Optimal. Leaf size=193 \[ \frac{e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}}+\frac{3 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right ) \sqrt{\coth ^2(a c+b c x)}}-\frac{2 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^2 \sqrt{\coth ^2(a c+b c x)}}-\frac{3 \tan ^{-1}\left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}} \]
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Rubi [A] time = 0.86382, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {6720, 2282, 390, 1158, 12, 288, 203} \[ \frac{e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}}+\frac{3 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right ) \sqrt{\coth ^2(a c+b c x)}}-\frac{2 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^2 \sqrt{\coth ^2(a c+b c x)}}-\frac{3 \tan ^{-1}\left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{b c \sqrt{\coth ^2(a c+b c x)}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 390
Rule 1158
Rule 12
Rule 288
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx &=\frac{\coth (a c+b c x) \int e^{c (a+b x)} \tanh ^3(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{\left (-1+x^2\right )^3}{\left (1+x^2\right )^3} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt{\coth ^2(a c+b c x)}}\\ &=\frac{\coth (a c+b c x) \operatorname{Subst}\left (\int \left (1-\frac{2 \left (1+3 x^4\right )}{\left (1+x^2\right )^3}\right ) \, 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+3 x^4}{\left (1+x^2\right )^3} \, 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 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt{\coth ^2(a c+b c x)}}+\frac{\coth (a c+b c x) \operatorname{Subst}\left (\int -\frac{12 x^2}{\left (1+x^2\right )^2} \, dx,x,e^{c (a+b x)}\right )}{2 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 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt{\coth ^2(a c+b c x)}}-\frac{(6 \coth (a c+b c x)) \operatorname{Subst}\left (\int \frac{x^2}{\left (1+x^2\right )^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 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt{\coth ^2(a c+b c x)}}+\frac{3 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right ) \sqrt{\coth ^2(a c+b c x)}}-\frac{(3 \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 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt{\coth ^2(a c+b c x)}}+\frac{3 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right ) \sqrt{\coth ^2(a c+b c x)}}-\frac{3 \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 = 0.297194, size = 104, normalized size = 0.54 \[ \frac{\left (e^{c (a+b x)} \left (5 e^{2 c (a+b x)}+e^{4 c (a+b x)}+2\right )-3 \left (e^{2 c (a+b x)}+1\right )^2 \tan ^{-1}\left (e^{c (a+b x)}\right )\right ) \coth (c (a+b x))}{b c \left (e^{2 c (a+b x)}+1\right )^2 \sqrt{\coth ^2(c (a+b x))}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.209, size = 301, normalized size = 1.6 \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{{{\rm e}^{c \left ( bx+a \right ) }} \left ( 3\,{{\rm e}^{2\,c \left ( bx+a \right ) }}+1 \right ) }{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \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{{\frac{3\,i}{2}} \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{{\frac{3\,i}{2}} \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 = 1.66612, size = 122, normalized size = 0.63 \begin{align*} -\frac{3 \, \arctan \left (e^{\left (b c x + a c\right )}\right )}{b c} + \frac{e^{\left (5 \, b c x + 5 \, a c\right )} + 5 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 2 \, e^{\left (b c x + a c\right )}}{b c{\left (e^{\left (4 \, b c x + 4 \, a c\right )} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.54231, size = 1175, normalized size = 6.09 \begin{align*} \frac{\cosh \left (b c x + a c\right )^{5} + 5 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{4} + \sinh \left (b c x + a c\right )^{5} + 5 \,{\left (2 \, \cosh \left (b c x + a c\right )^{2} + 1\right )} \sinh \left (b c x + a c\right )^{3} + 5 \, \cosh \left (b c x + a c\right )^{3} + 5 \,{\left (2 \, \cosh \left (b c x + a c\right )^{3} + 3 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} - 3 \,{\left (\cosh \left (b c x + a c\right )^{4} + 4 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{3} + \sinh \left (b c x + a c\right )^{4} + 2 \,{\left (3 \, \cosh \left (b c x + a c\right )^{2} + 1\right )} \sinh \left (b c x + a c\right )^{2} + 2 \, \cosh \left (b c x + a c\right )^{2} + 4 \,{\left (\cosh \left (b c x + a c\right )^{3} + \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right ) + 1\right )} \arctan \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right ) +{\left (5 \, \cosh \left (b c x + a c\right )^{4} + 15 \, \cosh \left (b c x + a c\right )^{2} + 2\right )} \sinh \left (b c x + a c\right ) + 2 \, \cosh \left (b c x + a c\right )}{b c \cosh \left (b c x + a c\right )^{4} + 4 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{3} + b c \sinh \left (b c x + a c\right )^{4} + 2 \, b c \cosh \left (b c x + a c\right )^{2} + 2 \,{\left (3 \, b c \cosh \left (b c x + a c\right )^{2} + b c\right )} \sinh \left (b c x + a c\right )^{2} + b c + 4 \,{\left (b c \cosh \left (b c x + a c\right )^{3} + b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21802, size = 176, normalized size = 0.91 \begin{align*} -\frac{{\left (3 \, \arctan \left (e^{\left (b c x + a c\right )}\right ) e^{\left (-a c\right )} \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - e^{\left (b c x\right )} \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - \frac{3 \, e^{\left (3 \, b c x + 2 \, a c\right )} \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + e^{\left (b c x\right )} \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{{\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{2}}\right )} e^{\left (a c\right )}}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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