Optimal. Leaf size=45 \[ \frac {\text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}+\frac {x \log \left (e^{2 (a+b x)}+1\right )}{b}-\frac {x^2}{2} \]
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Rubi [A] time = 0.08, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3718, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{2 b^2}+\frac {x \log \left (e^{2 (a+b x)}+1\right )}{b}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rubi steps
\begin {align*} \int x \tanh (a+b x) \, dx &=-\frac {x^2}{2}+2 \int \frac {e^{2 (a+b x)} x}{1+e^{2 (a+b x)}} \, dx\\ &=-\frac {x^2}{2}+\frac {x \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {\int \log \left (1+e^{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac {x^2}{2}+\frac {x \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^2}\\ &=-\frac {x^2}{2}+\frac {x \log \left (1+e^{2 (a+b x)}\right )}{b}+\frac {\text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}\\ \end {align*}
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Mathematica [C] time = 3.24, size = 149, normalized size = 3.31 \[ \frac {1}{2} \left (x^2 \tanh (a)+\frac {-b^2 x^2 \tanh (a) \sqrt {-\text {csch}^2(a)} e^{-\tanh ^{-1}(\coth (a))}-\text {Li}_2\left (e^{-2 \left (b x+\tanh ^{-1}(\coth (a))\right )}\right )+2 b x \log \left (1-e^{-2 \left (\tanh ^{-1}(\coth (a))+b x\right )}\right )+2 \tanh ^{-1}(\coth (a)) \left (\log \left (1-e^{-2 \left (\tanh ^{-1}(\coth (a))+b x\right )}\right )-\log \left (i \sinh \left (\tanh ^{-1}(\coth (a))+b x\right )\right )+b x\right )+i \pi b x-i \pi \log \left (e^{2 b x}+1\right )+i \pi \log (\cosh (b x))}{b^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.69, size = 141, normalized size = 3.13 \[ -\frac {b^{2} x^{2} + 2 \, a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + 2 \, a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - 2 \, {\left (b x + a\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left (b x + a\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 2 \, {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right )}{2 \, b^{2}} \]
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 \[ \int x \operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 70, normalized size = 1.56 \[ -\frac {x^{2}}{2}-\frac {2 a x}{b}-\frac {a^{2}}{b^{2}}+\frac {x \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}+\frac {\polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 40, normalized size = 0.89 \[ -\frac {1}{2} \, x^{2} + \frac {2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\mathrm {sinh}\left (a+b\,x\right )}{\mathrm {cosh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sinh {\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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