Optimal. Leaf size=42 \[ -\frac {1}{10} b x^5 \tanh ^{-1}(\tanh (a+b x))+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2+\frac {b^2 x^6}{60} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 30} \[ -\frac {1}{10} b x^5 \tanh ^{-1}(\tanh (a+b x))+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2+\frac {b^2 x^6}{60} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int x^3 \tanh ^{-1}(\tanh (a+b x))^2 \, dx &=\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2-\frac {1}{2} b \int x^4 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac {1}{10} b x^5 \tanh ^{-1}(\tanh (a+b x))+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{10} b^2 \int x^5 \, dx\\ &=\frac {b^2 x^6}{60}-\frac {1}{10} b x^5 \tanh ^{-1}(\tanh (a+b x))+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2\\ \end {align*}
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Mathematica [A] time = 0.03, size = 37, normalized size = 0.88 \[ \frac {1}{60} x^4 \left (-6 b x \tanh ^{-1}(\tanh (a+b x))+15 \tanh ^{-1}(\tanh (a+b x))^2+b^2 x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 24, normalized size = 0.57 \[ \frac {1}{6} \, b^{2} x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 24, normalized size = 0.57 \[ \frac {1}{6} \, b^{2} x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 38, normalized size = 0.90 \[ \frac {x^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{4}-\frac {b \left (\frac {x^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}{5}-\frac {x^{6} b}{30}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 36, normalized size = 0.86 \[ \frac {1}{60} \, b^{2} x^{6} - \frac {1}{10} \, b x^{5} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) + \frac {1}{4} \, x^{4} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 36, normalized size = 0.86 \[ \frac {b^2\,x^6}{60}-\frac {b\,x^5\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{10}+\frac {x^4\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.45, size = 78, normalized size = 1.86 \[ \begin {cases} \frac {x^{3} \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{3 b} - \frac {x^{2} \operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{4 b^{2}} + \frac {x \operatorname {atanh}^{5}{\left (\tanh {\left (a + b x \right )} \right )}}{10 b^{3}} - \frac {\operatorname {atanh}^{6}{\left (\tanh {\left (a + b x \right )} \right )}}{60 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {atanh}^{2}{\left (\tanh {\relax (a )} \right )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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