Optimal. Leaf size=18 \[ \frac {e^{n \tan ^{-1}(a x)}}{a c n} \]
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Rubi [A] time = 0.03, 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 = {5071} \[ \frac {e^{n \tan ^{-1}(a x)}}{a c n} \]
Antiderivative was successfully verified.
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Rule 5071
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] time = 0.01, size = 42, normalized size = 2.33 \[ \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a c n} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.42, size = 17, normalized size = 0.94 \[ \frac {e^{\left (n \arctan \left (a x\right )\right )}}{a c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 17, normalized size = 0.94 \[ \frac {e^{\left (n \arctan \left (a x\right )\right )}}{a c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 18, normalized size = 1.00 \[ \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{a c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 17, normalized size = 0.94 \[ \frac {e^{\left (n \arctan \left (a x\right )\right )}}{a c n} \]
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 \[ \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{a\,c\,n} \]
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 \[ \begin {cases} \tilde {\infty } x & \text {for}\: c = 0 \wedge n = 0 \\\tilde {\infty } \int e^{n \operatorname {atan}{\left (a x \right )}}\, dx & \text {for}\: c = 0 \\\frac {\operatorname {atan}{\left (a x \right )}}{a c} & \text {for}\: n = 0 \\\frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a c n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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