Optimal. Leaf size=49 \[ -\frac {e^{-a-b x}}{4 b}-\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{12 b} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2282, 12, 270} \[ -\frac {e^{-a-b x}}{4 b}-\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{12 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 2282
Rubi steps
\begin {align*} \int e^{a+b x} \sinh ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{4 x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2+\frac {1}{x^2}+x^2\right ) \, dx,x,e^{a+b x}\right )}{4 b}\\ &=-\frac {e^{-a-b x}}{4 b}-\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{12 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 0.80 \[ \frac {e^{-a-b x} \left (-6 e^{2 (a+b x)}+e^{4 (a+b x)}-3\right )}{12 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 54, normalized size = 1.10 \[ -\frac {\cosh \left (b x + a\right )^{2} - 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 3}{6 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 34, normalized size = 0.69 \[ \frac {e^{\left (3 \, b x + 3 \, a\right )} - 6 \, e^{\left (b x + a\right )} - 3 \, e^{\left (-b x - a\right )}}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 35, normalized size = 0.71 \[ \frac {\left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (b x +a \right )\right )}{3}\right ) \cosh \left (b x +a \right )+\frac {\left (\sinh ^{3}\left (b x +a \right )\right )}{3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 40, normalized size = 0.82 \[ \frac {e^{\left (3 \, b x + 3 \, a\right )}}{12 \, b} - \frac {e^{\left (b x + a\right )}}{2 \, b} - \frac {e^{\left (-b x - a\right )}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 36, normalized size = 0.73 \[ -\frac {6\,{\mathrm {e}}^{a+b\,x}+3\,{\mathrm {e}}^{-a-b\,x}-{\mathrm {e}}^{3\,a+3\,b\,x}}{12\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.15, size = 78, normalized size = 1.59 \[ \begin {cases} \frac {e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b} - \frac {2 e^{a} e^{b x} \cosh ^{2}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x e^{a} \sinh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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