\(\int \frac {1}{(c+a^2 c x^2) \arctan (a x)^3} \, dx\) [622]

Optimal result

Integrand size = 19, antiderivative size = 16 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=-\frac {1}{2 a c \arctan (a x)^2} \]

[Out]

-1/2/a/c/arctan(a*x)^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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {5004} \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=-\frac {1}{2 a c \arctan (a x)^2} \]

[In]

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

[Out]

-1/2*1/(a*c*ArcTan[a*x]^2)

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 {1}{2 a c \arctan (a x)^2} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

-1/2*1/(a*c*ArcTan[a*x]^2)

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(-\frac {1}{2 a c \arctan \left (a x \right )^{2}}\)	\(15\)
default	\(-\frac {1}{2 a c \arctan \left (a x \right )^{2}}\)	\(15\)
parallelrisch	\(-\frac {1}{2 a c \arctan \left (a x \right )^{2}}\)	\(15\)
risch	\(\frac {2}{a c \left (\ln \left (-i a x +1\right )-\ln \left (i a x +1\right )\right )^{2}}\)	\(30\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=-\frac {1}{2 \, a c \arctan \left (a x\right )^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=- \frac {1}{2 a c \operatorname {atan}^{2}{\left (a x \right )}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=-\frac {1}{2 \, a c \arctan \left (a x\right )^{2}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=-\frac {1}{2\,a\,c\,{\mathrm {atan}\left (a\,x\right )}^2} \]

[In]

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

[Out]

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