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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________