\(\int \frac {x^m (c+a^2 c x^2)^3}{\arctan (a x)^3} \, dx\) [674]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 10.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 17.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.32 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=c^{3} \left (\int \frac {x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x^{2} x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4} x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6} x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(x**m/atan(a*x)**3, x) + Integral(3*a**2*x**2*x**m/atan(a*x)**3, x) + Integral(3*a**4*x**4*x**m/
atan(a*x)**3, x) + Integral(a**6*x**6*x**m/atan(a*x)**3, x))

Maxima [N/A]

Not integrable

Time = 1.24 (sec) , antiderivative size = 399, normalized size of antiderivative = 18.14 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

1/2*(x*arctan(a*x)^2*integrate(((a^10*c^3*m^2 + 17*a^10*c^3*m + 72*a^10*c^3)*x^10 + (5*a^8*c^3*m^2 + 67*a^8*c^
3*m + 224*a^8*c^3)*x^8 + 2*(5*a^6*c^3*m^2 + 49*a^6*c^3*m + 120*a^6*c^3)*x^6 + c^3*m^2 + 2*(5*a^4*c^3*m^2 + 31*
a^4*c^3*m + 48*a^4*c^3)*x^4 - c^3*m + (5*a^2*c^3*m^2 + 13*a^2*c^3*m + 8*a^2*c^3)*x^2)*x^m/(x^2*arctan(a*x)), x
) - ((a^10*c^3*m + 8*a^10*c^3)*x^10 + (5*a^8*c^3*m + 32*a^8*c^3)*x^8 + 2*(5*a^6*c^3*m + 24*a^6*c^3)*x^6 + 2*(5
*a^4*c^3*m + 16*a^4*c^3)*x^4 + c^3*m + (5*a^2*c^3*m + 8*a^2*c^3)*x^2)*x^m*arctan(a*x) - (a^9*c^3*x^9 + 4*a^7*c
^3*x^7 + 6*a^5*c^3*x^5 + 4*a^3*c^3*x^3 + a*c^3*x)*x^m)/(a^2*x*arctan(a*x)^2)

Giac [N/A]

Not integrable

Time = 218.44 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int \frac {x^m\,{\left (c\,a^2\,x^2+c\right )}^3}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]

[In]

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

[Out]

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