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

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\frac{1}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right ) \]

[Out]

Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]

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Rubi [A]  time = 0.04321, 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 = {} \[ \int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]),x]

[Out]

Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]

Rubi steps

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

Mathematica [A]  time = 0.201257, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]),x]

[Out]

Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]

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Maple [A]  time = 0.483, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\arctan \left ( ax \right ) }{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/arctan(a*x)/(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{1}{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(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{1}{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(sqrt(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*} \int \frac{1}{\sqrt{c \left (a^{2} x^{2} + 1\right )} \operatorname{atan}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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