\(\int \frac {c+a^2 c x^2}{\arctan (a x)^{3/2}} \, dx\) [974]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^{3/2}} \, dx=\text {Int}\left (\frac {c+a^2 c x^2}{\arctan (a x)^{3/2}},x\right ) \]

[Out]

Unintegrable((a^2*c*x^2+c)/arctan(a*x)^(3/2),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 = {} \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^{3/2}} \, dx=\int \frac {c+a^2 c x^2}{\arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^{3/2}} \, dx=\int \frac {c+a^2 c x^2}{\arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.35 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^{3/2}} \, dx=c \left (\int \frac {a^{2} x^{2}}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 169.69 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.16 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^{3/2}} \, dx=\int { \frac {a^{2} c x^{2} + c}{\arctan \left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^{3/2}} \, dx=\int \frac {c\,a^2\,x^2+c}{{\mathrm {atan}\left (a\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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