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

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{x^m}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.104568, 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 = {} \[ \int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.469357, size = 0, normalized size = 0. \[ \int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.109, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(x^m/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/atan(a*x)**2/(a**2*c*x**2+c)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

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