Optimal. Leaf size=34 \[ \frac {x \tanh ^{-1}(\tanh (a+b x))^3}{3 b}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{12 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2168, 2157, 30} \[ \frac {x \tanh ^{-1}(\tanh (a+b x))^3}{3 b}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{12 b^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rule 2168
Rubi steps
\begin {align*} \int x \tanh ^{-1}(\tanh (a+b x))^2 \, dx &=\frac {x \tanh ^{-1}(\tanh (a+b x))^3}{3 b}-\frac {\int \tanh ^{-1}(\tanh (a+b x))^3 \, dx}{3 b}\\ &=\frac {x \tanh ^{-1}(\tanh (a+b x))^3}{3 b}-\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^2}\\ &=\frac {x \tanh ^{-1}(\tanh (a+b x))^3}{3 b}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{12 b^2}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 74, normalized size = 2.18 \[ \frac {(a+b x) \left (4 \left (2 a^2+a b x-b^2 x^2\right ) \tanh ^{-1}(\tanh (a+b x))-\left ((3 a-b x) (a+b x)^2\right )-6 (a-b x) \tanh ^{-1}(\tanh (a+b x))^2\right )}{12 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 24, normalized size = 0.71 \[ \frac {1}{4} \, b^{2} x^{4} + \frac {2}{3} \, a b x^{3} + \frac {1}{2} \, a^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 24, normalized size = 0.71 \[ \frac {1}{4} \, b^{2} x^{4} + \frac {2}{3} \, a b x^{3} + \frac {1}{2} \, a^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 38, normalized size = 1.12 \[ \frac {x^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{2}-b \left (-\frac {b \,x^{4}}{12}+\frac {x^{3} \arctanh \left (\tanh \left (b x +a \right )\right )}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 36, normalized size = 1.06 \[ \frac {1}{12} \, b^{2} x^{4} - \frac {1}{3} \, b x^{3} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) + \frac {1}{2} \, x^{2} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 36, normalized size = 1.06 \[ \frac {b^2\,x^4}{12}-\frac {b\,x^3\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{3}+\frac {x^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 41, normalized size = 1.21 \[ \begin {cases} \frac {x \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{3 b} - \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{12 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {atanh}^{2}{\left (\tanh {\relax (a )} \right )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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