Optimal. Leaf size=72 \[ \frac {x^3 \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^6}{10 b^2}+\frac {x \tanh ^{-1}(\tanh (a+b x))^7}{35 b^3}-\frac {\tanh ^{-1}(\tanh (a+b x))^8}{280 b^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2199, 2188, 30}
\begin {gather*} -\frac {\tanh ^{-1}(\tanh (a+b x))^8}{280 b^4}+\frac {x \tanh ^{-1}(\tanh (a+b x))^7}{35 b^3}-\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^6}{10 b^2}+\frac {x^3 \tanh ^{-1}(\tanh (a+b x))^5}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2188
Rule 2199
Rubi steps
\begin {align*} \int x^3 \tanh ^{-1}(\tanh (a+b x))^4 \, dx &=\frac {x^3 \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {3 \int x^2 \tanh ^{-1}(\tanh (a+b x))^5 \, dx}{5 b}\\ &=\frac {x^3 \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^6}{10 b^2}+\frac {\int x \tanh ^{-1}(\tanh (a+b x))^6 \, dx}{5 b^2}\\ &=\frac {x^3 \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^6}{10 b^2}+\frac {x \tanh ^{-1}(\tanh (a+b x))^7}{35 b^3}-\frac {\int \tanh ^{-1}(\tanh (a+b x))^7 \, dx}{35 b^3}\\ &=\frac {x^3 \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^6}{10 b^2}+\frac {x \tanh ^{-1}(\tanh (a+b x))^7}{35 b^3}-\frac {\text {Subst}\left (\int x^7 \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{35 b^4}\\ &=\frac {x^3 \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^6}{10 b^2}+\frac {x \tanh ^{-1}(\tanh (a+b x))^7}{35 b^3}-\frac {\tanh ^{-1}(\tanh (a+b x))^8}{280 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 71, normalized size = 0.99 \begin {gather*} \frac {1}{280} x^4 \left (b^4 x^4-8 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+28 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-56 b x \tanh ^{-1}(\tanh (a+b x))^3+70 \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 = 1.03 \[\frac {x^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{4}-b \left (\frac {x^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{5}-\frac {3 b \left (\frac {x^{6} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{6}-\frac {b \left (\frac {x^{7} \arctanh \left (\tanh \left (b x +a \right )\right )}{7}-\frac {b \,x^{8}}{56}\right )}{3}\right )}{5}\right )\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.42, size = 72, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, b x^{5} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} + \frac {1}{4} \, x^{4} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4} + \frac {1}{280} \, {\left (28 \, b x^{6} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} + {\left (b^{2} x^{8} - 8 \, b x^{7} \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 = 0.38, size = 45, normalized size = 0.62 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} + \frac {4}{7} \, a b^{3} x^{7} + a^{2} b^{2} x^{6} + \frac {4}{5} \, a^{3} b x^{5} + \frac {1}{4} \, a^{4} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.68, size = 75, normalized size = 1.04 \begin {gather*} \frac {b^{4} x^{8}}{280} - \frac {b^{3} x^{7} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{35} + \frac {b^{2} x^{6} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{10} - \frac {b x^{5} \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{5} + \frac {x^{4} \operatorname {atanh}^{4}{\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 = 0.39, size = 45, normalized size = 0.62 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} + \frac {4}{7} \, a b^{3} x^{7} + a^{2} b^{2} x^{6} + \frac {4}{5} \, a^{3} b x^{5} + \frac {1}{4} \, a^{4} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.02, size = 70, normalized size = 0.97 \begin {gather*} \frac {b^4\,x^8}{280}-\frac {b^3\,x^7\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{35}+\frac {b^2\,x^6\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{10}-\frac {b\,x^5\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{5}+\frac {x^4\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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