Optimal. Leaf size=46 \[ \frac {\log \left (1-e^{2 c (a+b x)}\right ) \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{b c} \]
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Rubi [A] time = 0.09, antiderivative size = 46, 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} \[ \frac {\log \left (1-e^{2 c (a+b x)}\right ) \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{b c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 260
Rule 2282
Rule 6720
Rubi steps
\begin {align*} \int e^{c (a+b x)} \sqrt {\text {csch}^2(a c+b c x)} \, dx &=\left (\sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \int e^{c (a+b x)} \text {csch}(a c+b c x) \, dx\\ &=\frac {\left (\sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname {Subst}\left (\int \frac {2 x}{-1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac {\left (2 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname {Subst}\left (\int \frac {x}{-1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac {\sqrt {\text {csch}^2(a c+b c x)} \log \left (1-e^{2 c (a+b x)}\right ) \sinh (a c+b c x)}{b c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.96 \[ \frac {\log \left (1-e^{2 c (a+b x)}\right ) \sinh (c (a+b x)) \sqrt {\text {csch}^2(c (a+b x))}}{b c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 42, normalized size = 0.91 \[ \frac {\log \left (\frac {2 \, \sinh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 48, normalized size = 1.04 \[ \frac {\log \left ({\left | e^{\left (2 \, b c x + 2 \, a c\right )} - 1 \right |}\right )}{b c \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 68, normalized size = 1.48 \[ \frac {\ln \left ({\mathrm e}^{2 b c x}-{\mathrm e}^{-2 a c}\right ) \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, {\mathrm e}^{-c \left (b x +a \right )}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 39, normalized size = 0.85 \[ \frac {\log \left (e^{\left (b c x + a c\right )} + 1\right )}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} - 1\right )}{b c} \]
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 \[ \int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,\sqrt {\frac {1}{{\mathrm {sinh}\left (a\,c+b\,c\,x\right )}^2}} \,d x \]
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 \[ e^{a c} \int \sqrt {\operatorname {csch}^{2}{\left (a c + b c x \right )}} e^{b c x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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