Optimal. Leaf size=47 \[ -\frac {\cosh (a+b x)}{b^2}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^2}+\frac {x \sinh (a+b x)}{b}-\frac {x \text {csch}(a+b x)}{b} \]
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Rubi [A] time = 0.05, antiderivative size = 47, 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 = {5450, 3296, 2638, 5419, 3770} \[ -\frac {\cosh (a+b x)}{b^2}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^2}+\frac {x \sinh (a+b x)}{b}-\frac {x \text {csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3770
Rule 5419
Rule 5450
Rubi steps
\begin {align*} \int x \cosh (a+b x) \coth ^2(a+b x) \, dx &=\int x \cosh (a+b x) \, dx+\int x \coth (a+b x) \text {csch}(a+b x) \, dx\\ &=-\frac {x \text {csch}(a+b x)}{b}+\frac {x \sinh (a+b x)}{b}+\frac {\int \text {csch}(a+b x) \, dx}{b}-\frac {\int \sinh (a+b x) \, dx}{b}\\ &=-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac {\cosh (a+b x)}{b^2}-\frac {x \text {csch}(a+b x)}{b}+\frac {x \sinh (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 66, normalized size = 1.40 \[ \frac {2 b x \sinh (a+b x)-2 \cosh (a+b x)+b x \tanh \left (\frac {1}{2} (a+b x)\right )-b x \coth \left (\frac {1}{2} (a+b x)\right )+2 \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 367, normalized size = 7.81 \[ \frac {{\left (b x - 1\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b x - 1\right )} \sinh \left (b x + a\right )^{4} - 6 \, b x \cosh \left (b x + a\right )^{2} + 6 \, {\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} + b x - 2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 4 \, {\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{3} - 3 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{2 \, {\left (b^{2} \cosh \left (b x + a\right )^{3} + 3 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{2} \sinh \left (b x + a\right )^{3} - b^{2} \cosh \left (b x + a\right ) + {\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} - b^{2}\right )} \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 144, normalized size = 3.06 \[ \frac {b x e^{\left (4 \, b x + 4 \, a\right )} - 6 \, b x e^{\left (2 \, b x + 2 \, a\right )} + b x - 2 \, e^{\left (3 \, b x + 3 \, a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, e^{\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, e^{\left (3 \, b x + 3 \, a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - 2 \, e^{\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - e^{\left (4 \, b x + 4 \, a\right )} + 1}{2 \, {\left (b^{2} e^{\left (3 \, b x + 3 \, a\right )} - b^{2} e^{\left (b x + a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 89, normalized size = 1.89 \[ \frac {\left (b x -1\right ) {\mathrm e}^{b x +a}}{2 b^{2}}-\frac {\left (b x +1\right ) {\mathrm e}^{-b x -a}}{2 b^{2}}-\frac {2 \,{\mathrm e}^{b x +a} x}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 109, normalized size = 2.32 \[ -\frac {6 \, b x e^{\left (b x + 2 \, a\right )} - {\left (b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} - {\left (b x + 1\right )} e^{\left (-b x\right )}}{2 \, {\left (b^{2} e^{\left (2 \, b x + 3 \, a\right )} - b^{2} e^{a}\right )}} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 95, normalized size = 2.02 \[ {\mathrm {e}}^{a+b\,x}\,\left (\frac {x}{2\,b}-\frac {1}{2\,b^2}\right )-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^4}}{b^2}\right )}{\sqrt {-b^4}}-{\mathrm {e}}^{-a-b\,x}\,\left (\frac {x}{2\,b}+\frac {1}{2\,b^2}\right )-\frac {2\,x\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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