Optimal. Leaf size=14 \[ \frac {\tan ^{-1}\left (\frac {b x}{a}\right )}{a b} \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {b x}{a}\right )}{a b} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rubi steps
\begin {align*} \int \frac {1}{a^2+b^2 x^2} \, dx &=\frac {\tan ^{-1}\left (\frac {b x}{a}\right )}{a b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {b x}{a}\right )}{a b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 15, normalized size = 1.07
method | result | size |
default | \(\frac {\arctan \left (\frac {b x}{a}\right )}{a b}\) | \(15\) |
risch | \(\frac {\arctan \left (\frac {b x}{a}\right )}{a b}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 5.30, size = 14, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (\frac {b x}{a}\right )}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 14, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (\frac {b x}{a}\right )}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.05, size = 26, normalized size = 1.86 \begin {gather*} \frac {- \frac {i \log {\left (- \frac {i a}{b} + x \right )}}{2} + \frac {i \log {\left (\frac {i a}{b} + x \right )}}{2}}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.46, size = 14, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (\frac {b x}{a}\right )}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} \frac {\mathrm {atan}\left (\frac {b\,x}{a}\right )}{a\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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