Optimal. Leaf size=59 \[ \frac{e^{2 a+2 b x}}{2 b}+\frac{e^{4 a+4 b x}}{16 b}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b}-\frac{x}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0603323, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 12, 446, 72} \[ \frac{e^{2 a+2 b x}}{2 b}+\frac{e^{4 a+4 b x}}{16 b}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b}-\frac{x}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 12
Rule 446
Rule 72
Rubi steps
\begin{align*} \int e^{2 (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 \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 \left (-1+x^2\right )} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^3}{(-1+x) x} \, dx,x,e^{2 a+2 b x}\right )}{8 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (4+\frac{8}{-1+x}-\frac{1}{x}+x\right ) \, dx,x,e^{2 a+2 b x}\right )}{8 b}\\ &=\frac{e^{2 a+2 b x}}{2 b}+\frac{e^{4 a+4 b x}}{16 b}-\frac{x}{4}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0424042, size = 48, normalized size = 0.81 \[ \frac{8 e^{2 (a+b x)}+e^{4 (a+b x)}+16 \log \left (1-e^{2 (a+b x)}\right )-4 b x}{16 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.115, size = 55, normalized size = 0.9 \begin{align*} -{\frac{x}{4}}+{\frac{{{\rm e}^{4\,bx+4\,a}}}{16\,b}}+{\frac{{{\rm e}^{2\,bx+2\,a}}}{2\,b}}-2\,{\frac{a}{b}}+{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0337, size = 95, normalized size = 1.61 \begin{align*} \frac{{\left (8 \, e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{16 \, b} + \frac{7 \,{\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.94596, size = 354, normalized size = 6. \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} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} + 4\right )} \sinh \left (b x + a\right )^{2} - 4 \, b x + 8 \, \cosh \left (b x + a\right )^{2} + 4 \,{\left (\cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 16 \, \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21171, size = 70, normalized size = 1.19 \begin{align*} -\frac{4 \, b x -{\left (e^{\left (4 \, b x + 8 \, a\right )} + 8 \, e^{\left (2 \, b x + 6 \, a\right )}\right )} e^{\left (-4 \, a\right )} - 16 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]