3.997 \(\int \frac{1}{x^4 (c+a^2 c x^2)^2 \tan ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{8 \text{Unintegrable}\left (\frac{1}{x^5 \left (a^2 c x^2+c\right )^2 \sqrt{\tan ^{-1}(a x)}},x\right )}{a}-12 a \text{Unintegrable}\left (\frac{1}{x^3 \left (a^2 c x^2+c\right )^2 \sqrt{\tan ^{-1}(a x)}},x\right )-\frac{2}{a c^2 x^4 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}} \]

[Out]

-2/(a*c^2*x^4*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (8*Unintegrable[1/(x^5*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]),
x])/a - 12*a*Unintegrable[1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]

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Rubi [A]  time = 0.201475, 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 = {} \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-2/(a*c^2*x^4*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (8*Defer[Int][1/(x^5*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]
)/a - 12*a*Defer[Int][1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2}{a c^2 x^4 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{8 \int \frac{1}{x^5 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{a}-(12 a) \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 6.77262, size = 0, normalized size = 0. \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]

[Out]

Integrate[1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x]

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Maple [A]  time = 0.66, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ({a}^{2}c{x}^{2}+c \right ) ^{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^4/(a^2*c*x^2+c)^2/arctan(a*x)^(3/2),x)

[Out]

int(1/x^4/(a^2*c*x^2+c)^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^4/(a^2*c*x^2+c)^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^4/(a^2*c*x^2+c)^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**4/(a**2*c*x**2+c)**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 )}^{2} x^{4} \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^4/(a^2*c*x^2+c)^2/arctan(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((a^2*c*x^2 + c)^2*x^4*arctan(a*x)^(3/2)), x)