\(\int \frac {\arctan (a x)^2}{c+a^2 c x^2} \, dx\) [286]

Optimal result

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

[Out]

1/3*arctan(a*x)^3/a/c

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

[In]

Int[ArcTan[a*x]^2/(c + a^2*c*x^2),x]

[Out]

ArcTan[a*x]^3/(3*a*c)

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 {\arctan (a x)^3}{3 a c} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

ArcTan[a*x]^3/(3*a*c)

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {\arctan \left (a x \right )^{3}}{3 a c}\)	\(15\)
default	\(\frac {\arctan \left (a x \right )^{3}}{3 a c}\)	\(15\)
parallelrisch	\(\frac {\arctan \left (a x \right )^{3}}{3 a c}\)	\(15\)
parts	\(\frac {\arctan \left (a x \right )^{3}}{3 a c}\)	\(15\)
risch	\(\frac {i \ln \left (i a x +1\right )^{3}}{24 c a}-\frac {i \ln \left (-i a x +1\right ) \ln \left (i a x +1\right )^{2}}{8 c a}+\frac {i \ln \left (-i a x +1\right )^{2} \ln \left (i a x +1\right )}{8 c a}-\frac {i \ln \left (-i a x +1\right )^{3}}{24 c a}\)	\(94\)

[In]

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

[Out]

1/3*arctan(a*x)^3/a/c

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {\arctan \left (a x\right )^{3}}{3 \, a c} \]

[In]

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

[Out]

1/3*arctan(a*x)^3/(a*c)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {\arctan \left (a x\right )^{3}}{3 \, a c} \]

[In]

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

[Out]

1/3*arctan(a*x)^3/(a*c)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {{\mathrm {atan}\left (a\,x\right )}^3}{3\,a\,c} \]

[In]

int(atan(a*x)^2/(c + a^2*c*x^2),x)

[Out]

atan(a*x)^3/(3*a*c)