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

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

[Out]

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

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Rubi [A]  time = 0.0362287, 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 = {} \[ \int \frac{x \left (c+a^2 c x^2\right )^2}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.893431, size = 0, normalized size = 0. \[ \int \frac{x \left (c+a^2 c x^2\right )^2}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.934, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(a^2*c*x^2+c)^2/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^{7} + 3 \, a^{4} c^{2} x^{5} + 3 \, a^{2} c^{2} x^{3} - c^{2}{\left (\int \frac{7 \, a^{6} x^{6}}{\arctan \left (a x\right )}\,{d x} + \int \frac{15 \, a^{4} x^{4}}{\arctan \left (a x\right )}\,{d x} + \int \frac{9 \, a^{2} x^{2}}{\arctan \left (a x\right )}\,{d x} + \int \frac{1}{\arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right ) + c^{2} x}{a \arctan \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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