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 = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 12
Rule 75
Rule 446
Rule 2282
Rubi steps
\begin {align*} \int e^{a+b x} \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^2}{8 x^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^2}{x^3} \, dx,x,e^{a+b x}\right )}{8 b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-1+x) (1+x)^2}{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*}
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Mathematica [A] time = 0.05, size = 43, normalized size = 0.75 \[ \frac {2 e^{-2 (a+b x)}+2 e^{2 (a+b x)}+e^{4 (a+b x)}-4 b x}{32 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 96, normalized size = 1.68 \[ \frac {3 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - \sinh \left (b x + a\right )^{3} - 2 \, {\left (2 \, b x - 1\right )} \cosh \left (b x + a\right ) + {\left (4 \, b x - 3 \, \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )}{32 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 57, normalized size = 1.00 \[ -\frac {4 \, b x - 2 \, {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 4 \, a - e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 53, normalized size = 0.93 \[ \frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{4}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {b x}{8}-\frac {a}{8}+\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{4}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 50, normalized size = 0.88 \[ -\frac {1}{8} \, x - \frac {a}{8 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 42, normalized size = 0.74 \[ \frac {\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{16}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{16}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{32}}{b}-\frac {x}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.66, size = 175, normalized size = 3.07 \[ \begin {cases} - \frac {x e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )}}{8} + \frac {x e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8} + \frac {x e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8} - \frac {x e^{a} e^{b x} \cosh ^{3}{\left (a + b x \right )}}{8} - \frac {e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )}}{8 b} + \frac {e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b} + \frac {e^{a} e^{b x} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text {for}\: b \neq 0 \\x e^{a} \sinh {\relax (a )} \cosh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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