Optimal. Leaf size=100 \[ \frac {e^{-3 a-3 b x}}{96 b}-\frac {e^{-a-b x}}{32 b}+\frac {e^{a+b x}}{16 b}-\frac {e^{3 a+3 b x}}{48 b}-\frac {e^{5 a+5 b x}}{160 b}+\frac {e^{7 a+7 b x}}{224 b} \]
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Rubi [A] time = 0.08, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2282, 12, 448} \[ \frac {e^{-3 a-3 b x}}{96 b}-\frac {e^{-a-b x}}{32 b}+\frac {e^{a+b x}}{16 b}-\frac {e^{3 a+3 b x}}{48 b}-\frac {e^{5 a+5 b x}}{160 b}+\frac {e^{7 a+7 b x}}{224 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 448
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^3 \left (1+x^2\right )^2}{32 x^4} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^3 \left (1+x^2\right )^2}{x^4} \, dx,x,e^{a+b x}\right )}{32 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (2-\frac {1}{x^4}+\frac {1}{x^2}-2 x^2-x^4+x^6\right ) \, dx,x,e^{a+b x}\right )}{32 b}\\ &=\frac {e^{-3 a-3 b x}}{96 b}-\frac {e^{-a-b x}}{32 b}+\frac {e^{a+b x}}{16 b}-\frac {e^{3 a+3 b x}}{48 b}-\frac {e^{5 a+5 b x}}{160 b}+\frac {e^{7 a+7 b x}}{224 b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 73, normalized size = 0.73 \[ \frac {e^{-3 (a+b x)} \left (-105 e^{2 (a+b x)}+210 e^{4 (a+b x)}-70 e^{6 (a+b x)}-21 e^{8 (a+b x)}+15 e^{10 (a+b x)}+35\right )}{3360 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 175, normalized size = 1.75 \[ \frac {25 \, \cosh \left (b x + a\right )^{5} + 125 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 10 \, \sinh \left (b x + a\right )^{5} - 2 \, {\left (50 \, \cosh \left (b x + a\right )^{2} - 21\right )} \sinh \left (b x + a\right )^{3} - 63 \, \cosh \left (b x + a\right )^{3} + {\left (250 \, \cosh \left (b x + a\right )^{3} - 189 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 2 \, {\left (25 \, \cosh \left (b x + a\right )^{4} - 63 \, \cosh \left (b x + a\right )^{2} + 70\right )} \sinh \left (b x + a\right ) + 70 \, \cosh \left (b x + a\right )}{1680 \, {\left (b \cosh \left (b x + a\right )^{2} - 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 80, normalized size = 0.80 \[ -\frac {35 \, {\left (3 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} - {\left (15 \, e^{\left (7 \, b x + 28 \, a\right )} - 21 \, e^{\left (5 \, b x + 26 \, a\right )} - 70 \, e^{\left (3 \, b x + 24 \, a\right )} + 210 \, e^{\left (b x + 22 \, a\right )}\right )} e^{\left (-21 \, a\right )}}{3360 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 108, normalized size = 1.08 \[ \frac {3 \sinh \left (b x +a \right )}{32 b}-\frac {\sinh \left (3 b x +3 a \right )}{32 b}-\frac {\sinh \left (5 b x +5 a \right )}{160 b}+\frac {\sinh \left (7 b x +7 a \right )}{224 b}+\frac {\cosh \left (b x +a \right )}{32 b}-\frac {\cosh \left (3 b x +3 a \right )}{96 b}-\frac {\cosh \left (5 b x +5 a \right )}{160 b}+\frac {\cosh \left (7 b x +7 a \right )}{224 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 78, normalized size = 0.78 \[ -\frac {{\left (21 \, e^{\left (-2 \, b x - 2 \, a\right )} + 70 \, e^{\left (-4 \, b x - 4 \, a\right )} - 210 \, e^{\left (-6 \, b x - 6 \, a\right )} - 15\right )} e^{\left (7 \, b x + 7 \, a\right )}}{3360 \, b} - \frac {3 \, e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{96 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 69, normalized size = 0.69 \[ \frac {210\,{\mathrm {e}}^{a+b\,x}-105\,{\mathrm {e}}^{-a-b\,x}+35\,{\mathrm {e}}^{-3\,a-3\,b\,x}-70\,{\mathrm {e}}^{3\,a+3\,b\,x}-21\,{\mathrm {e}}^{5\,a+5\,b\,x}+15\,{\mathrm {e}}^{7\,a+7\,b\,x}}{3360\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 163.25, size = 197, normalized size = 1.97 \[ \begin {cases} - \frac {4 e^{2 a} e^{2 b x} \sinh ^{5}{\left (a + b x \right )}}{35 b} + \frac {8 e^{2 a} e^{2 b x} \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{35 b} + \frac {2 e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{35 b} - \frac {e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{105 b} - \frac {4 e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{105 b} + \frac {2 e^{2 a} e^{2 b x} \cosh ^{5}{\left (a + b x \right )}}{105 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh ^{3}{\relax (a )} \cosh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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