Optimal. Leaf size=18 \[ \frac {e^{n \text {ArcTan}(a x)}}{a c n} \]
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Rubi [A]
time = 0.02, antiderivative size = 18, 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 = {5179}
\begin {gather*} \frac {e^{n \text {ArcTan}(a x)}}{a c n} \end {gather*}
Antiderivative was successfully verified.
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Rule 5179
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx &=\frac {e^{n \tan ^{-1}(a x)}}{a c n}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.01, size = 42, normalized size = 2.33 \begin {gather*} \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a c n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 18, normalized size = 1.00
method | result | size |
gosper | \(\frac {{\mathrm e}^{n \arctan \left (a x \right )}}{a c n}\) | \(18\) |
risch | \(\frac {\left (-i a x +1\right )^{\frac {i n}{2}} \left (i a x +1\right )^{-\frac {i n}{2}}}{c a n}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 17, normalized size = 0.94 \begin {gather*} \frac {e^{\left (n \arctan \left (a x\right )\right )}}{a c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.83, size = 17, normalized size = 0.94 \begin {gather*} \frac {e^{\left (n \arctan \left (a x\right )\right )}}{a c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (12) = 24\).
time = 0.60, size = 26, normalized size = 1.44 \begin {gather*} \begin {cases} \frac {x}{c} & \text {for}\: a = 0 \wedge n = 0 \\\frac {\operatorname {atan}{\left (a x \right )}}{a c} & \text {for}\: n = 0 \\\frac {x}{c} & \text {for}\: a = 0 \\\frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a c n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 17, normalized size = 0.94 \begin {gather*} \frac {e^{\left (n \arctan \left (a x\right )\right )}}{a c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.63, size = 17, normalized size = 0.94 \begin {gather*} \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{a\,c\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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