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

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

[Out]

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

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Rubi [A]  time = 0.0846464, 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 = {} \[ \int \frac{x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.332036, size = 0, normalized size = 0. \[ \int \frac{x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.437, 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 ) ^{2}}}\, 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)^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{m \arctan \left (a x\right ) \int \frac{x^{m}}{x \arctan \left (a x\right )}\,{d x} - x^{m}}{a c \arctan \left (a 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)^2,x, algorithm="maxima")

[Out]

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

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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 )^{2}}, 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)^2,x, algorithm="fricas")

[Out]

integral(x^m/((a^2*c*x^2 + c)*arctan(a*x)^2), 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}^{2}{\left (a x \right )} + \operatorname{atan}^{2}{\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)**2,x)

[Out]

Integral(x**m/(a**2*x**2*atan(a*x)**2 + atan(a*x)**2), 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 )^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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