\(\int \frac {1}{(c+a^2 c x^2) \arctan (a x)^{3/2}} \, dx\) [986]

Optimal result

Integrand size = 21, antiderivative size = 16 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c \sqrt {\arctan (a x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5004} \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c \sqrt {\arctan (a x)}} \]

[In]

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

[Out]

-2/(a*c*Sqrt[ArcTan[a*x]])

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{a c \sqrt {\arctan (a x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c \sqrt {\arctan (a x)}} \]

[In]

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

[Out]

-2/(a*c*Sqrt[ArcTan[a*x]])

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {2}{a c \sqrt {\arctan \left (a x \right )}}\)	\(15\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c \sqrt {\arctan \left (a x\right )}} \]

[In]

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

[Out]

-2/(a*c*sqrt(arctan(a*x)))

Sympy [A] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=- \frac {2}{a c \sqrt {\operatorname {atan}{\left (a x \right )}}} \]

[In]

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

[Out]

-2/(a*c*sqrt(atan(a*x)))

Maxima [F(-2)]

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

[In]

integrate(1/(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 [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c \sqrt {\arctan \left (a x\right )}} \]

[In]

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

[Out]

-2/(a*c*sqrt(arctan(a*x)))

Mupad [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=-\frac {2}{a\,c\,\sqrt {\mathrm {atan}\left (a\,x\right )}} \]

[In]

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

[Out]

-2/(a*c*atan(a*x)^(1/2))