Optimal. Leaf size=82 \[ \frac {\text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}-\frac {\tanh (a+b x)}{2 b^2}+\frac {x \log \left (e^{2 (a+b x)}+1\right )}{b}-\frac {x \tanh ^2(a+b x)}{2 b}+\frac {x}{2 b}-\frac {x^2}{2} \]
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Rubi [A] time = 0.12, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3720, 3473, 8, 3718, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{2 b^2}-\frac {\tanh (a+b x)}{2 b^2}+\frac {x \log \left (e^{2 (a+b x)}+1\right )}{b}-\frac {x \tanh ^2(a+b x)}{2 b}+\frac {x}{2 b}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3473
Rule 3718
Rule 3720
Rubi steps
\begin {align*} \int x \tanh ^3(a+b x) \, dx &=-\frac {x \tanh ^2(a+b x)}{2 b}+\frac {\int \tanh ^2(a+b x) \, dx}{2 b}+\int x \tanh (a+b x) \, dx\\ &=-\frac {x^2}{2}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}+2 \int \frac {e^{2 (a+b x)} x}{1+e^{2 (a+b x)}} \, dx+\frac {\int 1 \, dx}{2 b}\\ &=\frac {x}{2 b}-\frac {x^2}{2}+\frac {x \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}-\frac {\int \log \left (1+e^{2 (a+b x)}\right ) \, dx}{b}\\ &=\frac {x}{2 b}-\frac {x^2}{2}+\frac {x \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 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 b}-\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}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [C] time = 6.13, size = 231, normalized size = 2.82 \[ \frac {\text {csch}(a) \text {sech}(a) \left (b^2 x^2 e^{-\tanh ^{-1}(\coth (a))}-\frac {i \coth (a) \left (i \text {Li}_2\left (e^{2 i \left (i b x+i \tanh ^{-1}(\coth (a))\right )}\right )-b x \left (-\pi +2 i \tanh ^{-1}(\coth (a))\right )-2 \left (i \tanh ^{-1}(\coth (a))+i b x\right ) \log \left (1-e^{2 i \left (i \tanh ^{-1}(\coth (a))+i b x\right )}\right )+2 i \tanh ^{-1}(\coth (a)) \log \left (i \sinh \left (\tanh ^{-1}(\coth (a))+b x\right )\right )-\pi \log \left (e^{2 b x}+1\right )+\pi \log (\cosh (b x))\right )}{\sqrt {1-\coth ^2(a)}}\right )}{2 b^2 \sqrt {\text {csch}^2(a) \left (\sinh ^2(a)-\cosh ^2(a)\right )}}-\frac {\text {sech}(a) \sinh (b x) \text {sech}(a+b x)}{2 b^2}+\frac {x \text {sech}^2(a+b x)}{2 b}+\frac {1}{2} x^2 \tanh (a) \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.57, size = 1106, normalized size = 13.49 \[ \text {result too large to display} \]
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 \[ \int x \operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 111, normalized size = 1.35 \[ -\frac {x^{2}}{2}+\frac {2 b x \,{\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 b x +2 a}+1}{b^{2} \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}-\frac {2 a x}{b}-\frac {a^{2}}{b^{2}}+\frac {x \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}+\frac {\polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 131, normalized size = 1.60 \[ -x^{2} + \frac {b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} + b^{2} x^{2} + 2 \, {\left (b^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 2}{2 \, {\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} + \frac {2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int x \sinh ^{3}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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