Optimal. Leaf size=61 \[ \frac {1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac {1}{280} b^3 x^8 \]
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Rubi [A] time = 0.04, 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 = {2168, 30} \[ \frac {1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac {1}{280} b^3 x^8 \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
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 = 0.03, size = 54, normalized size = 0.89 \[ -\frac {1}{280} x^5 \left (-8 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+28 b x \coth ^{-1}(\tanh (a+b x))^2-56 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 52, normalized size = 0.85 \[ \frac {1}{8} \, b^{3} x^{8} + \frac {3}{7} \, a b^{2} x^{7} - \frac {1}{8} \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x^{6} - \frac {1}{20} \, {\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.14, size = 18111, normalized size = 296.90 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 54, normalized size = 0.89 \[ -\frac {1}{10} \, b x^{6} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {1}{5} \, x^{5} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{280} \, {\left (b^{2} x^{8} - 8 \, b x^{7} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )\right )} b \]
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 \[ -\frac {b^3\,x^8}{280}+\frac {b^2\,x^7\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{35}-\frac {b\,x^6\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{10}+\frac {x^5\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.88, size = 97, normalized size = 1.59 \[ \begin {cases} \frac {x^{4} \operatorname {acoth}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{4 b} - \frac {x^{3} \operatorname {acoth}^{5}{\left (\tanh {\left (a + b x \right )} \right )}}{5 b^{2}} + \frac {x^{2} \operatorname {acoth}^{6}{\left (\tanh {\left (a + b x \right )} \right )}}{10 b^{3}} - \frac {x \operatorname {acoth}^{7}{\left (\tanh {\left (a + b x \right )} \right )}}{35 b^{4}} + \frac {\operatorname {acoth}^{8}{\left (\tanh {\left (a + b x \right )} \right )}}{280 b^{5}} & \text {for}\: b \neq 0 \\\frac {x^{5} \operatorname {acoth}^{3}{\left (\tanh {\relax (a )} \right )}}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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