Optimal. Leaf size=53 \[ \frac {\coth ^{-1}(\tanh (a+b x))^6}{60 b^3}-\frac {x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac {x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b} \]
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Rubi [A] time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2168, 2157, 30} \[ \frac {\coth ^{-1}(\tanh (a+b x))^6}{60 b^3}-\frac {x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac {x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rule 2168
Rubi steps
\begin {align*} \int x^2 \coth ^{-1}(\tanh (a+b x))^3 \, dx &=\frac {x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\int x \coth ^{-1}(\tanh (a+b x))^4 \, dx}{2 b}\\ &=\frac {x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac {\int \coth ^{-1}(\tanh (a+b x))^5 \, dx}{10 b^2}\\ &=\frac {x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac {\operatorname {Subst}\left (\int x^5 \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{10 b^3}\\ &=\frac {x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac {\coth ^{-1}(\tanh (a+b x))^6}{60 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 1.02 \[ -\frac {1}{60} x^3 \left (-6 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+15 b x \coth ^{-1}(\tanh (a+b x))^2-20 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 52, normalized size = 0.98 \[ \frac {1}{6} \, b^{3} x^{6} + \frac {3}{5} \, a b^{2} x^{5} - \frac {3}{16} \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x^{4} - \frac {1}{12} \, {\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} x^{3} \]
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 \[ \int x^{2} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.13, size = 18111, normalized size = 341.72 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 54, normalized size = 1.02 \[ -\frac {1}{4} \, b x^{4} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {1}{3} \, x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{60} \, {\left (b^{2} x^{6} - 6 \, b x^{5} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 53, normalized size = 1.00 \[ -\frac {b^3\,x^6}{60}+\frac {b^2\,x^5\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{10}-\frac {b\,x^4\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{4}+\frac {x^3\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.38, size = 60, normalized size = 1.13 \[ \begin {cases} \frac {x^{2} \operatorname {acoth}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{4 b} - \frac {x \operatorname {acoth}^{5}{\left (\tanh {\left (a + b x \right )} \right )}}{10 b^{2}} + \frac {\operatorname {acoth}^{6}{\left (\tanh {\left (a + b x \right )} \right )}}{60 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {acoth}^{3}{\left (\tanh {\relax (a )} \right )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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