Optimal. Leaf size=38 \[ -\frac {e^{-2 \text {ArcTan}(a x)} (2-a x)}{5 a c \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.03, 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 = {5177}
\begin {gather*} -\frac {(2-a x) e^{-2 \text {ArcTan}(a x)}}{5 a c \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5177
Rubi steps
\begin {align*} \int \frac {e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac {e^{-2 \tan ^{-1}(a x)} (2-a x)}{5 a c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 37, normalized size = 0.97 \begin {gather*} \frac {e^{-2 \text {ArcTan}(a x)} (-2+a x)}{5 a c \sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 41, normalized size = 1.08
method | result | size |
gosper | \(\frac {\left (a^{2} x^{2}+1\right ) \left (a x -2\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{5 a \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.47, size = 44, normalized size = 1.16 \begin {gather*} \frac {\sqrt {a^{2} c x^{2} + c} {\left (a x - 2\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{5 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{- 2 \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 35, normalized size = 0.92 \begin {gather*} \frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {x}{5\,c}-\frac {2}{5\,a\,c}\right )}{\sqrt {c\,a^2\,x^2+c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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