Optimal. Leaf size=47 \[ \frac {\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x}{b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {x^2}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \]
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Rubi [A] time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2168, 2157, 29} \[ -\frac {x}{b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x^2}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2168
Rubi steps
\begin {align*} \int \frac {x^2}{\tanh ^{-1}(\tanh (a+b x))^3} \, dx &=-\frac {x^2}{2 b \tanh ^{-1}(\tanh (a+b x))^2}+\frac {\int \frac {x}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx}{b}\\ &=-\frac {x^2}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {x}{b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {\int \frac {1}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=-\frac {x^2}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {x}{b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}\\ &=-\frac {x^2}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {x}{b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 49, normalized size = 1.04 \[ \frac {-\frac {b^2 x^2}{\tanh ^{-1}(\tanh (a+b x))^2}-\frac {2 b x}{\tanh ^{-1}(\tanh (a+b x))}+2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )+3}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 61, normalized size = 1.30 \[ \frac {4 \, a b x + 3 \, a^{2} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 37, normalized size = 0.79 \[ \frac {\log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac {4 \, a x + \frac {3 \, a^{2}}{b}}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 136, normalized size = 2.89 \[ -\frac {a^{2}}{2 b^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{2 b^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}+\frac {2 a}{b^{3} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )-2 b x -2 a}{b^{3} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {\ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.22, size = 48, normalized size = 1.02 \[ \frac {4 \, a b x + 3 \, a^{2}}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac {\log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 46, normalized size = 0.98 \[ \frac {\ln \left (\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )\right )}{b^3}-\frac {\frac {b^2\,x^2}{2}+b\,x\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{b^3\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2} \]
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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