Optimal. Leaf size=23 \[ -\frac {b}{2 x}-\frac {\tanh ^{-1}(\coth (a+b x))}{2 x^2} \]
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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 {\tanh ^{-1}(\coth (a+b x))}{2 x^2}-\frac {b}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2199
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\coth (a+b x))}{x^3} \, dx &=-\frac {\tanh ^{-1}(\coth (a+b x))}{2 x^2}+\frac {1}{2} b \int \frac {1}{x^2} \, dx\\ &=-\frac {b}{2 x}-\frac {\tanh ^{-1}(\coth (a+b x))}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.78 \begin {gather*} -\frac {b x+\tanh ^{-1}(\coth (a+b x))}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 20, normalized size = 0.87
method | result | size |
default | \(-\frac {b}{2 x}-\frac {\arctanh \left (\coth \left (b x +a \right )\right )}{2 x^{2}}\) | \(20\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{b x +a}\right )}{2 x^{2}}-\frac {4 b x +2 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )+2 i \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}-2 i \pi +i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}-2 i \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right )^{3}+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{3}-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )}{8 x^{2}}\) | \(337\) |
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 {b}{2 \, x} - \frac {\operatorname {artanh}\left (\coth \left (b x + a\right )\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 11, normalized size = 0.48 \begin {gather*} -\frac {2 \, b x + a}{2 \, 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. 80 vs.
\(2 (19) = 38\).
time = 7.73, size = 80, normalized size = 3.48 \begin {gather*} \begin {cases} - \frac {\operatorname {atanh}{\left (\coth {\left (b x + \log {\left (- e^{- b x} \right )} \right )} \right )}}{2 x^{2}} & \text {for}\: a = \log {\left (- e^{- b x} \right )} \\- \frac {\operatorname {atanh}{\left (\coth {\left (b x + \log {\left (e^{- b x} \right )} \right )} \right )}}{2 x^{2}} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\- \frac {b}{2 x} - \frac {\operatorname {atanh}{\left (\frac {1}{\tanh {\left (a + b x \right )}} \right )}}{2 x^{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 = 0.40, size = 71, normalized size = 3.09 \begin {gather*} -\frac {b}{2 \, x} - \frac {\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 )}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.07, size = 16, normalized size = 0.70 \begin {gather*} -\frac {\mathrm {atanh}\left (\mathrm {coth}\left (a+b\,x\right )\right )+b\,x}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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