Optimal. Leaf size=74 \[ \frac {e^{2 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{4 b c}+\frac {1}{2} x \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x) \]
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Rubi [A] time = 0.10, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6720, 2282, 12, 14} \[ \frac {e^{2 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{4 b c}+\frac {1}{2} x \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2282
Rule 6720
Rubi steps
\begin {align*} \int e^{c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \, dx &=\left (\sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\right ) \int e^{c (a+b x)} \cosh (a c+b c x) \, dx\\ &=\frac {\left (\sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{2 x} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac {\left (\sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{x} \, dx,x,e^{c (a+b x)}\right )}{2 b c}\\ &=\frac {\left (\sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x}+x\right ) \, dx,x,e^{c (a+b x)}\right )}{2 b c}\\ &=\frac {e^{2 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{4 b c}+\frac {1}{2} x \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 0.65 \[ \frac {\left (e^{2 c (a+b x)}+2 b c x\right ) \sqrt {\cosh ^2(c (a+b x))} \text {sech}(c (a+b x))}{4 b c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 66, normalized size = 0.89 \[ \frac {{\left (2 \, b c x + 1\right )} \cosh \left (b c x + a c\right ) - {\left (2 \, b c x - 1\right )} \sinh \left (b c x + a c\right )}{4 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 23, normalized size = 0.31 \[ \frac {1}{2} \, x + \frac {e^{\left (2 \, b c x + 2 \, a c\right )}}{4 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{c \left (b x +a \right )} \sqrt {\frac {\cosh \left (2 b c x +2 a c \right )}{2}+\frac {1}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 29, normalized size = 0.39 \[ \frac {1}{2} \, x + \frac {a}{2 \, b} + \frac {e^{\left (2 \, b c x + 2 \, a c\right )}}{4 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 76, normalized size = 1.03 \[ \frac {\left (x\,{\mathrm {e}}^{a\,c+b\,c\,x}+\frac {{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}}{2\,b\,c}\right )\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.61, size = 204, normalized size = 2.76 \[ \begin {cases} 0 & \text {for}\: a = \frac {\log {\left (- i e^{- b c x} \right )}}{c} \vee a = \frac {\log {\left (i e^{- b c x} \right )}}{c} \\x & \text {for}\: c = 0 \\x \sqrt {\cosh ^{2}{\left (a c \right )}} e^{a c} & \text {for}\: b = 0 \\- \frac {x \sqrt {\cosh ^{2}{\left (a c + b c x \right )}} e^{a c} e^{b c x} \sinh {\left (a c + b c x \right )}}{2 \cosh {\left (a c + b c x \right )}} + \frac {x \sqrt {\cosh ^{2}{\left (a c + b c x \right )}} e^{a c} e^{b c x}}{2} + \frac {\sqrt {\cosh ^{2}{\left (a c + b c x \right )}} e^{a c} e^{b c x} \sinh {\left (a c + b c x \right )}}{b c \cosh {\left (a c + b c x \right )}} - \frac {\sqrt {\cosh ^{2}{\left (a c + b c x \right )}} e^{a c} e^{b c x}}{2 b c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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