\(\int (d x)^m (a+b \arctan (c x^3))^2 \, dx\) [128]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int (d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\text {Int}\left ((d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2,x\right ) \]

[Out]

Unintegrable((d*x)^m*(a+b*arctan(c*x^3))^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 = {} \[ \int (d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\int (d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\int (d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int (d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\int { {\left (b \arctan \left (c x^{3}\right ) + a\right )}^{2} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

integral((b^2*arctan(c*x^3)^2 + 2*a*b*arctan(c*x^3) + a^2)*(d*x)^m, x)

Sympy [F(-1)]

Timed out. \[ \int (d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 2.37 (sec) , antiderivative size = 304, normalized size of antiderivative = 16.89 \[ \int (d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\int { {\left (b \arctan \left (c x^{3}\right ) + a\right )}^{2} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

(d*x)^(m + 1)*a^2/(d*(m + 1)) + 1/16*(4*b^2*d^m*x*x^m*arctan(c*x^3)^2 - b^2*d^m*x*x^m*log(c^2*x^6 + 1)^2 + 16*
(m + 1)*integrate(1/16*(12*b^2*c^2*d^m*x^6*x^m*log(c^2*x^6 + 1) + 12*((b^2*c^2*d^m*m + b^2*c^2*d^m)*x^6 + b^2*
d^m*m + b^2*d^m)*x^m*arctan(c*x^3)^2 + ((b^2*c^2*d^m*m + b^2*c^2*d^m)*x^6 + b^2*d^m*m + b^2*d^m)*x^m*log(c^2*x
^6 + 1)^2 - 8*(3*b^2*c*d^m*x^3 - 4*(a*b*c^2*d^m*m + a*b*c^2*d^m)*x^6 - 4*a*b*d^m*m - 4*a*b*d^m)*x^m*arctan(c*x
^3))/((c^2*m + c^2)*x^6 + m + 1), x))/(m + 1)

Giac [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\int { {\left (b \arctan \left (c x^{3}\right ) + a\right )}^{2} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d x)^m \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {atan}\left (c\,x^3\right )\right )}^2 \,d x \]

[In]

int((d*x)^m*(a + b*atan(c*x^3))^2,x)

[Out]

int((d*x)^m*(a + b*atan(c*x^3))^2, x)