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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{x \left (a^2 c x^2+c\right )}{\tan ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0235311, 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 = {} \[ \int \frac{x \left (c+a^2 c x^2\right )}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.09627, size = 0, normalized size = 0. \[ \int \frac{x \left (c+a^2 c x^2\right )}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.871, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ({a}^{2}c{x}^{2}+c \right ) }{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{5} c x^{5} + 2 \, a^{3} c x^{3} - 2 \, a^{2} c{\left (\int \frac{15 \, a^{4} x^{5}}{\arctan \left (a x\right )}\,{d x} + \int \frac{22 \, a^{2} x^{3}}{\arctan \left (a x\right )}\,{d x} + \int \frac{7 \, x}{\arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )^{2} + a c x +{\left (5 \, a^{6} c x^{6} + 11 \, a^{4} c x^{4} + 7 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}{2 \, a^{2} \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^5*c*x^5 + 2*a^3*c*x^3 - 2*a^2*arctan(a*x)^2*integrate((15*a^4*c*x^5 + 22*a^2*c*x^3 + 7*c*x)/arctan(a*x
), x) + a*c*x + (5*a^6*c*x^6 + 11*a^4*c*x^4 + 7*a^2*c*x^2 + c)*arctan(a*x))/(a^2*arctan(a*x)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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