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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^3}{x \tan ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0489339, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right )^3}{x \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.21194, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right )^3}{x \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.163, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({a}^{2}c{x}^{2}+c \right ) ^{3}}{x \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)^3/x/arctan(a*x)^3,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{9} c^{3} x^{9} + 4 \, a^{7} c^{3} x^{7} + 6 \, a^{5} c^{3} x^{5} + 4 \, a^{3} c^{3} x^{3} - 2 \, c^{3} x^{2}{\left (\int \frac{28 \, a^{10} x^{7}}{\arctan \left (a x\right )}\,{d x} + \int \frac{81 \, a^{8} x^{5}}{\arctan \left (a x\right )}\,{d x} + \int \frac{76 \, a^{6} x^{3}}{\arctan \left (a x\right )}\,{d x} + \int \frac{22 \, a^{4} x}{\arctan \left (a x\right )}\,{d x} + \int \frac{1}{x^{3} \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )^{2} + a c^{3} x +{\left (7 \, a^{10} c^{3} x^{10} + 27 \, a^{8} c^{3} x^{8} + 38 \, a^{6} c^{3} x^{6} + 22 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \arctan \left (a x\right )}{2 \, a^{2} x^{2} \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)^3/x/arctan(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^9*c^3*x^9 + 4*a^7*c^3*x^7 + 6*a^5*c^3*x^5 + 4*a^3*c^3*x^3 + a*c^3*x - 2*x^2*arctan(a*x)^2*integrate((2
8*a^10*c^3*x^10 + 81*a^8*c^3*x^8 + 76*a^6*c^3*x^6 + 22*a^4*c^3*x^4 + c^3)/(x^3*arctan(a*x)), x) + (7*a^10*c^3*
x^10 + 27*a^8*c^3*x^8 + 38*a^6*c^3*x^6 + 22*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)*arctan(a*x))/(a^2*x^2*arctan(a*
x)^2)

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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}}{x \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)^3/x/arctan(a*x)^3,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)/(x*arctan(a*x)^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int \frac{1}{x \operatorname{atan}^{3}{\left (a x \right )}}\, dx + \int \frac{3 a^{2} x}{\operatorname{atan}^{3}{\left (a x \right )}}\, dx + \int \frac{3 a^{4} x^{3}}{\operatorname{atan}^{3}{\left (a x \right )}}\, dx + \int \frac{a^{6} x^{5}}{\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)**3/x/atan(a*x)**3,x)

[Out]

c**3*(Integral(1/(x*atan(a*x)**3), x) + Integral(3*a**2*x/atan(a*x)**3, x) + Integral(3*a**4*x**3/atan(a*x)**3
, x) + Integral(a**6*x**5/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 )}^{3}}{x \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)^3/x/arctan(a*x)^3,x, algorithm="giac")

[Out]

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