\(\int \frac {1}{x (c+a^2 c x^2)^3 \sqrt {\arctan (a x)}} \, dx\) [943]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.89 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 5.51 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {\int \frac {1}{a^{6} x^{7} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{4} x^{5} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{2} x^{3} \sqrt {\operatorname {atan}{\left (a x \right )}} + x \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(1/(a**6*x**7*sqrt(atan(a*x)) + 3*a**4*x**5*sqrt(atan(a*x)) + 3*a**2*x**3*sqrt(atan(a*x)) + x*sqrt(ata
n(a*x))), x)/c**3

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [N/A]

Not integrable

Time = 158.51 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x \sqrt {\arctan \left (a x\right )}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\int \frac {1}{x\,\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

[In]

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

[Out]

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