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/exp(2*arctan(a*x))+1/10*(2*a*x-1)/a/c^3/exp(2*arctan(a*x))/(a^2*x^2+1)^2-3/20*(-a*x+1)/a/c^3/exp(2
*arctan(a*x))/(a^2*x^2+1)

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Rubi [A]  time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, 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*}

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Mathematica [C]  time = 0.17, 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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fricas [A]  time = 0.44, size = 68, normalized size = 0.76 \[ -\frac {{\left (3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 14 \, a x + 13\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{40 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [A]  time = 0.04, size = 59, normalized size = 0.66 \[ -\frac {\left (3 a^{4} x^{4}-6 a^{3} x^{3}+12 a^{2} x^{2}-14 a x +13\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{40 \left (a^{2} x^{2}+1\right )^{2} c^{3} a} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \]

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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mupad [B]  time = 0.62, size = 79, normalized size = 0.89 \[ \frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x-1\right )}{20\,a\,c^3\,\left (a^2\,x^2+1\right )}-\frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{40\,a\,c^3}+\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{10\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-2*atan(a*x))/(c + a^2*c*x^2)^3,x)

[Out]

(3*exp(-2*atan(a*x))*(a*x - 1))/(20*a*c^3*(a^2*x^2 + 1)) - (3*exp(-2*atan(a*x)))/(40*a*c^3) + (exp(-2*atan(a*x
))*(2*a*x - 1))/(10*a*c^3*(a^2*x^2 + 1)^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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