Optimal. Leaf size=23 \[ -\frac {b}{2 x}-\frac {\tanh ^{-1}(\tanh (a+b x))}{2 x^2} \]
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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}(\tanh (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}(\tanh (a+b x))}{x^3} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))}{2 x^2}+\frac {1}{2} b \int \frac {1}{x^2} \, dx\\ &=-\frac {b}{2 x}-\frac {\tanh ^{-1}(\tanh (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}(\tanh (a+b x))}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 20, normalized size = 0.87
method | result | size |
default | \(-\frac {b}{2 x}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{2 x^{2}}\) | \(20\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{b x +a}\right )}{2 x^{2}}-\frac {4 b x -i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+2 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )}{8 x^{2}}\) | \(287\) |
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 (\tanh \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 = 0.33, 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 [A]
time = 0.19, size = 19, normalized size = 0.83 \begin {gather*} - \frac {b}{2 x} - \frac {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, 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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Mupad [B]
time = 0.95, size = 16, normalized size = 0.70 \begin {gather*} -\frac {\mathrm {atanh}\left (\mathrm {tanh}\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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