Optimal. Leaf size=22 \[ \frac{\sinh (a+b x)}{b}-\frac{\text{csch}(a+b x)}{b} \]
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Rubi [A] time = 0.0216343, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2590, 14} \[ \frac{\sinh (a+b x)}{b}-\frac{\text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2590
Rule 14
Rubi steps
\begin{align*} \int \cosh (a+b x) \coth ^2(a+b x) \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1-x^2}{x^2} \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (-1+\frac{1}{x^2}\right ) \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=-\frac{\text{csch}(a+b x)}{b}+\frac{\sinh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0142865, size = 22, normalized size = 1. \[ \frac{\sinh (a+b x)}{b}-\frac{\text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 32, normalized size = 1.5 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{\sinh \left ( bx+a \right ) }}+2\,\sinh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19467, size = 76, normalized size = 3.45 \begin{align*} -\frac{e^{\left (-b x - a\right )}}{2 \, b} - \frac{5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1}{2 \, b{\left (e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79687, size = 85, normalized size = 3.86 \begin{align*} \frac{\cosh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{2} - 3}{2 \, b \sinh \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (a + b x \right )} \coth ^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19667, size = 68, normalized size = 3.09 \begin{align*} -\frac{\frac{{\left (5 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-a\right )}}{e^{\left (3 \, b x + 2 \, a\right )} - e^{\left (b x\right )}} - e^{\left (b x + a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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