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \left (\frac {x^3}{\left (1+a^2 x^2\right ) \arctan (a x)^3}-\frac {3 x^2}{2 a \arctan (a x)^2}\right ) \, dx=\int { \frac {x^{3}}{{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3}} - \frac {3 \, x^{2}}{2 \, a \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^3/(a^2*x^2+1)/arctan(a*x)^3-3/2*x^2/a/arctan(a*x)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (\frac {x^3}{\left (1+a^2 x^2\right ) \arctan (a x)^3}-\frac {3 x^2}{2 a \arctan (a x)^2}\right ) \, dx=-\frac {x^3}{2\,a\,{\mathrm {atan}\left (a\,x\right )}^2} \]

[In]

int(x^3/(atan(a*x)^3*(a^2*x^2 + 1)) - (3*x^2)/(2*a*atan(a*x)^2),x)

[Out]

-x^3/(2*a*atan(a*x)^2)