Optimal. Leaf size=57 \[ -\frac{e^{-2 a-2 b x}}{16 b}-\frac{e^{2 a+2 b x}}{16 b}+\frac{e^{4 a+4 b x}}{32 b}-\frac{x}{8} \]
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Rubi [A] time = 0.052599, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2282, 12, 446, 75} \[ -\frac{e^{-2 a-2 b x}}{16 b}-\frac{e^{2 a+2 b x}}{16 b}+\frac{e^{4 a+4 b x}}{32 b}-\frac{x}{8} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 446
Rule 75
Rubi steps
\begin{align*} \int e^{a+b x} \cosh (a+b x) \sinh ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2 \left (1+x^2\right )}{8 x^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2 \left (1+x^2\right )}{x^3} \, dx,x,e^{a+b x}\right )}{8 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1-x)^2 (1+x)}{x^2} \, dx,x,e^{2 a+2 b x}\right )}{16 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+\frac{1}{x^2}-\frac{1}{x}+x\right ) \, dx,x,e^{2 a+2 b x}\right )}{16 b}\\ &=-\frac{e^{-2 a-2 b x}}{16 b}-\frac{e^{2 a+2 b x}}{16 b}+\frac{e^{4 a+4 b x}}{32 b}-\frac{x}{8}\\ \end{align*}
Mathematica [A] time = 0.0661357, size = 45, normalized size = 0.79 \[ -\frac{2 e^{-2 (a+b x)}+2 e^{2 (a+b x)}-e^{4 (a+b x)}+4 b x}{32 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 71, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{4}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{4}}-{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{8}}-{\frac{bx}{8}}-{\frac{a}{8}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03779, size = 68, normalized size = 1.19 \begin{align*} -\frac{1}{8} \, x - \frac{a}{8 \, b} + \frac{e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b} - \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53401, size = 261, normalized size = 4.58 \begin{align*} -\frac{\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, \sinh \left (b x + a\right )^{3} + 2 \,{\left (2 \, b x + 1\right )} \cosh \left (b x + a\right ) -{\left (4 \, b x + 9 \, \cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )}{32 \,{\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 [A] time = 26.8278, size = 177, normalized size = 3.11 \begin{align*} \begin{cases} - \frac{x e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )}}{8} + \frac{x e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8} + \frac{x e^{a} e^{b x} \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8} - \frac{x e^{a} e^{b x} \cosh ^{3}{\left (a + b x \right )}}{8} + \frac{3 e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )}}{8 b} - \frac{e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{4 b} + \frac{e^{a} e^{b x} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text{for}\: b \neq 0 \\x e^{a} \sinh ^{2}{\left (a \right )} \cosh{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13002, size = 77, normalized size = 1.35 \begin{align*} -\frac{4 \, b x - 2 \,{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 4 \, a - e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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