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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{x \left (a^2 c x^2+c\right )^3}{\tan ^{-1}(a x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0381323, 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 = {} \[ \int \frac{x \left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.532251, size = 0, normalized size = 0. \[ \int \frac{x \left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.16, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ({a}^{2}c{x}^{2}+c \right ) ^{3}}{\arctan \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x}{\arctan \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x)/arctan(a*x), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int \frac{x}{\operatorname{atan}{\left (a x \right )}}\, dx + \int \frac{3 a^{2} x^{3}}{\operatorname{atan}{\left (a x \right )}}\, dx + \int \frac{3 a^{4} x^{5}}{\operatorname{atan}{\left (a x \right )}}\, dx + \int \frac{a^{6} x^{7}}{\operatorname{atan}{\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)**3/atan(a*x),x)

[Out]

c**3*(Integral(x/atan(a*x), x) + Integral(3*a**2*x**3/atan(a*x), x) + Integral(3*a**4*x**5/atan(a*x), x) + Int
egral(a**6*x**7/atan(a*x), 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 )}^{3} x}{\arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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