Optimal. Leaf size=40 \[ -\frac {e^{2 a+2 b x}}{4 b}+\frac {e^{4 a+4 b x}}{16 b}+\frac {x}{4} \]
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Rubi [A] time = 0.04, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2282, 12, 266, 43} \[ -\frac {e^{2 a+2 b x}}{4 b}+\frac {e^{4 a+4 b x}}{16 b}+\frac {x}{4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \sinh ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{4 x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(1-x)^2}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2+\frac {1}{x}+x\right ) \, dx,x,e^{2 a+2 b x}\right )}{8 b}\\ &=-\frac {e^{2 a+2 b x}}{4 b}+\frac {e^{4 a+4 b x}}{16 b}+\frac {x}{4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.80 \[ \frac {-4 e^{2 (a+b x)}+e^{4 (a+b x)}+4 b x}{16 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 92, normalized size = 2.30 \[ \frac {{\left (4 \, b x + 1\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (4 \, b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (4 \, b x + 1\right )} \sinh \left (b x + a\right )^{2} - 4}{16 \, {\left (b \cosh \left (b x + a\right )^{2} - 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 30, normalized size = 0.75 \[ \frac {4 \, b x + e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 61, normalized size = 1.52 \[ \frac {x}{4}-\frac {\sinh \left (2 b x +2 a \right )}{4 b}+\frac {\sinh \left (4 b x +4 a \right )}{16 b}-\frac {\cosh \left (2 b x +2 a \right )}{4 b}+\frac {\cosh \left (4 b x +4 a \right )}{16 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 37, normalized size = 0.92 \[ \frac {1}{4} \, x - \frac {{\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{16 \, b} + \frac {a}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.86, size = 32, normalized size = 0.80 \[ \frac {x}{4}-\frac {\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{4}-\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{16}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.58, size = 139, normalized size = 3.48 \[ \begin {cases} \frac {x e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )}}{4} - \frac {x e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2} + \frac {x e^{2 a} e^{2 b x} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac {e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )}}{2 b} - \frac {e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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