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

Optimal. Leaf size=40 \[ -\frac{2 \text{Unintegrable}\left (\frac{1}{x^5 \tan ^{-1}(a x)^2},x\right )}{a c}-\frac{1}{2 a c x^4 \tan ^{-1}(a x)^2} \]

[Out]

-1/(2*a*c*x^4*ArcTan[a*x]^2) - (2*Unintegrable[1/(x^5*ArcTan[a*x]^2), x])/(a*c)

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Rubi [A]  time = 0.0819838, 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 = {} \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-1/(2*a*c*x^4*ArcTan[a*x]^2) - (2*Defer[Int][1/(x^5*ArcTan[a*x]^2), x])/(a*c)

Rubi steps

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

Mathematica [A]  time = 2.32177, size = 0, normalized size = 0. \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.974, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ({a}^{2}c{x}^{2}+c \right ) \left ( \arctan \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \, x^{5} \arctan \left (a x\right )^{2} \int \frac{3 \, a^{2} x^{2} + 5}{x^{6} \arctan \left (a x\right )}\,{d x} - a x + 4 \,{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{2 \, a^{2} c x^{5} \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*x^5*arctan(a*x)^2*integrate(2*(3*a^2*x^2 + 5)/(x^6*arctan(a*x)), x) - a*x + 4*(a^2*x^2 + 1)*arctan(a*x)
)/(a^2*c*x^5*arctan(a*x)^2)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{2} c x^{6} + c x^{4}\right )} \arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((a^2*c*x^6 + c*x^4)*arctan(a*x)^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{2} x^{6} \operatorname{atan}^{3}{\left (a x \right )} + x^{4} \operatorname{atan}^{3}{\left (a x \right )}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**2*x**6*atan(a*x)**3 + x**4*atan(a*x)**3), 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^{4} \arctan \left (a x\right )^{3}}\,{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)/arctan(a*x)^3,x, algorithm="giac")

[Out]

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