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

Optimal. Leaf size=41 \[ -\frac {2 \text {Int}\left (\frac {1}{x^5 \tan ^{-1}(a x)^2},x\right )}{a c}-\frac {1}{2 a c x^4 \tan ^{-1}(a x)^2} \]

[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 = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 2.53, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx \]

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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fricas [A]  time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{2} c x^{6} + c x^{4}\right )} \arctan \left (a x\right )^{3}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 2.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{3}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, x^{5} \arctan \left (a x\right )^{2} \int \frac {3 \, a^{2} x^{2} + 5}{x^{6} \arctan \left (a x\right )}\,{d x} - a x + 4 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{2 \, a^{2} c x^{5} \arctan \left (a x\right )^{2}} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^4\,{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{2} x^{6} \operatorname {atan}^{3}{\left (a x \right )} + x^{4} \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c} \]

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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