Optimal. Leaf size=47 \[ \frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x}{b^2 \coth ^{-1}(\tanh (a+b x))}-\frac {x^2}{2 b \coth ^{-1}(\tanh (a+b x))^2} \]
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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} \[ -\frac {x}{b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x^2}{2 b \coth ^{-1}(\tanh (a+b x))^2} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2168
Rubi steps
\begin {align*} \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))^3} \, dx &=-\frac {x^2}{2 b \coth ^{-1}(\tanh (a+b x))^2}+\frac {\int \frac {x}{\coth ^{-1}(\tanh (a+b x))^2} \, dx}{b}\\ &=-\frac {x^2}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {x}{b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {\int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=-\frac {x^2}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {x}{b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^3}\\ &=-\frac {x^2}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {x}{b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {\log \left (\coth ^{-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 \[ \frac {-\frac {b^2 x^2}{\coth ^{-1}(\tanh (a+b x))^2}-\frac {2 b x}{\coth ^{-1}(\tanh (a+b x))}+2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )+3}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 250, normalized size = 5.32 \[ \frac {64 \, a b^{3} x^{3} + 3 \, \pi ^{4} + 24 \, \pi ^{2} a^{2} + 48 \, a^{4} + 4 \, {\left (5 \, \pi ^{2} b^{2} + 44 \, a^{2} b^{2}\right )} x^{2} + 40 \, {\left (\pi ^{2} a b + 4 \, a^{3} b\right )} x + {\left (16 \, b^{4} x^{4} + 64 \, a b^{3} x^{3} + \pi ^{4} + 8 \, \pi ^{2} a^{2} + 16 \, a^{4} + 8 \, {\left (\pi ^{2} b^{2} + 12 \, a^{2} b^{2}\right )} x^{2} + 16 \, {\left (\pi ^{2} a b + 4 \, a^{3} b\right )} x\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{2 \, {\left (16 \, b^{7} x^{4} + 64 \, a b^{6} x^{3} + \pi ^{4} b^{3} + 8 \, \pi ^{2} a^{2} b^{3} + 16 \, a^{4} b^{3} + 8 \, {\left (\pi ^{2} b^{5} + 12 \, a^{2} b^{5}\right )} x^{2} + 16 \, {\left (\pi ^{2} a b^{4} + 4 \, a^{3} b^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 952, normalized size = 20.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.12, size = 94, normalized size = 2.00 \[ -\frac {8 \, {\left (3 \, \pi ^{2} + 12 i \, \pi a - 12 \, a^{2} + {\left (8 i \, \pi b - 16 \, a b\right )} x\right )}}{64 \, b^{5} x^{2} - 16 \, \pi ^{2} b^{3} - 64 i \, \pi a b^{3} + 64 \, a^{2} b^{3} + {\left (-64 i \, \pi b^{4} + 128 \, a b^{4}\right )} x} + \frac {\log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 46, normalized size = 0.98 \[ \frac {\ln \left (\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )\right )}{b^3}-\frac {\frac {b^2\,x^2}{2}+b\,x\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{b^3\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2} \]
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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