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

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

[Out]

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

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Rubi [A]  time = 0.06794, 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 = {} \[ \int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.5551, size = 0, normalized size = 0. \[ \int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.686, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}\sqrt{{a}^{2}c{x}^{2}+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} c x^{2} + c} 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)^(1/2)/arctan(a*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} 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)^(1/2)/arctan(a*x)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(c*(a**2*x**2 + 1))/atan(a*x)**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} c x^{2} + c} 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)^(1/2)/arctan(a*x)^2,x, algorithm="giac")

[Out]

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