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

Optimal. Leaf size=89 \[ -\frac{(1-4 a x) e^{-\tan ^{-1}(a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}-\frac{12 (1-2 a x) e^{-\tan ^{-1}(a x)}}{85 a c^3 \left (a^2 x^2+1\right )}-\frac{24 e^{-\tan ^{-1}(a x)}}{85 a c^3} \]

[Out]

-24/(85*a*c^3*E^ArcTan[a*x]) - (1 - 4*a*x)/(17*a*c^3*E^ArcTan[a*x]*(1 + a^2*x^2)^2) - (12*(1 - 2*a*x))/(85*a*c
^3*E^ArcTan[a*x]*(1 + a^2*x^2))

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Rubi [A]  time = 0.0825961, 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} \[ -\frac{(1-4 a x) e^{-\tan ^{-1}(a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}-\frac{12 (1-2 a x) e^{-\tan ^{-1}(a x)}}{85 a c^3 \left (a^2 x^2+1\right )}-\frac{24 e^{-\tan ^{-1}(a x)}}{85 a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-24/(85*a*c^3*E^ArcTan[a*x]) - (1 - 4*a*x)/(17*a*c^3*E^ArcTan[a*x]*(1 + a^2*x^2)^2) - (12*(1 - 2*a*x))/(85*a*c
^3*E^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^{-\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac{e^{-\tan ^{-1}(a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac{12 \int \frac{e^{-\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{17 c}\\ &=-\frac{e^{-\tan ^{-1}(a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}-\frac{12 e^{-\tan ^{-1}(a x)} (1-2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}+\frac{24 \int \frac{e^{-\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{85 c^2}\\ &=-\frac{24 e^{-\tan ^{-1}(a x)}}{85 a c^3}-\frac{e^{-\tan ^{-1}(a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}-\frac{12 e^{-\tan ^{-1}(a x)} (1-2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}\\ \end{align*}

Mathematica [C]  time = 0.1476, size = 91, normalized size = 1.02 \[ \frac{5 (4 a x-1) e^{-\tan ^{-1}(a x)}-12 (1-i a x)^{-\frac{i}{2}} (1+i a x)^{\frac{i}{2}} \left (a^2 x^2+1\right ) \left (2 a^2 x^2-2 a x+3\right )}{85 a c^3 \left (a^2 x^2+1\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((5*(-1 + 4*a*x))/E^ArcTan[a*x] - (12*(1 + I*a*x)^(I/2)*(1 + a^2*x^2)*(3 - 2*a*x + 2*a^2*x^2))/(1 - I*a*x)^(I/
2))/(85*a*c^3*(1 + a^2*x^2)^2)

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Maple [A]  time = 0.036, size = 57, normalized size = 0.6 \begin{align*} -{\frac{24\,{a}^{4}{x}^{4}-24\,{a}^{3}{x}^{3}+60\,{a}^{2}{x}^{2}-44\,ax+41}{85\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{{\rm e}^{\arctan \left ( ax \right ) }}a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/85*(24*a^4*x^4-24*a^3*x^3+60*a^2*x^2-44*a*x+41)/(a^2*x^2+1)^2/c^3/exp(arctan(a*x))/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (-\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(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.0356, size = 155, normalized size = 1.74 \begin{align*} -\frac{{\left (24 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 60 \, a^{2} x^{2} - 44 \, a x + 41\right )} e^{\left (-\arctan \left (a x\right )\right )}}{85 \,{\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(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/85*(24*a^4*x^4 - 24*a^3*x^3 + 60*a^2*x^2 - 44*a*x + 41)*e^(-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(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 (-\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(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

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