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]

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

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Rubi [A]  time = 0.09, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.15, size = 16, normalized size = 1.00 \[ -\frac {x^3}{2 a \tan ^{-1}(a x)^2} \]

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]

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

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fricas [A]  time = 0.45, size = 14, normalized size = 0.88 \[ -\frac {x^{3}}{2 \, a \arctan \left (a x\right )^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [F]  time = 2.44, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{3}}-\frac {3 x^{2}}{2 a \arctan \left (a x \right )^{2}}\, dx \]

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 = 0.48, size = 14, normalized size = 0.88 \[ -\frac {x^{3}}{2 \, a \arctan \left (a x\right )^{2}} \]

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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mupad [B]  time = 0.43, size = 14, normalized size = 0.88 \[ -\frac {x^3}{2\,a\,{\mathrm {atan}\left (a\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \left (- \frac {2 a x^{3}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\right )\, dx + \int \frac {3 x^{2} \operatorname {atan}{\left (a x \right )}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x^{4} \operatorname {atan}{\left (a x \right )}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{2 a} \]

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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