Optimal. Leaf size=91 \[ \frac{e^{-4 a-4 b x}}{128 b}-\frac{e^{-2 a-2 b x}}{64 b}-\frac{e^{2 a+2 b x}}{32 b}-\frac{e^{4 a+4 b x}}{128 b}+\frac{e^{6 a+6 b x}}{192 b}+\frac{x}{16} \]
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Rubi [A] time = 0.0743978, antiderivative size = 91, 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, 88} \[ \frac{e^{-4 a-4 b x}}{128 b}-\frac{e^{-2 a-2 b x}}{64 b}-\frac{e^{2 a+2 b x}}{32 b}-\frac{e^{4 a+4 b x}}{128 b}+\frac{e^{6 a+6 b x}}{192 b}+\frac{x}{16} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 446
Rule 88
Rubi steps
\begin{align*} \int e^{a+b x} \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^3 \left (1+x^2\right )^2}{32 x^5} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^3 \left (1+x^2\right )^2}{x^5} \, dx,x,e^{a+b x}\right )}{32 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-1+x)^3 (1+x)^2}{x^3} \, dx,x,e^{2 a+2 b x}\right )}{64 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2-\frac{1}{x^3}+\frac{1}{x^2}+\frac{2}{x}-x+x^2\right ) \, dx,x,e^{2 a+2 b x}\right )}{64 b}\\ &=\frac{e^{-4 a-4 b x}}{128 b}-\frac{e^{-2 a-2 b x}}{64 b}-\frac{e^{2 a+2 b x}}{32 b}-\frac{e^{4 a+4 b x}}{128 b}+\frac{e^{6 a+6 b x}}{192 b}+\frac{x}{16}\\ \end{align*}
Mathematica [A] time = 0.0900048, size = 67, normalized size = 0.74 \[ \frac{3 e^{-4 (a+b x)}-6 e^{-2 (a+b x)}-12 e^{2 (a+b x)}-3 e^{4 (a+b x)}+2 e^{6 (a+b x)}+24 b x}{384 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 107, normalized size = 1.2 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{6}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{8}}+{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{16}}+{\frac{bx}{16}}+{\frac{a}{16}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{6}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00758, size = 104, normalized size = 1.14 \begin{align*} -\frac{{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{128 \, b} + \frac{b x + a}{16 \, b} + \frac{2 \, e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} - 12 \, e^{\left (2 \, b x + 2 \, a\right )}}{384 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49023, size = 456, normalized size = 5.01 \begin{align*} \frac{5 \, \cosh \left (b x + a\right )^{5} + 25 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - \sinh \left (b x + a\right )^{5} -{\left (10 \, \cosh \left (b x + a\right )^{2} - 3\right )} \sinh \left (b x + a\right )^{3} - 9 \, \cosh \left (b x + a\right )^{3} +{\left (50 \, \cosh \left (b x + a\right )^{3} - 27 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 12 \,{\left (2 \, b x - 1\right )} \cosh \left (b x + a\right ) -{\left (5 \, \cosh \left (b x + a\right )^{4} + 24 \, b x - 9 \, \cosh \left (b x + a\right )^{2} + 12\right )} \sinh \left (b x + a\right )}{384 \,{\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \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 [A] time = 1.25396, size = 109, normalized size = 1.2 \begin{align*} \frac{24 \, b x - 3 \,{\left (6 \, e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 24 \, a + 2 \, e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} - 12 \, e^{\left (2 \, b x + 2 \, a\right )}}{384 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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