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

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

[Out]

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

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Rubi [A]  time = 0.0451765, 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 = {} \[ \int \frac{x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.520343, size = 0, normalized size = 0. \[ \int \frac{x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.123, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ({a}^{2}c{x}^{2}+c \right ) \arctan \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(a**2*x**2*atan(a*x) + atan(a*x)), x)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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