Optimal. Leaf size=42 \[ \frac {b^2 x^6}{60}-\frac {1}{10} b x^5 \tanh ^{-1}(\tanh (a+b x))+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2 \]
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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 = {2199, 30}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2199
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 = 0.02, size = 37, normalized size = 0.88 \begin {gather*} \frac {1}{60} x^4 \left (b^2 x^2-6 b x \tanh ^{-1}(\tanh (a+b x))+15 \tanh ^{-1}(\tanh (a+b x))^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 35.96, size = 38, normalized size = 0.90
method | result | size |
default | \(\frac {x^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{4}-\frac {b \left (\frac {x^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}{5}-\frac {b \,x^{6}}{30}\right )}{2}\) | \(38\) |
risch | \(\text {Expression too large to display}\) | \(2083\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 36, normalized size = 0.86 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 24, normalized size = 0.57 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 37, normalized size = 0.88 \begin {gather*} \frac {b^{2} x^{6}}{60} - \frac {b x^{5} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{10} + \frac {x^{4} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 24, normalized size = 0.57 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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