Optimal. Leaf size=15 \[ \frac {i \log (i+a x)}{a} \]
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Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5181, 31}
\begin {gather*} \frac {i \log (a x+i)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 5181
Rubi steps
\begin {align*} \int \frac {e^{i \tan ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx &=\int \frac {1}{1-i a x} \, dx\\ &=\frac {i \log (i+a x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {i \log (i+a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 26, normalized size = 1.73
method | result | size |
default | \(\frac {i \ln \left (a^{2} x^{2}+1\right )}{2 a}+\frac {\arctan \left (a x \right )}{a}\) | \(26\) |
meijerg | \(\frac {i \ln \left (a^{2} x^{2}+1\right )}{2 a}+\frac {\arctan \left (a x \right )}{a}\) | \(26\) |
risch | \(\frac {i \ln \left (a^{2} x^{2}+1\right )}{2 a}+\frac {\arctan \left (a x \right )}{a}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 24 vs. \(2 (11) = 22\).
time = 0.46, size = 24, normalized size = 1.60 \begin {gather*} \frac {\arctan \left (a x\right )}{a} + \frac {i \, \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.08, size = 15, normalized size = 1.00 \begin {gather*} \frac {i \, \log \left (\frac {a x + i}{a}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 8, normalized size = 0.53 \begin {gather*} \frac {i \log {\left (a x + i \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 12, normalized size = 0.80 \begin {gather*} \frac {i \, \log \left (-i \, a x + 1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 15, normalized size = 1.00 \begin {gather*} \frac {\ln \left (x+\frac {1{}\mathrm {i}}{a}\right )\,1{}\mathrm {i}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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