\(\int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx\) [655]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx=-\frac {\sqrt {c+a^2 c x^2}}{2 a c x \arctan (a x)^2}-\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2},x\right )}{2 a} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.14 (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 = {} \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx=\int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx \]

[In]

Int[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3),x]

[Out]

-1/2*Sqrt[c + a^2*c*x^2]/(a*c*x*ArcTan[a*x]^2) - Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(2*a
)

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+a^2 c x^2}}{2 a c x \arctan (a x)^2}-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{2 a} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx=\int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx \]

[In]

Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3),x]

[Out]

Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]

Maple [N/A] (verified)

Not integrable

Time = 8.91 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {1}{x \arctan \left (a x \right )^{3} \sqrt {a^{2} c \,x^{2}+c}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx=\int { \frac {1}{\sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx=\int \frac {1}{x \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx=\int { \frac {1}{\sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 88.76 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx=\int { \frac {1}{\sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

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

[Out]

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