\(\int \frac {x^m}{(c+a^2 c x^2) \arctan (a x)^3} \, dx\) [677]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=-\frac {x^m}{2 a c \arctan (a x)^2}+\frac {m \text {Int}\left (\frac {x^{-1+m}}{\arctan (a x)^2},x\right )}{2 a c} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = -\frac {x^m}{2 a c \arctan (a x)^2}+\frac {m \int \frac {x^{-1+m}}{\arctan (a x)^2} \, dx}{2 a c} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 4.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{3}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 6.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\frac {\int \frac {x^{m}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c} \]

[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

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 91, normalized size of antiderivative = 4.14 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 155.56 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int \frac {x^m}{{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

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

[Out]

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