Optimal. Leaf size=16 \[ \frac {\coth ^{-1}(\tanh (a+b x))^4}{4 b} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2157, 30} \[ \frac {\coth ^{-1}(\tanh (a+b x))^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rubi steps
\begin {align*} \int \coth ^{-1}(\tanh (a+b x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b}\\ &=\frac {\coth ^{-1}(\tanh (a+b x))^4}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {\coth ^{-1}(\tanh (a+b x))^4}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 49, normalized size = 3.06 \[ \frac {1}{4} \, b^{3} x^{4} + a b^{2} x^{3} - \frac {3}{8} \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x^{2} - \frac {1}{4} \, {\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} x \]
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 \[ \int \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 15, normalized size = 0.94 \[ \frac {\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{4}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 51, normalized size = 3.19 \[ -\frac {3}{2} \, b x^{2} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + x \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{4} \, {\left (b^{2} x^{4} - 4 \, b x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 47, normalized size = 2.94 \[ \frac {x\,\left (2\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )-b\,x\right )\,\left (b^2\,x^2-2\,b\,x\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+2\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 20, normalized size = 1.25 \[ \begin {cases} \frac {\operatorname {acoth}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{4 b} & \text {for}\: b \neq 0 \\x \operatorname {acoth}^{3}{\left (\tanh {\relax (a )} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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