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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{x \left (a^2 c x^2+c\right )^3}{\tan ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0355532, 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 = {} \[ \int \frac{x \left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.01826, size = 0, normalized size = 0. \[ \int \frac{x \left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.348, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ({a}^{2}c{x}^{2}+c \right ) ^{3}}{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(a^2*c*x^2+c)^3/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 \, a^{2} c^{3}{\left (\int \frac{45 \, a^{8} x^{9}}{\arctan \left (a x\right )}\,{d x} + \int \frac{148 \, a^{6} x^{7}}{\arctan \left (a x\right )}\,{d x} + \int \frac{174 \, a^{4} x^{5}}{\arctan \left (a x\right )}\,{d x} + \int \frac{84 \, a^{2} x^{3}}{\arctan \left (a x\right )}\,{d x} + \int \frac{13 \, x}{\arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )^{2} + a c^{3} x +{\left (9 \, a^{10} c^{3} x^{10} + 37 \, a^{8} c^{3} x^{8} + 58 \, a^{6} c^{3} x^{6} + 42 \, a^{4} c^{3} x^{4} + 13 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )}{2 \, a^{2} \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a^2*c*x^2+c)^3/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*a^2*arctan(a*x)^2*integrate((4
5*a^8*c^3*x^9 + 148*a^6*c^3*x^7 + 174*a^4*c^3*x^5 + 84*a^2*c^3*x^3 + 13*c^3*x)/arctan(a*x), x) + (9*a^10*c^3*x
^10 + 37*a^8*c^3*x^8 + 58*a^6*c^3*x^6 + 42*a^4*c^3*x^4 + 13*a^2*c^3*x^2 + c^3)*arctan(a*x))/(a^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^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + 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(x*(a^2*c*x^2+c)^3/arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral((a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + 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{x}{\operatorname{atan}^{3}{\left (a x \right )}}\, dx + \int \frac{3 a^{2} x^{3}}{\operatorname{atan}^{3}{\left (a x \right )}}\, dx + \int \frac{3 a^{4} x^{5}}{\operatorname{atan}^{3}{\left (a x \right )}}\, dx + \int \frac{a^{6} x^{7}}{\operatorname{atan}^{3}{\left (a x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(x/atan(a*x)**3, x) + Integral(3*a**2*x**3/atan(a*x)**3, x) + Integral(3*a**4*x**5/atan(a*x)**3,
x) + Integral(a**6*x**7/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(x*(a^2*c*x^2+c)^3/arctan(a*x)^3,x, algorithm="giac")

[Out]

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