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

Optimal. Leaf size=18 \[ -\frac {e^{-2 \text {ArcTan}(a x)}}{2 a c} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5179} \begin {gather*} -\frac {e^{-2 \text {ArcTan}(a x)}}{2 a c} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.01, size = 34, normalized size = 1.89 \begin {gather*} -\frac {(1-i a x)^{-i} (1+i a x)^i}{2 a c} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [A]
time = 0.06, size = 18, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Maxima [A]
time = 0.27, size = 23, normalized size = 1.28 \begin {gather*} -\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{a^{3} c x^{2} + a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 3.82, size = 15, normalized size = 0.83 \begin {gather*} -\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 8.60, size = 19, normalized size = 1.06 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]
time = 0.40, size = 15, normalized size = 0.83 \begin {gather*} -\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 0.53, size = 15, normalized size = 0.83 \begin {gather*} -\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{2\,a\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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