Optimal. Leaf size=74 \[ \frac{e^{2 c (a+b x)} \text{csch}(a c+b c x)}{4 b c \sqrt{\text{csch}^2(a c+b c x)}}-\frac{x \text{csch}(a c+b c x)}{2 \sqrt{\text{csch}^2(a c+b c x)}} \]
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Rubi [A] time = 0.114765, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6720, 2282, 12, 14} \[ \frac{e^{2 c (a+b x)} \text{csch}(a c+b c x)}{4 b c \sqrt{\text{csch}^2(a c+b c x)}}-\frac{x \text{csch}(a c+b c x)}{2 \sqrt{\text{csch}^2(a c+b c x)}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 12
Rule 14
Rubi steps
\begin{align*} \int \frac{e^{c (a+b x)}}{\sqrt{\text{csch}^2(a c+b c x)}} \, dx &=\frac{\text{csch}(a c+b c x) \int e^{c (a+b x)} \sinh (a c+b c x) \, dx}{\sqrt{\text{csch}^2(a c+b c x)}}\\ &=\frac{\text{csch}(a c+b c x) \operatorname{Subst}\left (\int \frac{-1+x^2}{2 x} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt{\text{csch}^2(a c+b c x)}}\\ &=\frac{\text{csch}(a c+b c x) \operatorname{Subst}\left (\int \frac{-1+x^2}{x} \, dx,x,e^{c (a+b x)}\right )}{2 b c \sqrt{\text{csch}^2(a c+b c x)}}\\ &=\frac{\text{csch}(a c+b c x) \operatorname{Subst}\left (\int \left (-\frac{1}{x}+x\right ) \, dx,x,e^{c (a+b x)}\right )}{2 b c \sqrt{\text{csch}^2(a c+b c x)}}\\ &=\frac{e^{2 c (a+b x)} \text{csch}(a c+b c x)}{4 b c \sqrt{\text{csch}^2(a c+b c x)}}-\frac{x \text{csch}(a c+b c x)}{2 \sqrt{\text{csch}^2(a c+b c x)}}\\ \end{align*}
Mathematica [A] time = 0.050228, size = 48, normalized size = 0.65 \[ \frac{\left (e^{2 c (a+b x)}-2 b c x\right ) \text{csch}(c (a+b x))}{4 b c \sqrt{\text{csch}^2(c (a+b x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.175, size = 106, normalized size = 1.4 \begin{align*} -{\frac{x{{\rm e}^{c \left ( bx+a \right ) }}}{2\,{{\rm e}^{2\,c \left ( bx+a \right ) }}-2}{\frac{1}{\sqrt{{\frac{{{\rm e}^{2\,c \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}}}}}}+{\frac{{{\rm e}^{3\,c \left ( bx+a \right ) }}}{ \left ( 4\,{{\rm e}^{2\,c \left ( bx+a \right ) }}-4 \right ) cb}{\frac{1}{\sqrt{{\frac{{{\rm e}^{2\,c \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61341, size = 49, normalized size = 0.66 \begin{align*} -\frac{b c x + a c}{2 \, b c} + \frac{e^{\left (2 \, b c x + 2 \, a c\right )}}{4 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49332, size = 165, normalized size = 2.23 \begin{align*} -\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 )}} \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{\operatorname{csch}^{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 = 1.13629, size = 96, normalized size = 1.3 \begin{align*} -\frac{1}{2} \, x \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) + \frac{e^{\left (2 \, b c x + 2 \, a c\right )} \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )}{4 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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