\(\int \frac {(d x)^m}{(a+b \arctan (c x^2))^2} \, dx\) [95]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2} \, dx=\text {Int}\left (\frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2} \, dx=\int \frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2} \, dx=\int \frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.00 \[ \int \frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 124, normalized size of antiderivative = 6.89 \[ \int \frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-1/2*((c^2*d^m*x^4 + d^m)*x^m - 2*(b^2*c*x*arctan(c*x^2) + a*b*c*x)*integrate(1/2*((c^2*d^m*m + 3*c^2*d^m)*x^4
+ d^m*m - d^m)*x^m/(b^2*c*x^2*arctan(c*x^2) + a*b*c*x^2), x))/(b^2*c*x*arctan(c*x^2) + a*b*c*x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d x)^m}{\left (a+b \arctan \left (c x^2\right )\right )^2} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2} \,d x \]

[In]

int((d*x)^m/(a + b*atan(c*x^2))^2,x)

[Out]

int((d*x)^m/(a + b*atan(c*x^2))^2, x)