Optimal. Leaf size=23 \[ -\frac {b x^3}{6}+\frac {1}{2} x^2 \tanh ^{-1}(\coth (a+b x)) \]
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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 = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6376, 30}
\begin {gather*} \frac {1}{2} x^2 \tanh ^{-1}(\coth (a+b x))-\frac {b x^3}{6} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 6376
Rubi steps
\begin {align*} \int x \tanh ^{-1}(\coth (a+b x)) \, dx &=\frac {1}{2} x^2 \tanh ^{-1}(\coth (a+b x))-\frac {1}{2} b \int x^2 \, dx\\ &=-\frac {b x^3}{6}+\frac {1}{2} x^2 \tanh ^{-1}(\coth (a+b x))\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 0.87 \begin {gather*} -\frac {1}{6} x^2 \left (b x-3 \tanh ^{-1}(\coth (a+b x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 20, normalized size = 0.87
method | result | size |
default | \(-\frac {b \,x^{3}}{6}+\frac {x^{2} \arctanh \left (\coth \left (b x +a \right )\right )}{2}\) | \(20\) |
risch | \(\frac {x^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{2}-\frac {b \,x^{3}}{6}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right )^{3}}{4}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )}{8}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}}{8}-\frac {i \pi \,x^{2}}{4}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}}{4}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}}{8}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}}{8}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{3}}{8}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}}{4}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )}{8}\) | \(363\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 19, normalized size = 0.83 \begin {gather*} -\frac {1}{6} \, b x^{3} + \frac {1}{2} \, x^{2} \operatorname {artanh}\left (\coth \left (b x + a\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 13, normalized size = 0.57 \begin {gather*} \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \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. 160 vs.
\(2 (19) = 38\).
time = 4.15, size = 160, normalized size = 6.96 \begin {gather*} \begin {cases} - \frac {x \log {\left (- e^{- b x} \right )} \operatorname {atanh}{\left (\coth {\left (b x + \log {\left (- e^{- b x} \right )} \right )} \right )}}{b} - \frac {\log {\left (- e^{- b x} \right )}^{2} \operatorname {atanh}{\left (\coth {\left (b x + \log {\left (- e^{- b x} \right )} \right )} \right )}}{2 b^{2}} & \text {for}\: a = \log {\left (- e^{- b x} \right )} \\- \frac {x \log {\left (e^{- b x} \right )} \operatorname {atanh}{\left (\coth {\left (b x + \log {\left (e^{- b x} \right )} \right )} \right )}}{b} - \frac {\log {\left (e^{- b x} \right )}^{2} \operatorname {atanh}{\left (\coth {\left (b x + \log {\left (e^{- b x} \right )} \right )} \right )}}{2 b^{2}} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\- \frac {b x^{3}}{6} + \frac {x^{2} \operatorname {atanh}{\left (\frac {1}{\tanh {\left (a + b x \right )}} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (19) = 38\).
time = 0.38, size = 71, normalized size = 3.09 \begin {gather*} -\frac {1}{6} \, b x^{3} + \frac {1}{4} \, x^{2} \log \left (-\frac {\frac {e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} + 1}{\frac {e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 19, normalized size = 0.83 \begin {gather*} \frac {x^2\,\mathrm {atanh}\left (\mathrm {coth}\left (a+b\,x\right )\right )}{2}-\frac {b\,x^3}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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