Optimal. Leaf size=57 \[ \frac {e^{-2 a-2 b x}}{32 b}-\frac {e^{4 a+4 b x}}{32 b}+\frac {e^{6 a+6 b x}}{96 b}+\frac {x}{8} \]
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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2282, 12, 446, 75} \[ \frac {e^{-2 a-2 b x}}{32 b}-\frac {e^{4 a+4 b x}}{32 b}+\frac {e^{6 a+6 b x}}{96 b}+\frac {x}{8} \]
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 446
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \cosh (a+b x) \sinh ^3(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1-x^2\right ) \left (1-x^2\right )^3}{16 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 )^3}{x^3} \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-1-x) (1-x)^3}{x^2} \, dx,x,e^{2 a+2 b x}\right )}{32 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{x^2}+\frac {2}{x}-2 x+x^2\right ) \, dx,x,e^{2 a+2 b x}\right )}{32 b}\\ &=\frac {e^{-2 a-2 b x}}{32 b}-\frac {e^{4 a+4 b x}}{32 b}+\frac {e^{6 a+6 b x}}{96 b}+\frac {x}{8}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 43, normalized size = 0.75 \[ \frac {3 e^{-2 (a+b x)}-3 e^{4 (a+b x)}+e^{6 (a+b x)}+12 b x}{96 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 152, normalized size = 2.67 \[ \frac {4 \, \cosh \left (b x + a\right )^{4} - 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 4 \, \sinh \left (b x + a\right )^{4} + 3 \, {\left (4 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} + 3 \, {\left (4 \, b x + 8 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, {\left (4 \, \cosh \left (b x + a\right )^{3} + 3 \, {\left (4 \, b x + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \, {\left (b \cosh \left (b x + a\right )^{2} - 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 60, normalized size = 1.05 \[ \frac {12 \, b x - 3 \, {\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (e^{\left (6 \, b x + 12 \, a\right )} - 3 \, e^{\left (4 \, b x + 10 \, a\right )}\right )} e^{\left (-6 \, a\right )}}{96 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 89, normalized size = 1.56 \[ \frac {x}{8}-\frac {\sinh \left (2 b x +2 a \right )}{32 b}-\frac {\sinh \left (4 b x +4 a \right )}{32 b}+\frac {\sinh \left (6 b x +6 a \right )}{96 b}+\frac {\cosh \left (2 b x +2 a \right )}{32 b}-\frac {\cosh \left (4 b x +4 a \right )}{32 b}+\frac {\cosh \left (6 b x +6 a \right )}{96 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 52, normalized size = 0.91 \[ -\frac {{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{96 \, b} + \frac {b x + a}{8 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 42, normalized size = 0.74 \[ \frac {x}{8}+\frac {\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{32}-\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{32}+\frac {{\mathrm {e}}^{6\,a+6\,b\,x}}{96}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 58.18, size = 235, normalized size = 4.12 \[ \begin {cases} - \frac {x e^{2 a} e^{2 b x} \sinh ^{4}{\left (a + b x \right )}}{8} + \frac {x e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4} - \frac {x e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{4} + \frac {x e^{2 a} e^{2 b x} \cosh ^{4}{\left (a + b x \right )}}{8} + \frac {7 e^{2 a} e^{2 b x} \sinh ^{4}{\left (a + b x \right )}}{48 b} - \frac {e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{6 b} + \frac {e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b} - \frac {e^{2 a} e^{2 b x} \cosh ^{4}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh ^{3}{\relax (a )} \cosh {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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