Optimal. Leaf size=16 \[ \frac {\tanh ^{-1}(\tanh (a+b x))^2}{2 b} \]
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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 = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2157, 30} \[ \frac {\tanh ^{-1}(\tanh (a+b x))^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rubi steps
\begin {align*} \int \tanh ^{-1}(\tanh (a+b x)) \, dx &=\frac {\operatorname {Subst}\left (\int x \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b}\\ &=\frac {\tanh ^{-1}(\tanh (a+b x))^2}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.12 \[ x \tanh ^{-1}(\tanh (a+b x))-\frac {b x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 10, normalized size = 0.62 \[ \frac {1}{2} \, b x^{2} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 10, normalized size = 0.62 \[ \frac {1}{2} \, b x^{2} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 15, normalized size = 0.94 \[ \frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 16, normalized size = 1.00 \[ -\frac {1}{2} \, b x^{2} + x \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 16, normalized size = 1.00 \[ x\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )-\frac {b\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 19, normalized size = 1.19 \[ \begin {cases} \frac {\operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{2 b} & \text {for}\: b \neq 0 \\x \operatorname {atanh}{\left (\tanh {\relax (a )} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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