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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x^m}{\left (a^2 c x^2+c\right )^2 \tan ^{-1}(a x)^3},x\right ) \]

[Out]

Unintegrable(x^m/(a^2*c*x^2+c)^2/arctan(a*x)^3,x)

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.00, size = 0, normalized size = 0.00 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 1.57, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^m}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m/(a**4*x**4*atan(a*x)**3 + 2*a**2*x**2*atan(a*x)**3 + atan(a*x)**3), x)/c**2

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