Optimal. Leaf size=34 \[ -\frac {1}{2 b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {x}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \]
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Rubi [A] time = 0.01, 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 = {2168, 2157, 30} \[ -\frac {1}{2 b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {x}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rule 2168
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 {\operatorname {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 = 0.05, size = 27, normalized size = 0.79 \[ -\frac {\tanh ^{-1}(\tanh (a+b x))+b x}{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 32, normalized size = 0.94 \[ -\frac {2 \, b x + a}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 18, normalized size = 0.53 \[ -\frac {2 \, b x + a}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 43, normalized size = 1.26 \[ -\frac {b x -\arctanh \left (\tanh \left (b x +a \right )\right )}{2 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {1}{b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.19, size = 32, normalized size = 0.94 \[ -\frac {2 \, b x + a}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
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 \[ -\frac {\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+b\,x}{2\,b^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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