Optimal. Leaf size=23 \[ -\frac {b x^4}{12}+\frac {1}{3} x^3 \coth ^{-1}(\coth (a+b x)) \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2199, 30}
\begin {gather*} \frac {1}{3} x^3 \coth ^{-1}(\coth (a+b x))-\frac {b x^4}{12} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2199
Rubi steps
\begin {align*} \int x^2 \coth ^{-1}(\coth (a+b x)) \, dx &=\frac {1}{3} x^3 \coth ^{-1}(\coth (a+b x))-\frac {1}{3} b \int x^3 \, dx\\ &=-\frac {b x^4}{12}+\frac {1}{3} x^3 \coth ^{-1}(\coth (a+b x))\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 0.87 \begin {gather*} -\frac {1}{12} x^3 \left (b x-4 \coth ^{-1}(\coth (a+b x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 20, normalized size = 0.87
method | result | size |
default | \(-\frac {b \,x^{4}}{12}+\frac {x^{3} \mathrm {arccoth}\left (\coth \left (b x +a \right )\right )}{3}\) | \(20\) |
risch | \(\frac {x^{3} \ln \left ({\mathrm e}^{b x +a}\right )}{3}-\frac {b \,x^{4}}{12}-\frac {i \pi \,x^{3} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}}{12}-\frac {i \pi \,x^{3} \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )}{12}+\frac {i \pi \,x^{3} \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}}{6}-\frac {i \pi \,x^{3} \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{3}}{12}-\frac {i \pi \,x^{3} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )}{12}+\frac {i \pi \,x^{3} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}}{12}+\frac {i \pi \,x^{3} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}}{12}\) | \(304\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 13, normalized size = 0.57 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 13, normalized size = 0.57 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (19) = 38\).
time = 7.06, size = 76, normalized size = 3.30 \begin {gather*} \begin {cases} \frac {x^{3} \operatorname {acoth}{\left (\coth {\left (b x + \log {\left (- e^{- b x} \right )} \right )} \right )}}{3} & \text {for}\: a = \log {\left (- e^{- b x} \right )} \\\frac {x^{3} \operatorname {acoth}{\left (\coth {\left (b x + \log {\left (e^{- b x} \right )} \right )} \right )}}{3} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\- \frac {b x^{4}}{12} + \frac {x^{3} \operatorname {acoth}{\left (\frac {1}{\tanh {\left (a + b x \right )}} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 13, normalized size = 0.57 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 19, normalized size = 0.83 \begin {gather*} \frac {x^3\,\mathrm {acoth}\left (\mathrm {coth}\left (a+b\,x\right )\right )}{3}-\frac {b\,x^4}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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