Optimal. Leaf size=23 \[ \frac {1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))-\frac {b x^4}{12} \]
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Rubi [A] time = 0.01, antiderivative size = 23, 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 = {2168, 30} \[ \frac {1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))-\frac {b x^4}{12} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int x^2 \tanh ^{-1}(\tanh (a+b x)) \, dx &=\frac {1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))-\frac {1}{3} b \int x^3 \, dx\\ &=-\frac {b x^4}{12}+\frac {1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.87 \[ -\frac {1}{12} x^3 \left (b x-4 \tanh ^{-1}(\tanh (a+b x))\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.37, size = 13, normalized size = 0.57 \[ \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 13, normalized size = 0.57 \[ \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 20, normalized size = 0.87 \[ -\frac {b \,x^{4}}{12}+\frac {x^{3} \arctanh \left (\tanh \left (b x +a \right )\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 19, normalized size = 0.83 \[ -\frac {1}{12} \, b x^{4} + \frac {1}{3} \, x^{3} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 19, normalized size = 0.83 \[ \frac {x^3\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{3}-\frac {b\,x^4}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 19, normalized size = 0.83 \[ - \frac {b x^{4}}{12} + \frac {x^{3} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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