Optimal. Leaf size=34 \[ \frac {x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\coth ^{-1}(\tanh (a+b x))^5}{20 b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2168, 2157, 30} \[ \frac {x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\coth ^{-1}(\tanh (a+b x))^5}{20 b^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rule 2168
Rubi steps
\begin {align*} \int x \coth ^{-1}(\tanh (a+b x))^3 \, dx &=\frac {x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\int \coth ^{-1}(\tanh (a+b x))^4 \, dx}{4 b}\\ &=\frac {x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\operatorname {Subst}\left (\int x^4 \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{4 b^2}\\ &=\frac {x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\coth ^{-1}(\tanh (a+b x))^5}{20 b^2}\\ \end {align*}
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Mathematica [B] time = 0.07, size = 99, normalized size = 2.91 \[ \frac {(a+b x) \left (10 \left (2 a^2+a b x-b^2 x^2\right ) \coth ^{-1}(\tanh (a+b x))^2+(4 a-b x) (a+b x)^3-5 (3 a-b x) (a+b x)^2 \coth ^{-1}(\tanh (a+b x))-10 (a-b x) \coth ^{-1}(\tanh (a+b x))^3\right )}{20 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 52, normalized size = 1.53 \[ \frac {1}{5} \, b^{3} x^{5} + \frac {3}{4} \, a b^{2} x^{4} - \frac {1}{4} \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x^{3} - \frac {1}{8} \, {\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} x^{2} \]
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 \[ \int x \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.13, size = 18111, normalized size = 532.68 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 54, normalized size = 1.59 \[ -\frac {1}{2} \, b x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{20} \, {\left (b^{2} x^{5} - 5 \, b x^{4} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 53, normalized size = 1.56 \[ -\frac {b^3\,x^5}{20}+\frac {b^2\,x^4\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{4}-\frac {b\,x^3\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{2}+\frac {x^2\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.21, size = 41, normalized size = 1.21 \[ \begin {cases} \frac {x \operatorname {acoth}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{4 b} - \frac {\operatorname {acoth}^{5}{\left (\tanh {\left (a + b x \right )} \right )}}{20 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {acoth}^{3}{\left (\tanh {\relax (a )} \right )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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