Optimal. Leaf size=66 \[ \frac {e^{-a-b x}}{8 b}+\frac {3 e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{8 b}+\frac {e^{5 a+5 b x}}{40 b} \]
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Rubi [A] time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2282, 12, 270} \[ \frac {e^{-a-b x}}{8 b}+\frac {3 e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{8 b}+\frac {e^{5 a+5 b x}}{40 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \sinh ^3(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^3}{8 x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^3}{x^2} \, dx,x,e^{a+b x}\right )}{8 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (3-\frac {1}{x^2}-3 x^2+x^4\right ) \, dx,x,e^{a+b x}\right )}{8 b}\\ &=\frac {e^{-a-b x}}{8 b}+\frac {3 e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{8 b}+\frac {e^{5 a+5 b x}}{40 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.76 \[ \frac {e^{-a-b x} \left (15 e^{2 (a+b x)}-5 e^{4 (a+b x)}+e^{6 (a+b x)}+5\right )}{40 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 105, normalized size = 1.59 \[ \frac {3 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 2 \, \sinh \left (b x + a\right )^{3} - 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 5\right )} \sinh \left (b x + a\right ) + 5 \, \cosh \left (b x + a\right )}{20 \, {\left (b \cosh \left (b x + a\right )^{2} - 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 53, normalized size = 0.80 \[ \frac {{\left (e^{\left (5 \, b x + 10 \, a\right )} - 5 \, e^{\left (3 \, b x + 8 \, a\right )} + 15 \, e^{\left (b x + 6 \, a\right )}\right )} e^{\left (-5 \, a\right )} + 5 \, e^{\left (-b x - a\right )}}{40 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 80, normalized size = 1.21 \[ \frac {\sinh \left (b x +a \right )}{4 b}-\frac {\sinh \left (3 b x +3 a \right )}{8 b}+\frac {\sinh \left (5 b x +5 a \right )}{40 b}+\frac {\cosh \left (b x +a \right )}{2 b}-\frac {\cosh \left (3 b x +3 a \right )}{8 b}+\frac {\cosh \left (5 b x +5 a \right )}{40 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 53, normalized size = 0.80 \[ -\frac {{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 15 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (5 \, b x + 5 \, a\right )}}{40 \, b} + \frac {e^{\left (-b x - a\right )}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 45, normalized size = 0.68 \[ \frac {15\,{\mathrm {e}}^{a+b\,x}+5\,{\mathrm {e}}^{-a-b\,x}-5\,{\mathrm {e}}^{3\,a+3\,b\,x}+{\mathrm {e}}^{5\,a+5\,b\,x}}{40\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.58, size = 124, normalized size = 1.88 \[ \begin {cases} \frac {2 e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )}}{5 b} + \frac {e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{5 b} - \frac {4 e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{5 b} + \frac {2 e^{2 a} e^{2 b x} \cosh ^{3}{\left (a + b x \right )}}{5 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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