Optimal. Leaf size=80 \[ \frac {b^4 x^9}{630}-\frac {1}{70} b^3 x^8 \tanh ^{-1}(\tanh (a+b x))+\frac {2}{35} b^2 x^7 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4 \]
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Rubi [A]
time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 5, 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}{70} b^3 x^8 \tanh ^{-1}(\tanh (a+b x))+\frac {2}{35} b^2 x^7 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4+\frac {b^4 x^9}{630} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2199
Rubi steps
\begin {align*} \int x^4 \tanh ^{-1}(\tanh (a+b x))^4 \, dx &=\frac {1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4-\frac {1}{5} (4 b) \int x^5 \tanh ^{-1}(\tanh (a+b x))^3 \, dx\\ &=-\frac {2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4+\frac {1}{5} \left (2 b^2\right ) \int x^6 \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=\frac {2}{35} b^2 x^7 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4-\frac {1}{35} \left (4 b^3\right ) \int x^7 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac {1}{70} b^3 x^8 \tanh ^{-1}(\tanh (a+b x))+\frac {2}{35} b^2 x^7 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4+\frac {1}{70} b^4 \int x^8 \, dx\\ &=\frac {b^4 x^9}{630}-\frac {1}{70} b^3 x^8 \tanh ^{-1}(\tanh (a+b x))+\frac {2}{35} b^2 x^7 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 71, normalized size = 0.89 \begin {gather*} \frac {1}{630} x^5 \left (b^4 x^4-9 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+36 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-84 b x \tanh ^{-1}(\tanh (a+b x))^3+126 \tanh ^{-1}(\tanh (a+b x))^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 74, normalized size = 0.92 \[\frac {x^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{5}-\frac {4 b \left (\frac {x^{6} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{6}-\frac {b \left (\frac {x^{7} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{7}-\frac {2 b \left (\frac {x^{8} \arctanh \left (\tanh \left (b x +a \right )\right )}{8}-\frac {b \,x^{9}}{72}\right )}{7}\right )}{2}\right )}{5}\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.42, size = 72, normalized size = 0.90 \begin {gather*} -\frac {2}{15} \, b x^{6} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} + \frac {1}{5} \, x^{5} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4} + \frac {1}{630} \, {\left (36 \, b x^{7} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} + {\left (b^{2} x^{9} - 9 \, b x^{8} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )\right )} b\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 46, normalized size = 0.58 \begin {gather*} \frac {1}{9} \, b^{4} x^{9} + \frac {1}{2} \, a b^{3} x^{8} + \frac {6}{7} \, a^{2} b^{2} x^{7} + \frac {2}{3} \, a^{3} b x^{6} + \frac {1}{5} \, a^{4} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.04, size = 78, normalized size = 0.98 \begin {gather*} \frac {b^{4} x^{9}}{630} - \frac {b^{3} x^{8} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{70} + \frac {2 b^{2} x^{7} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{35} - \frac {2 b x^{6} \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{15} + \frac {x^{5} \operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 46, normalized size = 0.58 \begin {gather*} \frac {1}{9} \, b^{4} x^{9} + \frac {1}{2} \, a b^{3} x^{8} + \frac {6}{7} \, a^{2} b^{2} x^{7} + \frac {2}{3} \, a^{3} b x^{6} + \frac {1}{5} \, a^{4} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.06, size = 77, normalized size = 0.96 \begin {gather*} \frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^5\,\left (126\,b^4\,x^4-84\,b^3\,x^3\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+36\,b^2\,x^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2-9\,b\,x\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3+{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4\right )}{630\,b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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