Optimal. Leaf size=16 \[ \frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2188, 30}
\begin {gather*} \frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2188
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b}\\ &=\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 15, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {2 \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{b}\) | \(15\) |
default | \(\frac {2 \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{b}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, \sqrt {b x + a}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, \sqrt {b x + a}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 23.21, size = 24, normalized size = 1.50 \begin {gather*} \begin {cases} \frac {2 \sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {\operatorname {atanh}{\left (\tanh {\left (a \right )} \right )}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, \sqrt {b x + a}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 52, normalized size = 3.25 \begin {gather*} \frac {2\,\sqrt {\frac {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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