Optimal. Leaf size=83 \[ \frac{e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}}-\frac{2 \tan ^{-1}\left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}} \]
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Rubi [A] time = 0.194186, 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, 203} \[ \frac{e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}}-\frac{2 \tan ^{-1}\left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 388
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{c (a+b x)}}{\sqrt{\coth ^2(a c+b c x)}} \, dx &=\frac{\coth (a c+b c x) \int e^{c (a+b x)} \tanh (a c+b c x) \, dx}{\sqrt{\coth ^2(a c+b c x)}}\\ &=\frac{\coth (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{\coth ^2(a c+b c x)}}\\ &=\frac{e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}}-\frac{(2 \coth (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{\coth ^2(a c+b c x)}}\\ &=\frac{e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}}-\frac{2 \tan ^{-1}\left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}}\\ \end{align*}
Mathematica [A] time = 0.117547, size = 51, normalized size = 0.61 \[ \frac{\left (e^{c (a+b x)}-2 \tan ^{-1}\left (e^{c (a+b x)}\right )\right ) \coth (c (a+b x))}{b c \sqrt{\coth ^2(c (a+b x))}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.231, size = 218, normalized size = 2.6 \begin{align*}{\frac{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ){{\rm e}^{c \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) cb}{\frac{1}{\sqrt{{\frac{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}}}}}}+{\frac{i \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) \ln \left ({{\rm e}^{c \left ( bx+a \right ) }}-i \right ) }{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) cb}{\frac{1}{\sqrt{{\frac{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}}}}}}-{\frac{i \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) \ln \left ({{\rm e}^{c \left ( bx+a \right ) }}+i \right ) }{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) cb}{\frac{1}{\sqrt{{\frac{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}{ \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 = 1.71976, size = 47, normalized size = 0.57 \begin{align*} -\frac{2 \, \arctan \left (e^{\left (b c x + a c\right )}\right )}{b c} + \frac{e^{\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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Fricas [A] time = 2.25958, size = 132, normalized size = 1.59 \begin{align*} -\frac{2 \, \arctan \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right ) - \cosh \left (b c x + a c\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{\coth ^{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 = 1.18302, size = 81, normalized size = 0.98 \begin{align*} -\frac{2 \, \arctan \left (e^{\left (b c x + a c\right )}\right ) \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - e^{\left (b c x + a c\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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