Optimal. Leaf size=44 \[ \frac {\cosh (a c+b c x) \log \left (1+e^{2 c (a+b x)}\right ) \sqrt {\text {sech}^2(a c+b c x)}}{b c} \]
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Rubi [A]
time = 0.06, 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 = {6852, 2320, 12,
266} \begin {gather*} \frac {\log \left (e^{2 c (a+b x)}+1\right ) \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 266
Rule 2320
Rule 6852
Rubi steps
\begin {align*} \int e^{c (a+b x)} \sqrt {\text {sech}^2(a c+b c x)} \, dx &=\left (\cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \int e^{c (a+b x)} \text {sech}(a c+b c x) \, dx\\ &=\frac {\left (\cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {2 x}{1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac {\left (2 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac {\cosh (a c+b c x) \log \left (1+e^{2 c (a+b x)}\right ) \sqrt {\text {sech}^2(a c+b c x)}}{b c}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 42, normalized size = 0.95 \begin {gather*} \frac {\cosh (c (a+b x)) \log \left (1+e^{2 c (a+b x)}\right ) \sqrt {\text {sech}^2(c (a+b x))}}{b c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.22, size = 66, normalized size = 1.50
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{2 b c x}+{\mathrm e}^{-2 a c}\right ) \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, {\mathrm e}^{-c \left (b x +a \right )}}{c b}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 21, normalized size = 0.48 \begin {gather*} \frac {\log \left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 42, normalized size = 0.95 \begin {gather*} \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 {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*} e^{a c} \int \sqrt {\operatorname {sech}^{2}{\left (a c + b c x \right )}} e^{b c x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 20, normalized size = 0.45 \begin {gather*} \frac {\log \left (e^{\left (2 \, b c x\right )} + e^{\left (-2 \, a c\right )}\right )}{b c} \end {gather*}
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 \begin {gather*} \int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,\sqrt {\frac {1}{{\mathrm {cosh}\left (a\,c+b\,c\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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