Optimal. Leaf size=43 \[ \frac{\sinh ^2(a+b x)}{2 b}-\frac{\text{csch}^2(a+b x)}{2 b}+\frac{2 \log (\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.0420806, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2590, 266, 43} \[ \frac{\sinh ^2(a+b x)}{2 b}-\frac{\text{csch}^2(a+b x)}{2 b}+\frac{2 \log (\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2590
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \cosh ^2(a+b x) \coth ^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^3} \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(1-x)^2}{x^2} \, dx,x,-\sinh ^2(a+b x)\right )}{2 b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^2}-\frac{2}{x}\right ) \, dx,x,-\sinh ^2(a+b x)\right )}{2 b}\\ &=-\frac{\text{csch}^2(a+b x)}{2 b}+\frac{2 \log (\sinh (a+b x))}{b}+\frac{\sinh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0479868, size = 35, normalized size = 0.81 \[ -\frac{-\sinh ^2(a+b x)+\text{csch}^2(a+b x)-4 \log (\sinh (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 48, normalized size = 1.1 \begin{align*}{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{2\,b \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}}+2\,{\frac{\ln \left ( \sinh \left ( bx+a \right ) \right ) }{b}}-{\frac{ \left ({\rm coth} \left (bx+a\right ) \right ) ^{2}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.07772, size = 162, normalized size = 3.77 \begin{align*} \frac{2 \,{\left (b x + a\right )}}{b} + \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} + \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{2 \, e^{\left (-2 \, b x - 2 \, a\right )} + 15 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1}{8 \, b{\left (e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86623, size = 2049, normalized size = 47.65 \begin{align*} \text{result too large to display} \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 [B] time = 1.26816, size = 134, normalized size = 3.12 \begin{align*} -\frac{16 \, b x -{\left (8 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + \frac{8 \,{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 3\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} - e^{\left (2 \, b x + 2 \, a\right )} - 16 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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