Optimal. Leaf size=35 \[ \frac {e^{\text {ArcTan}(a x)} (1+a x)}{2 a c \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5177}
\begin {gather*} \frac {(a x+1) e^{\text {ArcTan}(a x)}}{2 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^{\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*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} \frac {e^{\text {ArcTan}(a x)} (1+a x)}{2 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 = 37, normalized size = 1.06
method | result | size |
gosper | \(\frac {\left (a^{2} x^{2}+1\right ) \left (a x +1\right ) {\mathrm e}^{\arctan \left (a x \right )}}{2 a \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(37\) |
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 = 3.11, size = 42, normalized size = 1.20 \begin {gather*} \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 {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^{\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 = 0.60, size = 33, normalized size = 0.94 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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