Optimal. Leaf size=83 \[ \frac{e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt{\tanh ^2(a c+b c x)}}-\frac{2 \tanh ^{-1}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x)}{b c \sqrt{\tanh ^2(a c+b c x)}} \]
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Rubi [A] time = 0.199188, antiderivative size = 83, 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, 388, 206} \[ \frac{e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt{\tanh ^2(a c+b c x)}}-\frac{2 \tanh ^{-1}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x)}{b c \sqrt{\tanh ^2(a c+b c x)}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 388
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{c (a+b x)}}{\sqrt{\tanh ^2(a c+b c x)}} \, dx &=\frac{\tanh (a c+b c x) \int e^{c (a+b x)} \coth (a c+b c x) \, dx}{\sqrt{\tanh ^2(a c+b c x)}}\\ &=\frac{\tanh (a c+b c x) \operatorname{Subst}\left (\int \frac{-1-x^2}{1-x^2} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt{\tanh ^2(a c+b c x)}}\\ &=\frac{e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt{\tanh ^2(a c+b c x)}}-\frac{(2 \tanh (a c+b c x)) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt{\tanh ^2(a c+b c x)}}\\ &=\frac{e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt{\tanh ^2(a c+b c x)}}-\frac{2 \tanh ^{-1}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x)}{b c \sqrt{\tanh ^2(a c+b c x)}}\\ \end{align*}
Mathematica [A] time = 0.123081, size = 51, normalized size = 0.61 \[ \frac{\left (e^{c (a+b x)}-2 \tanh ^{-1}\left (e^{c (a+b x)}\right )\right ) \tanh (c (a+b x))}{b c \sqrt{\tanh ^2(c (a+b x))}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.227, size = 213, normalized size = 2.6 \begin{align*}{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ){{\rm e}^{c \left ( bx+a \right ) }}}{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) cb}{\frac{1}{\sqrt{{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}}}}}}+{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) \ln \left ({{\rm e}^{c \left ( bx+a \right ) }}-1 \right ) }{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) cb}{\frac{1}{\sqrt{{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}}}}}}-{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) \ln \left ({{\rm e}^{c \left ( bx+a \right ) }}+1 \right ) }{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) cb}{\frac{1}{\sqrt{{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77196, size = 76, normalized size = 0.92 \begin{align*} \frac{e^{\left (b c x + a c\right )}}{b c} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03521, size = 196, normalized size = 2.36 \begin{align*} \frac{\cosh \left (b c x + a c\right ) - \log \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right ) + 1\right ) + \log \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right ) - 1\right ) + \sinh \left (b c x + a c\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{\tanh ^{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 = 1.30181, size = 119, normalized size = 1.43 \begin{align*} \frac{e^{\left (b c x + a c\right )} \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - \log \left (e^{\left (b c x + a c\right )} + 1\right ) \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + \log \left ({\left | e^{\left (b c x + a c\right )} - 1 \right |}\right ) \mathrm{sgn}\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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