\(\int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx\) [916]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\text {Int}\left (\frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 19, 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 = {} \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((a^2*c*x^2+c)/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 = 0.76 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53 \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=c \left (\int \frac {a^{2} x^{2}}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx + \int \frac {1}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((a^2*c*x^2+c)/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 = 75.74 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.16 \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\int { \frac {a^{2} c x^{2} + c}{\sqrt {\arctan \left (a x\right )}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\int \frac {c\,a^2\,x^2+c}{\sqrt {\mathrm {atan}\left (a\,x\right )}} \,d x \]

[In]

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

[Out]

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