Optimal. Leaf size=42 \[ -\frac {\log (\cosh (a+b x))}{b^3}+\frac {x \tanh (a+b x)}{b^2}-\frac {x^2 \text {sech}^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5418, 4184, 3475} \[ \frac {x \tanh (a+b x)}{b^2}-\frac {\log (\cosh (a+b x))}{b^3}-\frac {x^2 \text {sech}^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4184
Rule 5418
Rubi steps
\begin {align*} \int x^2 \text {sech}^2(a+b x) \tanh (a+b x) \, dx &=-\frac {x^2 \text {sech}^2(a+b x)}{2 b}+\frac {\int x \text {sech}^2(a+b x) \, dx}{b}\\ &=-\frac {x^2 \text {sech}^2(a+b x)}{2 b}+\frac {x \tanh (a+b x)}{b^2}-\frac {\int \tanh (a+b x) \, dx}{b^2}\\ &=-\frac {\log (\cosh (a+b x))}{b^3}-\frac {x^2 \text {sech}^2(a+b x)}{2 b}+\frac {x \tanh (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 55, normalized size = 1.31 \[ -\frac {\log (\cosh (a+b x))}{b^3}+\frac {x \tanh (a)}{b^2}+\frac {x \text {sech}(a) \sinh (b x) \text {sech}(a+b x)}{b^2}-\frac {x^2 \text {sech}^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 378, normalized size = 9.00 \[ \frac {2 \, b x \cosh \left (b x + a\right )^{4} + 8 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 2 \, b x \sinh \left (b x + a\right )^{4} - 2 \, {\left (b^{2} x^{2} - b x\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - 6 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \, {\left (2 \, b x \cosh \left (b x + a\right )^{3} - {\left (b^{2} x^{2} - b x\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b^{3} \cosh \left (b x + a\right )^{4} + 4 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{3} \sinh \left (b x + a\right )^{4} + 2 \, b^{3} \cosh \left (b x + a\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (b x + a\right )^{2} + b^{3}\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b^{3} \cosh \left (b x + a\right )^{3} + b^{3} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 142, normalized size = 3.38 \[ -\frac {2 \, b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} - 2 \, b x e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b x e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (4 \, b x + 4 \, a\right )} \log \left (-e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, e^{\left (2 \, b x + 2 \, a\right )} \log \left (-e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + \log \left (-e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}{b^{3} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{3} e^{\left (2 \, b x + 2 \, a\right )} + b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 73, normalized size = 1.74 \[ \frac {2 x}{b^{2}}+\frac {2 a}{b^{3}}-\frac {2 x \left (b x \,{\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 b x +2 a}+1\right )}{b^{2} \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}-\frac {\ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 94, normalized size = 2.24 \[ -\frac {2 \, {\left ({\left (b x^{2} e^{\left (2 \, a\right )} - x e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} - x e^{\left (4 \, b x + 4 \, a\right )}\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} - \frac {\log \left ({\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 102, normalized size = 2.43 \[ \frac {\frac {x^2}{b}-\frac {x^2\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b}}{2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1}-\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1\right )}{b^3}+\frac {2\,x}{b^2}-\frac {b\,x^2+2\,x}{b^2\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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 \[ \int x^{2} \sinh {\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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