\(\int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx\) [645]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx=\text {Int}\left (\frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx=\int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx=\int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 32.55 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 77.58 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\arctan (a x)^3} \, dx=\int \frac {\sqrt {c\,a^2\,x^2+c}}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]

[In]

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

[Out]

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