Optimal. Leaf size=73 \[ \frac{5 e^{a+b x}}{2 b}+\frac{e^{3 a+3 b x}}{6 b}+\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{4 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.0537078, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2282, 12, 390, 385, 206} \[ \frac{5 e^{a+b x}}{2 b}+\frac{e^{3 a+3 b x}}{6 b}+\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{4 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 390
Rule 385
Rule 206
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \cosh (a+b x) \coth ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{2 \left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (5+x^2-\frac{4 \left (1-3 x^2\right )}{\left (1-x^2\right )^2}\right ) \, dx,x,e^{a+b x}\right )}{2 b}\\ &=\frac{5 e^{a+b x}}{2 b}+\frac{e^{3 a+3 b x}}{6 b}-\frac{2 \operatorname{Subst}\left (\int \frac{1-3 x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{5 e^{a+b x}}{2 b}+\frac{e^{3 a+3 b x}}{6 b}+\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{5 e^{a+b x}}{2 b}+\frac{e^{3 a+3 b x}}{6 b}+\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{4 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end{align*}
Mathematica [C] time = 2.07236, size = 220, normalized size = 3.01 \[ \frac{e^{-5 (a+b x)} \left (256 e^{8 (a+b x)} \left (e^{2 (a+b x)}+1\right )^3 \, _5F_4\left (\frac{3}{2},2,2,2,2;1,1,1,\frac{11}{2};e^{2 (a+b x)}\right )-21 \left (91925 e^{2 (a+b x)}+61158 e^{4 (a+b x)}-20166 e^{6 (a+b x)}-15061 e^{8 (a+b x)}+753 e^{10 (a+b x)}+36015\right )-\frac{315 \left (-5328 e^{2 (a+b x)}-1821 e^{4 (a+b x)}+3264 e^{6 (a+b x)}+1149 e^{8 (a+b x)}-240 e^{10 (a+b x)}+e^{12 (a+b x)}-2401\right ) \tanh ^{-1}\left (\sqrt{e^{2 (a+b x)}}\right )}{\sqrt{e^{2 (a+b x)}}}\right )}{60480 b} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.119, size = 79, normalized size = 1.1 \begin{align*}{\frac{{{\rm e}^{3\,bx+3\,a}}}{6\,b}}+{\frac{5\,{{\rm e}^{bx+a}}}{2\,b}}-2\,{\frac{{{\rm e}^{bx+a}}}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}-2\,{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{b}}+2\,{\frac{\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04947, size = 117, normalized size = 1.6 \begin{align*} -\frac{2 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{2 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac{14 \, e^{\left (-2 \, b x - 2 \, a\right )} - 27 \, e^{\left (-4 \, b x - 4 \, a\right )} + 1}{6 \, b{\left (e^{\left (-3 \, b x - 3 \, a\right )} - e^{\left (-5 \, b x - 5 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86758, size = 797, normalized size = 10.92 \begin{align*} \frac{\cosh \left (b x + a\right )^{5} + 5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{5} + 2 \,{\left (5 \, \cosh \left (b x + a\right )^{2} + 7\right )} \sinh \left (b x + a\right )^{3} + 14 \, \cosh \left (b x + a\right )^{3} + 2 \,{\left (5 \, \cosh \left (b x + a\right )^{3} + 21 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 12 \,{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\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 )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) +{\left (5 \, \cosh \left (b x + a\right )^{4} + 42 \, \cosh \left (b x + a\right )^{2} - 27\right )} \sinh \left (b x + a\right ) - 27 \, \cosh \left (b x + a\right )}{6 \,{\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b\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 = 1.19178, size = 101, normalized size = 1.38 \begin{align*} \frac{{\left (e^{\left (3 \, b x + 15 \, a\right )} + 15 \, e^{\left (b x + 13 \, a\right )}\right )} e^{\left (-12 \, a\right )} - \frac{12 \, e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - 12 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 12 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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