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

Optimal. Leaf size=38 \[ \frac {\text {Int}\left (\frac {x}{\tan ^{-1}(a x)^2},x\right )}{a c}-\frac {x^2}{2 a c \tan ^{-1}(a x)^2} \]

[Out]

-1/2*x^2/a/c/arctan(a*x)^2+Unintegrable(x/arctan(a*x)^2,x)/a/c

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2/((a^2*c*x^2 + c)*arctan(a*x)^3), x)

________________________________________________________________________________________

giac [A]  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^2/(a^2*c*x^2+c)/arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.88, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {-2 \, \mathit {sage}_{0} x \arctan \left (a x\right )^{2} + a x^{2} + 2 \, {\left (a^{2} x^{3} + x\right )} \arctan \left (a x\right )}{2 \, a^{2} c \arctan \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________