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

Optimal. Leaf size=53 \[ \frac{(a x+1) e^{2 \tan ^{-1}(a x)}}{4 a c^2 \left (a^2 x^2+1\right )}+\frac{e^{2 \tan ^{-1}(a x)}}{8 a c^2} \]

[Out]

E^(2*ArcTan[a*x])/(8*a*c^2) + (E^(2*ArcTan[a*x])*(1 + a*x))/(4*a*c^2*(1 + a^2*x^2))

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Rubi [A]  time = 0.057281, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5070, 5071} \[ \frac{(a x+1) e^{2 \tan ^{-1}(a x)}}{4 a c^2 \left (a^2 x^2+1\right )}+\frac{e^{2 \tan ^{-1}(a x)}}{8 a c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*ArcTan[a*x])/(8*a*c^2) + (E^(2*ArcTan[a*x])*(1 + a*x))/(4*a*c^2*(1 + a^2*x^2))

Rule 5070

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((n - 2*a*(p + 1)*x)*(c + d*x^2
)^(p + 1)*E^(n*ArcTan[a*x]))/(a*c*(n^2 + 4*(p + 1)^2)), x] + Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 + 4*(p + 1)^2)
), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1]
&&  !IntegerQ[I*n] && NeQ[n^2 + 4*(p + 1)^2, 0] && IntegerQ[2*p]

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)}}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac{e^{2 \tan ^{-1}(a x)} (1+a x)}{4 a c^2 \left (1+a^2 x^2\right )}+\frac{\int \frac{e^{2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{4 c}\\ &=\frac{e^{2 \tan ^{-1}(a x)}}{8 a c^2}+\frac{e^{2 \tan ^{-1}(a x)} (1+a x)}{4 a c^2 \left (1+a^2 x^2\right )}\\ \end{align*}

Mathematica [C]  time = 0.0184911, size = 55, normalized size = 1.04 \[ \frac{(1-i a x)^i (1+i a x)^{-i} \left (a^2 x^2+2 a x+3\right )}{8 c^2 \left (a^3 x^2+a\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - I*a*x)^I*(3 + 2*a*x + a^2*x^2))/(8*c^2*(1 + I*a*x)^I*(a + a^3*x^2))

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Maple [A]  time = 0.037, size = 40, normalized size = 0.8 \begin{align*}{\frac{{{\rm e}^{2\,\arctan \left ( ax \right ) }} \left ({a}^{2}{x}^{2}+2\,ax+3 \right ) }{ \left ( 8\,{a}^{2}{x}^{2}+8 \right ) a{c}^{2}}} \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)^2,x)

[Out]

1/8*exp(2*arctan(a*x))*(a^2*x^2+2*a*x+3)/(a^2*x^2+1)/a/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.92883, size = 90, normalized size = 1.7 \begin{align*} \frac{{\left (a^{2} x^{2} + 2 \, a x + 3\right )} e^{\left (2 \, \arctan \left (a x\right )\right )}}{8 \,{\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \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)^2,x, algorithm="fricas")

[Out]

1/8*(a^2*x^2 + 2*a*x + 3)*e^(2*arctan(a*x))/(a^3*c^2*x^2 + a*c^2)

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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)**2,x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \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)^2,x, algorithm="giac")

[Out]

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