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

Optimal. Leaf size=42 \[ -\frac{2 \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{\tan ^{-1}(a x)}},x\right )}{a c}-\frac{2}{a c x \sqrt{\tan ^{-1}(a x)}} \]

[Out]

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

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Rubi [A]  time = 0.0764422, 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 = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2)),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.200762, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2)),x]

[Out]

Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2)), x]

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Maple [A]  time = 0.145, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({a}^{2}c{x}^{2}+c \right ) } \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a^2*c*x^2+c)/arctan(a*x)^(3/2),x)

[Out]

int(1/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(1/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(1/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{1}{a^{2} x^{3} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )} + x \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(1/x/(a**2*c*x**2+c)/atan(a*x)**(3/2),x)

[Out]

Integral(1/(a**2*x**3*atan(a*x)**(3/2) + x*atan(a*x)**(3/2)), x)/c

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^2*c*x^2+c)/arctan(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((a^2*c*x^2 + c)*x*arctan(a*x)^(3/2)), x)