Optimal. Leaf size=61 \[ -\frac {1}{280} b^3 x^8+\frac {1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3 \]
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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, 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}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac {1}{280} b^3 x^8 \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2199
Rubi steps
\begin {align*} \int x^4 \coth ^{-1}(\tanh (a+b x))^3 \, dx &=\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac {1}{5} (3 b) \int x^5 \coth ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3+\frac {1}{5} b^2 \int x^6 \coth ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac {1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac {1}{35} b^3 \int x^7 \, dx\\ &=-\frac {1}{280} b^3 x^8+\frac {1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 54, normalized size = 0.89 \begin {gather*} -\frac {1}{280} x^5 \left (b^3 x^3-8 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+28 b x \coth ^{-1}(\tanh (a+b x))^2-56 \coth ^{-1}(\tanh (a+b x))^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{4} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 54, normalized size = 0.89 \begin {gather*} -\frac {1}{10} \, b x^{6} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {1}{5} \, x^{5} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{280} \, {\left (b^{2} x^{8} - 8 \, b x^{7} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 52, normalized size = 0.85 \begin {gather*} \frac {1}{8} \, b^{3} x^{8} + \frac {3}{7} \, a b^{2} x^{7} - \frac {1}{8} \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x^{6} - \frac {1}{20} \, {\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.62, size = 56, normalized size = 0.92 \begin {gather*} - \frac {b^{3} x^{8}}{280} + \frac {b^{2} x^{7} \operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}{35} - \frac {b x^{6} \operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{10} + \frac {x^{5} \operatorname {acoth}^{3}{\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 [C] Result contains complex when optimal does not.
time = 0.42, size = 77, normalized size = 1.26 \begin {gather*} \frac {1}{8} \, b^{3} x^{8} - \frac {3}{14} \, {\left (-i \, \pi b^{2} - 2 \, a b^{2}\right )} x^{7} - \frac {1}{8} \, {\left (\pi ^{2} b - 4 i \, \pi a b - 4 \, a^{2} b\right )} x^{6} - \frac {1}{40} \, {\left (i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 53, normalized size = 0.87 \begin {gather*} -\frac {b^3\,x^8}{280}+\frac {b^2\,x^7\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{35}-\frac {b\,x^6\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{10}+\frac {x^5\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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