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 = 0.0589133, antiderivative size = 69, normalized size of antiderivative = 1., 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 2282
Rule 12
Rule 270
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*}
Mathematica [A] time = 0.0375875, 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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Maple [B] time = 0.013, size = 120, normalized size = 1.7 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{7}}-{\frac{3\, \left ( \cosh \left ( bx+a \right ) \right ) ^{4}\sinh \left ( bx+a \right ) }{35}}+{\frac{3\,\sinh \left ( bx+a \right ) }{35} \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) }+{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{7}}-{\frac{2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{35}}-{\frac{2\,\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{35}}-{\frac{2\,\cosh \left ( bx+a \right ) }{35}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04281, size = 73, normalized size = 1.06 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53266, size = 420, normalized size = 6.09 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18104, size = 70, normalized size = 1.01 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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