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

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

[Out]

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

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Rubi [A]  time = 0.0541744, 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 = {} \[ \int x^m \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.76929, size = 0, normalized size = 0. \[ \int x^m \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.588, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \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^m*(a^2*c*x^2+c)^2*arctan(a*x)^2,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} x^{m} \arctan \left (a x\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*x^m*arctan(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a**2*c*x**2+c)**2*atan(a*x)**2,x)

[Out]

c**2*(Integral(x**m*atan(a*x)**2, x) + Integral(2*a**2*x**2*x**m*atan(a*x)**2, x) + Integral(a**4*x**4*x**m*at
an(a*x)**2, x))

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{2} x^{m} \arctan \left (a x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="giac")

[Out]

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