Optimal. Leaf size=38 \[ -\frac {(1-a x) e^{-\tan ^{-1}(a x)}}{2 a c \sqrt {a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, 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]
[Out]
Rule 5069
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 = 0.02, 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]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 44, normalized size = 1.16 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (a x - 1\right )} e^{\left (-\arctan \left (a x\right )\right )}}{2 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 39, normalized size = 1.03 \[ \frac {\left (a^{2} x^{2}+1\right ) \left (a x -1\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{2 a \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (-\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.63, size = 35, normalized size = 0.92 \[ \frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (\frac {x}{2\,c}-\frac {1}{2\,a\,c}\right )}{\sqrt {c\,a^2\,x^2+c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{- \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________