Optimal. Leaf size=38 \[ \frac{\cosh ^3(a+b x)}{3 b}-\frac{2 \cosh (a+b x)}{b}-\frac{\text{sech}(a+b x)}{b} \]
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Rubi [A] time = 0.040545, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2590, 270} \[ \frac{\cosh ^3(a+b x)}{3 b}-\frac{2 \cosh (a+b x)}{b}-\frac{\text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2590
Rule 270
Rubi steps
\begin{align*} \int \sinh ^3(a+b x) \tanh ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2+\frac{1}{x^2}+x^2\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac{2 \cosh (a+b x)}{b}+\frac{\cosh ^3(a+b x)}{3 b}-\frac{\text{sech}(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0334827, size = 40, normalized size = 1.05 \[ -\frac{7 \cosh (a+b x)}{4 b}+\frac{\cosh (3 (a+b x))}{12 b}-\frac{\text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 50, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{4}}{3\,\cosh \left ( bx+a \right ) }}+{\frac{4\, \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3\,\cosh \left ( bx+a \right ) }}-{\frac{8\,\cosh \left ( bx+a \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06418, size = 107, normalized size = 2.82 \begin{align*} -\frac{21 \, e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} - \frac{20 \, e^{\left (-2 \, b x - 2 \, a\right )} + 69 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1}{24 \, 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 [A] time = 1.82863, size = 177, normalized size = 4.66 \begin{align*} \frac{\cosh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} - 10\right )} \sinh \left (b x + a\right )^{2} - 20 \, \cosh \left (b x + a\right )^{2} - 45}{24 \, 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 ^{3}{\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 = 1.2644, size = 103, normalized size = 2.71 \begin{align*} -\frac{{\left (21 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} -{\left (e^{\left (3 \, b x + 24 \, a\right )} - 21 \, e^{\left (b x + 22 \, a\right )}\right )} e^{\left (-21 \, a\right )} + \frac{48 \, e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} + 1}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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