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

Optimal. Leaf size=21 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^2}{\tan ^{-1}(a x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0240531, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right )^2}{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + a^2*c*x^2)^2/ArcTan[a*x],x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.433371, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right )^2}{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.861, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}{\arctan \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a^2*c*x^2+c)^2/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 )}^{2}}{\arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)/arctan(a*x), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int \frac{2 a^{2} x^{2}}{\operatorname{atan}{\left (a x \right )}}\, dx + \int \frac{a^{4} x^{4}}{\operatorname{atan}{\left (a x \right )}}\, dx + \int \frac{1}{\operatorname{atan}{\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)**2/atan(a*x),x)

[Out]

c**2*(Integral(2*a**2*x**2/atan(a*x), x) + Integral(a**4*x**4/atan(a*x), x) + Integral(1/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 )}^{2}}{\arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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