Optimal. Leaf size=34 \[ -\frac {x}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {1}{2 b^2 \tanh ^{-1}(\tanh (a+b x))} \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2199, 2188, 30}
\begin {gather*} -\frac {1}{2 b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {x}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2188
Rule 2199
Rubi steps
\begin {align*} \int \frac {x}{\tanh ^{-1}(\tanh (a+b x))^3} \, dx &=-\frac {x}{2 b \tanh ^{-1}(\tanh (a+b x))^2}+\frac {\int \frac {1}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx}{2 b}\\ &=-\frac {x}{2 b \tanh ^{-1}(\tanh (a+b x))^2}+\frac {\text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{2 b^2}\\ &=-\frac {x}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {1}{2 b^2 \tanh ^{-1}(\tanh (a+b x))}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 27, normalized size = 0.79 \begin {gather*} -\frac {b x+\tanh ^{-1}(\tanh (a+b x))}{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 43, normalized size = 1.26
method | result | size |
default | \(-\frac {1}{b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )}-\frac {b x -\arctanh \left (\tanh \left (b x +a \right )\right )}{2 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}\) | \(43\) |
risch | \(-\frac {2 i \left (\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 )-\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}+\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 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}-\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}+\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+4 i b x +4 i \ln \left ({\mathrm e}^{b x +a}\right )\right )}{b^{2} \left (\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 )-\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}+\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 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}-\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}+\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+4 i \ln \left ({\mathrm e}^{b x +a}\right )\right )^{2}}\) | \(540\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.84, size = 32, normalized size = 0.94 \begin {gather*} -\frac {2 \, b x + a}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 32, normalized size = 0.94 \begin {gather*} -\frac {2 \, b x + a}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 37.24, size = 42, normalized size = 1.24 \begin {gather*} \begin {cases} - \frac {x}{2 b \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}} - \frac {1}{2 b^{2} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 \operatorname {atanh}^{3}{\left (\tanh {\left (a \right )} \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 18, normalized size = 0.53 \begin {gather*} -\frac {2 \, b x + a}{2 \, {\left (b x + a\right )}^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 25, normalized size = 0.74 \begin {gather*} -\frac {\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+b\,x}{2\,b^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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