Optimal. Leaf size=14 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b} \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {214}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rubi steps
\begin {align*} \int \frac {1}{a^2-b^2 x^2} \, dx &=\frac {\tanh ^{-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 {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(30\) vs.
\(2(14)=28\).
time = 0.05, size = 31, normalized size = 2.21
method | result | size |
default | \(-\frac {\ln \left (-b x +a \right )}{2 a b}+\frac {\ln \left (b x +a \right )}{2 a b}\) | \(31\) |
norman | \(-\frac {\ln \left (-b x +a \right )}{2 a b}+\frac {\ln \left (b x +a \right )}{2 a b}\) | \(31\) |
risch | \(-\frac {\ln \left (-b x +a \right )}{2 a b}+\frac {\ln \left (b x +a \right )}{2 a b}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (14) = 28\).
time = 3.27, size = 31, normalized size = 2.21 \begin {gather*} \frac {\log \left (b x + a\right )}{2 \, a b} - \frac {\log \left (b x - a\right )}{2 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 25, normalized size = 1.79 \begin {gather*} \frac {\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b} \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. 20 vs.
\(2 (8) = 16\).
time = 0.05, size = 20, normalized size = 1.43 \begin {gather*} - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{2} - \frac {\log {\left (\frac {a}{b} + x \right )}}{2}}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs.
\(2 (14) = 28\).
time = 1.61, size = 33, normalized size = 2.36 \begin {gather*} \frac {\log \left ({\left | b x + a \right |}\right )}{2 \, a b} - \frac {\log \left ({\left | b x - a \right |}\right )}{2 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 14, normalized size = 1.00 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{a\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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