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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^2}{x \tan ^{-1}(a x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0489749, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right )^2}{x \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.10079, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right )^2}{x \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.873, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}{x \left ( \arctan \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{6} c^{2} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} - c^{2} x{\left (\int \frac{5 \, a^{6} x^{4}}{\arctan \left (a x\right )}\,{d x} + \int \frac{9 \, a^{4} x^{2}}{\arctan \left (a x\right )}\,{d x} + \int \frac{3 \, a^{2}}{\arctan \left (a x\right )}\,{d x} + \int -\frac{1}{x^{2} \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right ) + c^{2}}{a x \arctan \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 - x*arctan(a*x)*integrate((5*a^6*c^2*x^6 + 9*a^4*c^2*x^4 + 3*a^2
*c^2*x^2 - c^2)/(x^2*arctan(a*x)), x) + c^2)/(a*x*arctan(a*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}}{x \arctan \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int \frac{1}{x \operatorname{atan}^{2}{\left (a x \right )}}\, dx + \int \frac{2 a^{2} x}{\operatorname{atan}^{2}{\left (a x \right )}}\, dx + \int \frac{a^{4} x^{3}}{\operatorname{atan}^{2}{\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/x/atan(a*x)**2,x)

[Out]

c**2*(Integral(1/(x*atan(a*x)**2), x) + Integral(2*a**2*x/atan(a*x)**2, x) + Integral(a**4*x**3/atan(a*x)**2,
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}}{x \arctan \left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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