Optimal. Leaf size=69 \[ \frac {e^{-5 a-5 b x}}{320 b}-\frac {3 e^{-a-b x}}{64 b}-\frac {e^{3 a+3 b x}}{64 b}+\frac {e^{7 a+7 b x}}{448 b} \]
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Rubi [A] time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2282, 12, 270} \[ \frac {e^{-5 a-5 b x}}{320 b}-\frac {3 e^{-a-b x}}{64 b}-\frac {e^{3 a+3 b x}}{64 b}+\frac {e^{7 a+7 b x}}{448 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 2282
Rubi steps
\begin {align*} \int e^{a+b x} \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^3}{64 x^6} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^3}{x^6} \, dx,x,e^{a+b x}\right )}{64 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{x^6}+\frac {3}{x^2}-3 x^2+x^6\right ) \, dx,x,e^{a+b x}\right )}{64 b}\\ &=\frac {e^{-5 a-5 b x}}{320 b}-\frac {3 e^{-a-b x}}{64 b}-\frac {e^{3 a+3 b x}}{64 b}+\frac {e^{7 a+7 b x}}{448 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 51, normalized size = 0.74 \[ \frac {e^{-5 (a+b x)} \left (-105 e^{4 (a+b x)}-35 e^{8 (a+b x)}+5 e^{12 (a+b x)}+7\right )}{2240 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 154, normalized size = 2.23 \[ \frac {3 \, \cosh \left (b x + a\right )^{6} - 10 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 45 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \, \sinh \left (b x + a\right )^{6} + 5 \, {\left (9 \, \cosh \left (b x + a\right )^{4} - 7\right )} \sinh \left (b x + a\right )^{2} - 35 \, \cosh \left (b x + a\right )^{2} - {\left (3 \, \cosh \left (b x + a\right )^{5} - 35 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{560 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 52, normalized size = 0.75 \[ -\frac {7 \, {\left (15 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )} e^{\left (-5 \, b x - 5 \, a\right )} - 5 \, e^{\left (7 \, b x + 7 \, a\right )} + 35 \, e^{\left (3 \, b x + 3 \, a\right )}}{2240 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 88, normalized size = 1.28 \[ \frac {\frac {\left (\cosh ^{4}\left (b x +a \right )\right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{7}-\frac {3 \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{35}+\frac {3 \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{35}+\frac {\left (\sinh ^{2}\left (b x +a \right )\right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{7}-\frac {2 \left (\cosh ^{5}\left (b x +a \right )\right )}{35}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 54, normalized size = 0.78 \[ -\frac {{\left (15 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{320 \, b} + \frac {e^{\left (7 \, b x + 7 \, a\right )} - 7 \, e^{\left (3 \, b x + 3 \, a\right )}}{448 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.10, size = 50, normalized size = 0.72 \[ -\frac {105\,{\mathrm {e}}^{-a-b\,x}+35\,{\mathrm {e}}^{3\,a+3\,b\,x}-7\,{\mathrm {e}}^{-5\,a-5\,b\,x}-5\,{\mathrm {e}}^{7\,a+7\,b\,x}}{2240\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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