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

Optimal. Leaf size=21 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^2}{\tan ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0216738, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right )^2}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.698005, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right )^2}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.929, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}{ \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)^2/arctan(a*x)^3,x)

[Out]

int((a^2*c*x^2+c)^2/arctan(a*x)^3,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} - 6 \, a \arctan \left (a x\right )^{2} \int \frac{7 \, a^{6} c^{2} x^{6} + 15 \, a^{4} c^{2} x^{4} + 9 \, a^{2} c^{2} x^{2} + c^{2}}{\arctan \left (a x\right )}\,{d x} + c^{2} + 6 \,{\left (a^{7} c^{2} x^{7} + 3 \, a^{5} c^{2} x^{5} + 3 \, a^{3} c^{2} x^{3} + a c^{2} x\right )} \arctan \left (a x\right )}{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)^2/arctan(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 - 2*a*arctan(a*x)^2*integrate(3*(7*a^6*c^2*x^6 + 15*a^4*c^2*
x^4 + 9*a^2*c^2*x^2 + c^2)/arctan(a*x), x) + c^2 + 6*(a^7*c^2*x^7 + 3*a^5*c^2*x^5 + 3*a^3*c^2*x^3 + a*c^2*x)*a
rctan(a*x))/(a*arctan(a*x)^2)

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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 )^{3}}, 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)^3,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)/arctan(a*x)^3, 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}^{3}{\left (a x \right )}}\, dx + \int \frac{a^{4} x^{4}}{\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)**2/atan(a*x)**3,x)

[Out]

c**2*(Integral(2*a**2*x**2/atan(a*x)**3, x) + Integral(a**4*x**4/atan(a*x)**3, x) + Integral(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 )}^{2}}{\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)^2/arctan(a*x)^3,x, algorithm="giac")

[Out]

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