Optimal. Leaf size=44 \[ \frac{\log \left (e^{2 c (a+b x)}+1\right ) \cosh (a c+b c x)}{b c \sqrt{\cosh ^2(a c+b c x)}} \]
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Rubi [A] time = 0.113316, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6720, 2282, 12, 260} \[ \frac{\log \left (e^{2 c (a+b x)}+1\right ) \cosh (a c+b c x)}{b c \sqrt{\cosh ^2(a c+b c x)}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 12
Rule 260
Rubi steps
\begin{align*} \int \frac{e^{c (a+b x)}}{\sqrt{\cosh ^2(a c+b c x)}} \, dx &=\frac{\cosh (a c+b c x) \int e^{c (a+b x)} \text{sech}(a c+b c x) \, dx}{\sqrt{\cosh ^2(a c+b c x)}}\\ &=\frac{\cosh (a c+b c x) \operatorname{Subst}\left (\int \frac{2 x}{1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt{\cosh ^2(a c+b c x)}}\\ &=\frac{(2 \cosh (a c+b c x)) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt{\cosh ^2(a c+b c x)}}\\ &=\frac{\cosh (a c+b c x) \log \left (1+e^{2 c (a+b x)}\right )}{b c \sqrt{\cosh ^2(a c+b c x)}}\\ \end{align*}
Mathematica [A] time = 0.0541603, size = 42, normalized size = 0.95 \[ \frac{\log \left (e^{2 c (a+b x)}+1\right ) \cosh (c (a+b x))}{b c \sqrt{\cosh ^2(c (a+b x))}} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{c \left ( bx+a \right ) }}{\frac{1}{\sqrt{ \left ( \cosh \left ( bcx+ac \right ) \right ) ^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.7139, size = 28, normalized size = 0.64 \begin{align*} \frac{\log \left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82579, size = 97, normalized size = 2.2 \begin{align*} \frac{\log \left (\frac{2 \, \cosh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{a c} \int \frac{e^{b c x}}{\sqrt{\cosh ^{2}{\left (a c + b c x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23292, size = 27, normalized size = 0.61 \begin{align*} \frac{\log \left (e^{\left (2 \, b c x\right )} + e^{\left (-2 \, a c\right )}\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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