\(\int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx\) [291]

Optimal result

Integrand size = 21, antiderivative size = 18 \[ \int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx=-\frac {e^{-2 \arctan (a x)}}{2 a c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, 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 = {5179} \[ \int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx=-\frac {e^{-2 \arctan (a x)}}{2 a c} \]

[In]

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

[Out]

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

Rule 5179

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcTan[a*x])/(a*c*n), x] /; Fre
eQ[{a, c, d, n}, x] && EqQ[d, a^2*c]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx=-\frac {(1-i a x)^{-i} (1+i a x)^i}{2 a c} \]

[In]

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

[Out]

-1/2*(1 + I*a*x)^I/(a*c*(1 - I*a*x)^I)

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

method	result	size
gosper	\(-\frac {{\mathrm e}^{-2 \arctan \left (a x \right )}}{2 a c}\)	\(18\)
parallelrisch	\(-\frac {{\mathrm e}^{-2 \arctan \left (a x \right )}}{2 a c}\)	\(18\)
risch	\(-\frac {\left (-i a x +1\right )^{-i} \left (i a x +1\right )^{i}}{2 a c}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx=-\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 6.38 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx=\begin {cases} - \frac {e^{- 2 \operatorname {atan}{\left (a x \right )}}}{2 a c} & \text {for}\: a \neq 0 \\\frac {x}{c} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-exp(-2*atan(a*x))/(2*a*c), Ne(a, 0)), (x/c, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx=-\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{a^{3} c x^{2} + a c} \]

[In]

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

[Out]

-e^(-2*arctan(a*x))/(a^3*c*x^2 + a*c)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx=-\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx=-\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{2\,a\,c} \]

[In]

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

[Out]

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