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

Optimal. Leaf size=38 \[ -\frac{(1-a x) e^{-\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}} \]

[Out]

-(1 - a*x)/(2*a*c*E^ArcTan[a*x]*Sqrt[c + a^2*c*x^2])

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Rubi [A]  time = 0.038571, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {5069} \[ -\frac{(1-a x) e^{-\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 - a*x)/(2*a*c*E^ArcTan[a*x]*Sqrt[c + a^2*c*x^2])

Rule 5069

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

Rubi steps

\begin{align*} \int \frac{e^{-\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac{e^{-\tan ^{-1}(a x)} (1-a x)}{2 a c \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0202819, size = 37, normalized size = 0.97 \[ \frac{(a x-1) e^{-\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 + a*x)/(2*a*c*E^ArcTan[a*x]*Sqrt[c + a^2*c*x^2])

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Maple [A]  time = 0.035, size = 39, normalized size = 1. \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}+1 \right ) \left ( ax-1 \right ) }{2\,a{{\rm e}^{\arctan \left ( ax \right ) }}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \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)^(3/2),x)

[Out]

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

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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 )}^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.03495, size = 100, normalized size = 2.63 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x - 1\right )} e^{\left (-\arctan \left (a x\right )\right )}}{2 \,{\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(1/exp(arctan(a*x))/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

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

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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)**(3/2),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 )}^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="giac")

[Out]

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