Optimal. Leaf size=26 \[ x \cosh (a-c)+\frac {\log (\cosh (c+b x)) \sinh (a-c)}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5746, 3556, 8}
\begin {gather*} \frac {\sinh (a-c) \log (\cosh (b x+c))}{b}+x \cosh (a-c) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3556
Rule 5746
Rubi steps
\begin {align*} \int \cosh (a+b x) \text {sech}(c+b x) \, dx &=\cosh (a-c) \int 1 \, dx+\sinh (a-c) \int \tanh (c+b x) \, dx\\ &=x \cosh (a-c)+\frac {\log (\cosh (c+b x)) \sinh (a-c)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 26, normalized size = 1.00 \begin {gather*} x \cosh (a-c)+\frac {\log (\cosh (c+b x)) \sinh (a-c)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(145\) vs.
\(2(26)=52\).
time = 0.78, size = 146, normalized size = 5.62
method | result | size |
risch | \(x \,{\mathrm e}^{a -c}-x \,{\mathrm e}^{-a -c} {\mathrm e}^{2 a}+x \,{\mathrm e}^{-a -c} {\mathrm e}^{2 c}-\frac {{\mathrm e}^{-a -c} {\mathrm e}^{2 a} a}{b}+\frac {{\mathrm e}^{-a -c} {\mathrm e}^{2 c} a}{b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 a -2 c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 a -2 c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 51, normalized size = 1.96 \begin {gather*} \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}{2 \, b} + \frac {{\left (b x + a\right )} e^{\left (a - c\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs.
\(2 (26) = 52\).
time = 0.35, size = 86, normalized size = 3.31 \begin {gather*} \frac {2 \, b x + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \log \left (\frac {2 \, \cosh \left (b x + c\right )}{\cosh \left (b x + c\right ) - \sinh \left (b x + c\right )}\right )}{2 \, {\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )}} \end {gather*}
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 \begin {gather*} \int \cosh {\left (a + b x \right )} \operatorname {sech}{\left (b x + c \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 50, normalized size = 1.92 \begin {gather*} \frac {2 \, b x e^{\left (-a + c\right )} + {\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (2 \, b x + 2 \, c\right )} + 1\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 64, normalized size = 2.46 \begin {gather*} x\,{\mathrm {e}}^{c-a}+\frac {{\mathrm {e}}^{2\,c-2\,a}\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\right )\,\left (2\,b\,{\mathrm {e}}^{3\,a-3\,c}-2\,b\,{\mathrm {e}}^{a-c}\right )}{4\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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