3.293 \(\int \frac{e^{-2 \tan ^{-1}(a x)}}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=89 \[ -\frac{(1-2 a x) e^{-2 \tan ^{-1}(a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}-\frac{3 (1-a x) e^{-2 \tan ^{-1}(a x)}}{20 a c^3 \left (a^2 x^2+1\right )}-\frac{3 e^{-2 \tan ^{-1}(a x)}}{40 a c^3} \]

[Out]

-3/(40*a*c^3*E^(2*ArcTan[a*x])) - (1 - 2*a*x)/(10*a*c^3*E^(2*ArcTan[a*x])*(1 + a^2*x^2)^2) - (3*(1 - a*x))/(20
*a*c^3*E^(2*ArcTan[a*x])*(1 + a^2*x^2))

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Rubi [A]  time = 0.0875743, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5070, 5071} \[ -\frac{(1-2 a x) e^{-2 \tan ^{-1}(a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}-\frac{3 (1-a x) e^{-2 \tan ^{-1}(a x)}}{20 a c^3 \left (a^2 x^2+1\right )}-\frac{3 e^{-2 \tan ^{-1}(a x)}}{40 a c^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^3),x]

[Out]

-3/(40*a*c^3*E^(2*ArcTan[a*x])) - (1 - 2*a*x)/(10*a*c^3*E^(2*ArcTan[a*x])*(1 + a^2*x^2)^2) - (3*(1 - a*x))/(20
*a*c^3*E^(2*ArcTan[a*x])*(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 )^3} \, dx &=-\frac{e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \int \frac{e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{5 c}\\ &=-\frac{e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}-\frac{3 e^{-2 \tan ^{-1}(a x)} (1-a x)}{20 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \int \frac{e^{-2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{20 c^2}\\ &=-\frac{3 e^{-2 \tan ^{-1}(a x)}}{40 a c^3}-\frac{e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}-\frac{3 e^{-2 \tan ^{-1}(a x)} (1-a x)}{20 a c^3 \left (1+a^2 x^2\right )}\\ \end{align*}

Mathematica [C]  time = 0.14501, size = 85, normalized size = 0.96 \[ \frac{(8 a x-4) e^{-2 \tan ^{-1}(a x)}-3 (1-i a x)^{-i} (1+i a x)^i \left (a^2 x^2+1\right ) \left (a^2 x^2-2 a x+3\right )}{40 a c^3 \left (a^2 x^2+1\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^3),x]

[Out]

((-4 + 8*a*x)/E^(2*ArcTan[a*x]) - (3*(1 + I*a*x)^I*(1 + a^2*x^2)*(3 - 2*a*x + a^2*x^2))/(1 - I*a*x)^I)/(40*a*c
^3*(1 + a^2*x^2)^2)

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Maple [A]  time = 0.038, size = 59, normalized size = 0.7 \begin{align*} -{\frac{3\,{a}^{4}{x}^{4}-6\,{a}^{3}{x}^{3}+12\,{a}^{2}{x}^{2}-14\,ax+13}{40\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{{\rm e}^{2\,\arctan \left ( ax \right ) }}a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(3*a^4*x^4-6*a^3*x^3+12*a^2*x^2-14*a*x+13)/(a^2*x^2+1)^2/c^3/exp(2*arctan(a*x))/a

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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 )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

integrate(e^(-2*arctan(a*x))/(a^2*c*x^2 + c)^3, x)

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Fricas [A]  time = 2.0834, size = 155, normalized size = 1.74 \begin{align*} -\frac{{\left (3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 14 \, a x + 13\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{40 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/40*(3*a^4*x^4 - 6*a^3*x^3 + 12*a^2*x^2 - 14*a*x + 13)*e^(-2*arctan(a*x))/(a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c
^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(2*atan(a*x))/(a**2*c*x**2+c)**3,x)

[Out]

Timed out

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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 )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

integrate(e^(-2*arctan(a*x))/(a^2*c*x^2 + c)^3, x)