3.643 \(\int (\frac{x^3}{(1+a^2 x^2) \tan ^{-1}(a x)^3}-\frac{3 x^2}{2 a \tan ^{-1}(a x)^2}) \, dx\)

Optimal. Leaf size=16 \[ -\frac{x^3}{2 a \tan ^{-1}(a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.090723, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {4926} \[ -\frac{x^3}{2 a \tan ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4926

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*} \int \left (\frac{x^3}{\left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3}-\frac{3 x^2}{2 a \tan ^{-1}(a x)^2}\right ) \, dx &=-\frac{3 \int \frac{x^2}{\tan ^{-1}(a x)^2} \, dx}{2 a}+\int \frac{x^3}{\left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3} \, dx\\ &=-\frac{x^3}{2 a \tan ^{-1}(a x)^2}\\ \end{align*}

Mathematica [A]  time = 0.151827, size = 16, normalized size = 1. \[ -\frac{x^3}{2 a \tan ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.108, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ({a}^{2}{x}^{2}+1 \right ) \left ( \arctan \left ( ax \right ) \right ) ^{3}}}-{\frac{3\,{x}^{2}}{2\,a \left ( \arctan \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.36384, size = 19, normalized size = 1.19 \begin{align*} -\frac{x^{3}}{2 \, a \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.58476, size = 38, normalized size = 2.38 \begin{align*} -\frac{x^{3}}{2 \, a \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int - \frac{2 a x^{3}}{a^{2} x^{2} \operatorname{atan}^{3}{\left (a x \right )} + \operatorname{atan}^{3}{\left (a x \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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