Optimal. Leaf size=28 \[ \frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^2}-\frac {x}{b \coth ^{-1}(\tanh (a+b x))} \]
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Rubi [A] time = 0.01, antiderivative size = 28, 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, 29} \[ \frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^2}-\frac {x}{b \coth ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2168
Rubi steps
\begin {align*} \int \frac {x}{\coth ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac {x}{b \coth ^{-1}(\tanh (a+b x))}+\frac {\int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=-\frac {x}{b \coth ^{-1}(\tanh (a+b x))}+\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^2}\\ &=-\frac {x}{b \coth ^{-1}(\tanh (a+b x))}+\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 27, normalized size = 0.96 \[ \frac {-\frac {b x}{\coth ^{-1}(\tanh (a+b x))}+\log \left (\coth ^{-1}(\tanh (a+b x))\right )+1}{b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 97, normalized size = 3.46 \[ \frac {8 \, a b x + 2 \, \pi ^{2} + 8 \, a^{2} + {\left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{2 \, {\left (4 \, b^{4} x^{2} + 8 \, a b^{3} x + \pi ^{2} b^{2} + 4 \, a^{2} b^{2}\right )}} \]
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 \[ \int \frac {x}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 625, normalized size = 22.32 \[ -\frac {4 i x}{b \left (-2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+\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}+2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+\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}+2 \pi +4 i \ln \left ({\mathrm e}^{b x +a}\right )\right )}+\frac {\ln \left (\ln \left ({\mathrm e}^{b x +a}\right )-\frac {i \pi \left (-2 \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+\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 )-\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}+2 \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )-2 \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}+\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}-\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}+\mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+2\right )}{4}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.76, size = 47, normalized size = 1.68 \[ \frac {4 \, {\left (-i \, \pi + 2 \, a\right )}}{8 \, b^{3} x - 4 i \, \pi b^{2} + 8 \, a b^{2}} + \frac {\log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 28, normalized size = 1.00 \[ \frac {\ln \left (\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )\right )}{b^2}-\frac {x}{b\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )} \]
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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