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 = 0.11, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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 12
Rule 260
Rule 2282
Rule 6720
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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 0.56, size = 42, normalized size = 0.95 \[ \frac {\log \left (\frac {2 \, \cosh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 20, normalized size = 0.45 \[ \frac {\log \left (e^{\left (2 \, b c x\right )} + e^{\left (-2 \, a c\right )}\right )}{b c} \]
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 \[ \int \frac {2 \,{\mathrm e}^{c \left (b x +a \right )}}{\sqrt {2 \cosh \left (2 b c x +2 a c \right )+2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 21, normalized size = 0.48 \[ \frac {\log \left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{\sqrt {{\mathrm {cosh}\left (a\,c+b\,c\,x\right )}^2}} \,d x \]
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 \[ e^{a c} \int \frac {e^{b c x}}{\sqrt {\cosh ^{2}{\left (a c + b c x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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