3.985 \(\int \frac{x}{(c+a^2 c x^2) \tan ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=36 \[ \frac{2 \text{Unintegrable}\left (\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right )}{a c}-\frac{2 x}{a c \sqrt{\tan ^{-1}(a x)}} \]

[Out]

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

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Rubi [A]  time = 0.0480899, 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 = {} \[ \int \frac{x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x}{a c \sqrt{\tan ^{-1}(a x)}}+\frac{2 \int \frac{1}{\sqrt{\tan ^{-1}(a x)}} \, dx}{a c}\\ \end{align*}

Mathematica [A]  time = 1.01886, size = 0, normalized size = 0. \[ \int \frac{x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)),x]

[Out]

Integrate[x/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)), x]

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Maple [A]  time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{{a}^{2}c{x}^{2}+c} \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, 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)^(3/2),x)

[Out]

int(x/(a^2*c*x^2+c)/arctan(a*x)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a^2*c*x^2+c)/arctan(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a^2*c*x^2+c)/arctan(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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}^{\frac{3}{2}}{\left (a x \right )} + \operatorname{atan}^{\frac{3}{2}}{\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)**(3/2),x)

[Out]

Integral(x/(a**2*x**2*atan(a*x)**(3/2) + atan(a*x)**(3/2)), 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 )^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="giac")

[Out]

integrate(x/((a^2*c*x^2 + c)*arctan(a*x)^(3/2)), x)