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

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

[Out]

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

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Rubi [A]  time = 0.0213235, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.720318, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({a}^{2}c{x}^{2}+c \right ) ^{3}}{ \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)^3/arctan(a*x)^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3} - 8 \, a \arctan \left (a x\right ) \int \frac{a^{7} c^{3} x^{7} + 3 \, a^{5} c^{3} x^{5} + 3 \, a^{3} c^{3} x^{3} + a c^{3} x}{\arctan \left (a x\right )}\,{d x}}{a \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)^3/arctan(a*x)^2,x, algorithm="maxima")

[Out]

-(a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 + 4*a^2*c^3*x^2 + c^3 - a*arctan(a*x)*integrate(8*(a^7*c^3*x^7 +
3*a^5*c^3*x^5 + 3*a^3*c^3*x^3 + a*c^3*x)/arctan(a*x), x))/(a*arctan(a*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^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}{\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)^3/arctan(a*x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int \frac{3 a^{2} x^{2}}{\operatorname{atan}^{2}{\left (a x \right )}}\, dx + \int \frac{3 a^{4} x^{4}}{\operatorname{atan}^{2}{\left (a x \right )}}\, dx + \int \frac{a^{6} x^{6}}{\operatorname{atan}^{2}{\left (a x \right )}}\, dx + \int \frac{1}{\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)**3/atan(a*x)**2,x)

[Out]

c**3*(Integral(3*a**2*x**2/atan(a*x)**2, x) + Integral(3*a**4*x**4/atan(a*x)**2, x) + Integral(a**6*x**6/atan(
a*x)**2, x) + Integral(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 )}^{3}}{\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)^3/arctan(a*x)^2,x, algorithm="giac")

[Out]

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