\(\int \frac {x \sqrt {c+a^2 c x^2}}{\arctan (a x)} \, dx\) [496]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {x \sqrt {c+a^2 c x^2}}{\arctan (a x)} \, dx=\int \frac {x \sqrt {c+a^2 c x^2}}{\arctan (a x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x \sqrt {c+a^2 c x^2}}{\arctan (a x)} \, dx=\int \frac {x \sqrt {c+a^2 c x^2}}{\arctan (a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 5.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \sqrt {c+a^2 c x^2}}{\arctan (a x)} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} x}{\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {x \sqrt {c+a^2 c x^2}}{\arctan (a x)} \, dx=\int \frac {x \sqrt {c \left (a^{2} x^{2} + 1\right )}}{\operatorname {atan}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \sqrt {c+a^2 c x^2}}{\arctan (a x)} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} x}{\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 33.37 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x \sqrt {c+a^2 c x^2}}{\arctan (a x)} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} x}{\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \sqrt {c+a^2 c x^2}}{\arctan (a x)} \, dx=\int \frac {x\,\sqrt {c\,a^2\,x^2+c}}{\mathrm {atan}\left (a\,x\right )} \,d x \]

[In]

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

[Out]

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