\(\int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx\) [463]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\text {Int}\left (x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3,x\right ) \]

[Out]

Unintegrable(x^m*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 24, 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 = {} \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx \]

[In]

Int[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]

[Out]

Defer[Int][x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x]

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

Mathematica [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx \]

[In]

Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]

[Out]

Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x]

Maple [N/A] (verified)

Not integrable

Time = 3.66 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

int(x^m*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2),x)

[Out]

int(x^m*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{m} \arctan \left (a x\right )^{3} \,d x } \]

[In]

integrate(x^m*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 71.72 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x^{m} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]

[In]

integrate(x**m*atan(a*x)**3*(a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(x**m*sqrt(c*(a**2*x**2 + 1))*atan(a*x)**3, x)

Maxima [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{m} \arctan \left (a x\right )^{3} \,d x } \]

[In]

integrate(x^m*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^m*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x^m\,{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c} \,d x \]

[In]

int(x^m*atan(a*x)^3*(c + a^2*c*x^2)^(1/2),x)

[Out]

int(x^m*atan(a*x)^3*(c + a^2*c*x^2)^(1/2), x)