3.1030 \(\int \frac{1}{x^2 (c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{4 \text{Unintegrable}\left (\frac{1}{x^3 \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )}{a}-6 a \text{Unintegrable}\left (\frac{1}{x \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )-\frac{2}{a c x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}} \]

[Out]

-2/(a*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*Unintegrable[1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[
a*x]]), x])/a - 6*a*Unintegrable[1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]

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Rubi [A]  time = 0.35991, 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 = {} \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-2/(a*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*
x]]), x])/a - 6*a*Defer[Int][1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2}{a c x^2 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{4 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a}-(6 a) \int \frac{1}{x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 19.4375, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.81, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}} \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^2/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(3/2),x)

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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