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

Optimal. Leaf size=18 \[ \frac{e^{2 \tan ^{-1}(a x)}}{2 a c} \]

[Out]

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

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Rubi [A]  time = 0.0281226, antiderivative size = 18, normalized size of antiderivative = 1., 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 = {5071} \[ \frac{e^{2 \tan ^{-1}(a x)}}{2 a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5071

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*}

Mathematica [C]  time = 0.006442, size = 34, normalized size = 1.89 \[ \frac{(1-i a x)^i (1+i a x)^{-i}}{2 a c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(1 - I*a*x)^I/(2*a*c*(1 + I*a*x)^I)

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Maple [A]  time = 0.034, size = 16, normalized size = 0.9 \begin{align*}{\frac{{{\rm e}^{2\,\arctan \left ( ax \right ) }}}{2\,ac}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.5258, size = 20, normalized size = 1.11 \begin{align*} \frac{e^{\left (2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.00773, size = 39, normalized size = 2.17 \begin{align*} \frac{e^{\left (2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.08264, size = 20, normalized size = 1.11 \begin{align*} \frac{e^{\left (2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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