Optimal. Leaf size=46 \[ -\frac {\sinh (a+b x)}{b^2}-\frac {\tan ^{-1}(\sinh (a+b x))}{b^2}+\frac {x \cosh (a+b x)}{b}+\frac {x \text {sech}(a+b x)}{b} \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5449, 3296, 2637, 5418, 3770} \[ -\frac {\sinh (a+b x)}{b^2}-\frac {\tan ^{-1}(\sinh (a+b x))}{b^2}+\frac {x \cosh (a+b x)}{b}+\frac {x \text {sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3770
Rule 5418
Rule 5449
Rubi steps
\begin {align*} \int x \sinh (a+b x) \tanh ^2(a+b x) \, dx &=\int x \sinh (a+b x) \, dx-\int x \text {sech}(a+b x) \tanh (a+b x) \, dx\\ &=\frac {x \cosh (a+b x)}{b}+\frac {x \text {sech}(a+b x)}{b}-\frac {\int \cosh (a+b x) \, dx}{b}-\frac {\int \text {sech}(a+b x) \, dx}{b}\\ &=-\frac {\tan ^{-1}(\sinh (a+b x))}{b^2}+\frac {x \cosh (a+b x)}{b}+\frac {x \text {sech}(a+b x)}{b}-\frac {\sinh (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 50, normalized size = 1.09 \[ -\frac {\sinh (a+b x)}{b^2}-\frac {2 \tan ^{-1}\left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{b^2}+\frac {x \cosh (a+b x)}{b}+\frac {x \text {sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 283, normalized size = 6.15 \[ \frac {{\left (b x - 1\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b x - 1\right )} \sinh \left (b x + a\right )^{4} + 6 \, b x \cosh \left (b x + a\right )^{2} + 6 \, {\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + b x\right )} \sinh \left (b x + a\right )^{2} + b x - 4 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 4 \, {\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{3} + 3 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{2 \, {\left (b^{2} \cosh \left (b x + a\right )^{3} + 3 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{2} \sinh \left (b x + a\right )^{3} + b^{2} \cosh \left (b x + a\right ) + {\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} + b^{2}\right )} \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 102, normalized size = 2.22 \[ \frac {b x e^{\left (4 \, b x + 4 \, a\right )} + 6 \, b x e^{\left (2 \, b x + 2 \, a\right )} + b x - 4 \, \arctan \left (e^{\left (b x + a\right )}\right ) e^{\left (3 \, b x + 3 \, a\right )} - 4 \, \arctan \left (e^{\left (b x + a\right )}\right ) e^{\left (b x + a\right )} - e^{\left (4 \, b x + 4 \, a\right )} + 1}{2 \, {\left (b^{2} e^{\left (3 \, b x + 3 \, a\right )} + b^{2} e^{\left (b x + a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 94, normalized size = 2.04 \[ \frac {\left (b x -1\right ) {\mathrm e}^{b x +a}}{2 b^{2}}+\frac {\left (b x +1\right ) {\mathrm e}^{-b x -a}}{2 b^{2}}+\frac {2 x \,{\mathrm e}^{b x +a}}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )}+\frac {i \ln \left ({\mathrm e}^{b x +a}-i\right )}{b^{2}}-\frac {i \ln \left ({\mathrm e}^{b x +a}+i\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 81, normalized size = 1.76 \[ \frac {6 \, b x e^{\left (b x + 2 \, a\right )} + {\left (b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b x + 1\right )} e^{\left (-b x\right )}}{2 \, {\left (b^{2} e^{\left (2 \, b x + 3 \, a\right )} + b^{2} e^{a}\right )}} - \frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 90, normalized size = 1.96 \[ {\mathrm {e}}^{-a-b\,x}\,\left (\frac {x}{2\,b}+\frac {1}{2\,b^2}\right )-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^4}}{b^2}\right )}{\sqrt {b^4}}+{\mathrm {e}}^{a+b\,x}\,\left (\frac {x}{2\,b}-\frac {1}{2\,b^2}\right )+\frac {2\,x\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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 \[ \int x \sinh ^{3}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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