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

Optimal. Leaf size=45 \[ \frac{m \text{Unintegrable}\left (\frac{x^{m-1}}{\tan ^{-1}(a x)^2},x\right )}{2 a c}-\frac{x^m}{2 a c \tan ^{-1}(a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0816005, 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 = {} \[ \int \frac{x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.374935, size = 0, normalized size = 0. \[ \int \frac{x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.422, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \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(x^m/(a^2*c*x^2+c)/arctan(a*x)^3,x)

[Out]

int(x^m/(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{m x{\left (\int \frac{a^{2} m x^{m}}{\arctan \left (a x\right )}\,{d x} + \int \frac{a^{2} x^{m}}{\arctan \left (a x\right )}\,{d x} + \int \frac{m x^{m}}{x^{2} \arctan \left (a x\right )}\,{d x} + \int -\frac{x^{m}}{x^{2} \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )^{2} - a x x^{m} -{\left (a^{2} m x^{2} + m\right )} x^{m} \arctan \left (a x\right )}{2 \, a^{2} c x \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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