Optimal. Leaf size=21 \[ \frac{\cosh (a+b x)}{b}+\frac{\text{sech}(a+b x)}{b} \]
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Rubi [A] time = 0.0264534, antiderivative size = 21, 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} \[ \frac{\cosh (a+b x)}{b}+\frac{\text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2590
Rule 14
Rubi steps
\begin{align*} \int \sinh (a+b x) \tanh ^2(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{x^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-1+\frac{1}{x^2}\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac{\cosh (a+b x)}{b}+\frac{\text{sech}(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0307261, size = 21, normalized size = 1. \[ \frac{\cosh (a+b x)}{b}+\frac{\text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 32, normalized size = 1.5 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{\cosh \left ( bx+a \right ) }}+2\,\cosh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02232, size = 73, normalized size = 3.48 \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 = 1.75055, size = 85, normalized size = 4.05 \begin{align*} \frac{\cosh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{2} + 3}{2 \, b \cosh \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 \sinh{\left (a + b x \right )} \tanh ^{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 = 1.18766, size = 62, normalized size = 2.95 \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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