Optimal. Leaf size=31 \[ \frac{\sinh ^6(a+b x)}{6 b}+\frac{\sinh ^4(a+b x)}{4 b} \]
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Rubi [A] time = 0.036688, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2564, 14} \[ \frac{\sinh ^6(a+b x)}{6 b}+\frac{\sinh ^4(a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (1-x^2\right ) \, dx,x,i \sinh (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^3-x^5\right ) \, dx,x,i \sinh (a+b x)\right )}{b}\\ &=\frac{\sinh ^4(a+b x)}{4 b}+\frac{\sinh ^6(a+b x)}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0121394, size = 35, normalized size = 1.13 \[ \frac{1}{8} \left (\frac{\cosh (6 (a+b x))}{24 b}-\frac{3 \cosh (2 (a+b x))}{8 b}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 52, 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 ) ^{2}}{6}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01174, size = 76, normalized size = 2.45 \begin{align*} -\frac{{\left (9 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} - \frac{9 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75977, size = 197, normalized size = 6.35 \begin{align*} \frac{\cosh \left (b x + a\right )^{6} + 15 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{6} + 3 \,{\left (5 \, \cosh \left (b x + a\right )^{4} - 3\right )} \sinh \left (b x + a\right )^{2} - 9 \, \cosh \left (b x + a\right )^{2}}{192 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.74648, size = 42, normalized size = 1.35 \begin{align*} \begin{cases} - \frac{\sinh ^{6}{\left (a + b x \right )}}{12 b} + \frac{\sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b} & \text{for}\: b \neq 0 \\x \sinh ^{3}{\left (a \right )} \cosh ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16595, size = 66, normalized size = 2.13 \begin{align*} \frac{{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}^{3} - 12 \, e^{\left (2 \, b x + 2 \, a\right )} - 12 \, e^{\left (-2 \, b x - 2 \, a\right )}}{384 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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