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

Optimal. Leaf size=19 \[ \text{Unintegrable}\left (\frac{a^2 c x^2+c}{\tan ^{-1}(a x)^3},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0126611, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + a^2*c*x^2)/ArcTan[a*x]^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.15084, size = 0, normalized size = 0. \[ \int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.793, size = 0, normalized size = 0. \begin{align*} \int{\frac{{a}^{2}c{x}^{2}+c}{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{4} c x^{4} + 2 \, a^{2} c x^{2} - 4 \, a \arctan \left (a x\right )^{2} \int \frac{5 \, a^{4} c x^{4} + 6 \, a^{2} c x^{2} + c}{\arctan \left (a x\right )}\,{d x} + 4 \,{\left (a^{5} c x^{5} + 2 \, a^{3} c x^{3} + a c x\right )} \arctan \left (a x\right ) + c}{2 \, a \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^4*c*x^4 + 2*a^2*c*x^2 - 2*a*arctan(a*x)^2*integrate(2*(5*a^4*c*x^4 + 6*a^2*c*x^2 + c)/arctan(a*x), x)
+ 4*(a^5*c*x^5 + 2*a^3*c*x^3 + a*c*x)*arctan(a*x) + c)/(a*arctan(a*x)^2)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} c x^{2} + c}{\arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} c \left (\int \frac{a^{2} x^{2}}{\operatorname{atan}^{3}{\left (a x \right )}}\, dx + \int \frac{1}{\operatorname{atan}^{3}{\left (a x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a^{2} c x^{2} + c}{\arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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