3.177 \(\int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac {\tan ^{-1}(a x)^2}{2 a c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {4884} \[ \frac {\tan ^{-1}(a x)^2}{2 a c} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]/(c + a^2*c*x^2),x]

[Out]

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

Rule 4884

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*} \int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx &=\frac {\tan ^{-1}(a x)^2}{2 a c}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ \frac {\tan ^{-1}(a x)^2}{2 a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 14, normalized size = 0.88 \[ \frac {\arctan \left (a x\right )^{2}}{2 \, a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.17, size = 35, normalized size = 2.19 \[ -\frac {2 \, \pi \arctan \left (a x\right ) \left \lfloor \frac {\arctan \left (a x\right )}{\pi } + \frac {1}{2} \right \rfloor - \arctan \left (a x\right )^{2}}{2 \, a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.03, size = 15, normalized size = 0.94 \[ \frac {\arctan \left (a x \right )^{2}}{2 a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 14, normalized size = 0.88 \[ \frac {\arctan \left (a x\right )^{2}}{2 \, a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.38, size = 14, normalized size = 0.88 \[ \frac {{\mathrm {atan}\left (a\,x\right )}^2}{2\,a\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)/(c + a^2*c*x^2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 2.37, size = 36, normalized size = 2.25 \[ \begin {cases} 0 & \text {for}\: a = 0 \\\tilde {\infty } \left (\begin {cases} 0 & \text {for}\: a = 0 \\\frac {a x \operatorname {atan}{\left (a x \right )} - \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2}}{a} & \text {otherwise} \end {cases}\right ) & \text {for}\: c = 0 \\\frac {\operatorname {atan}^{2}{\left (a x \right )}}{2 a c} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((0, Eq(a, 0)), (zoo*Piecewise((0, Eq(a, 0)), ((a*x*atan(a*x) - log(a**2*x**2 + 1)/2)/a, True)), Eq(c
, 0)), (atan(a*x)**2/(2*a*c), True))

________________________________________________________________________________________