Optimal. Leaf size=59 \[ \frac{e^{-a-b x}}{4 b}+\frac{e^{a+b x}}{b}+\frac{e^{3 a+3 b x}}{12 b}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.0537115, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2282, 12, 461, 207} \[ \frac{e^{-a-b x}}{4 b}+\frac{e^{a+b x}}{b}+\frac{e^{3 a+3 b x}}{12 b}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 461
Rule 207
Rubi steps
\begin{align*} \int e^{a+b x} \cosh ^2(a+b x) \coth (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{4 x^2 \left (-1+x^2\right )} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^2 \left (-1+x^2\right )} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (4-\frac{1}{x^2}+x^2+\frac{8}{-1+x^2}\right ) \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac{e^{-a-b x}}{4 b}+\frac{e^{a+b x}}{b}+\frac{e^{3 a+3 b x}}{12 b}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{-a-b x}}{4 b}+\frac{e^{a+b x}}{b}+\frac{e^{3 a+3 b x}}{12 b}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.347232, size = 68, normalized size = 1.15 \[ \frac{e^{-a-b x} \left (12 e^{2 (a+b x)}+e^{4 (a+b x)}-24 \sqrt{e^{2 (a+b x)}} \tanh ^{-1}\left (\sqrt{e^{2 (a+b x)}}\right )+3\right )}{12 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 50, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( bx+a \right ) +{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{3}}+\cosh \left ( bx+a \right ) -2\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984571, size = 88, normalized size = 1.49 \begin{align*} \frac{e^{\left (3 \, b x + 3 \, a\right )} + 12 \, e^{\left (b x + a\right )}}{12 \, b} + \frac{e^{\left (-b x - a\right )}}{4 \, b} - \frac{\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (b x + a\right )} - 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.83499, size = 520, normalized size = 8.81 \begin{align*} \frac{\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 6 \,{\left (\cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{2} + 12 \, \cosh \left (b x + a\right )^{2} - 12 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 12 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 4 \,{\left (\cosh \left (b x + a\right )^{3} + 6 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3}{12 \,{\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\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 = 1.17613, size = 99, normalized size = 1.68 \begin{align*} \frac{e^{\left (-b x - a\right )}}{4 \, b} - \frac{\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac{\log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} + \frac{b^{2} e^{\left (3 \, b x + 3 \, a\right )} + 12 \, b^{2} e^{\left (b x + a\right )}}{12 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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