Optimal. Leaf size=67 \[ \frac{\sinh ^5(a+b x) \cosh (a+b x)}{6 b}-\frac{5 \sinh ^3(a+b x) \cosh (a+b x)}{24 b}+\frac{5 \sinh (a+b x) \cosh (a+b x)}{16 b}-\frac{5 x}{16} \]
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Rubi [A] time = 0.0325209, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 8} \[ \frac{\sinh ^5(a+b x) \cosh (a+b x)}{6 b}-\frac{5 \sinh ^3(a+b x) \cosh (a+b x)}{24 b}+\frac{5 \sinh (a+b x) \cosh (a+b x)}{16 b}-\frac{5 x}{16} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sinh ^6(a+b x) \, dx &=\frac{\cosh (a+b x) \sinh ^5(a+b x)}{6 b}-\frac{5}{6} \int \sinh ^4(a+b x) \, dx\\ &=-\frac{5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac{\cosh (a+b x) \sinh ^5(a+b x)}{6 b}+\frac{5}{8} \int \sinh ^2(a+b x) \, dx\\ &=\frac{5 \cosh (a+b x) \sinh (a+b x)}{16 b}-\frac{5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac{\cosh (a+b x) \sinh ^5(a+b x)}{6 b}-\frac{5 \int 1 \, dx}{16}\\ &=-\frac{5 x}{16}+\frac{5 \cosh (a+b x) \sinh (a+b x)}{16 b}-\frac{5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac{\cosh (a+b x) \sinh ^5(a+b x)}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0384124, size = 43, normalized size = 0.64 \[ \frac{45 \sinh (2 (a+b x))-9 \sinh (4 (a+b x))+\sinh (6 (a+b x))-60 a-60 b x}{192 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 49, normalized size = 0.7 \begin{align*}{\frac{1}{b} \left ( \left ({\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{5}}{6}}-{\frac{5\, \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{24}}+{\frac{5\,\sinh \left ( bx+a \right ) }{16}} \right ) \cosh \left ( bx+a \right ) -{\frac{5\,bx}{16}}-{\frac{5\,a}{16}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04283, size = 116, normalized size = 1.73 \begin{align*} -\frac{{\left (9 \, e^{\left (-2 \, b x - 2 \, a\right )} - 45 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} - \frac{5 \,{\left (b x + a\right )}}{16 \, b} - \frac{45 \, e^{\left (-2 \, b x - 2 \, a\right )} - 9 \, e^{\left (-4 \, b x - 4 \, 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 [A] time = 1.99368, size = 248, normalized size = 3.7 \begin{align*} \frac{3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 2 \,{\left (5 \, \cosh \left (b x + a\right )^{3} - 9 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 30 \, b x + 3 \,{\left (\cosh \left (b x + a\right )^{5} - 6 \, \cosh \left (b x + a\right )^{3} + 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.71063, size = 139, normalized size = 2.07 \begin{align*} \begin{cases} \frac{5 x \sinh ^{6}{\left (a + b x \right )}}{16} - \frac{15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} + \frac{15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{16} - \frac{5 x \cosh ^{6}{\left (a + b x \right )}}{16} + \frac{11 \sinh ^{5}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{16 b} - \frac{5 \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{6 b} + \frac{5 \sinh{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{16 b} & \text{for}\: b \neq 0 \\x \sinh ^{6}{\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 = 1.41203, size = 124, normalized size = 1.85 \begin{align*} -\frac{120 \, b x -{\left (110 \, e^{\left (6 \, b x + 6 \, a\right )} - 45 \, e^{\left (4 \, b x + 4 \, a\right )} + 9 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-6 \, b x - 6 \, a\right )} + 120 \, a - e^{\left (6 \, b x + 6 \, a\right )} + 9 \, e^{\left (4 \, b x + 4 \, a\right )} - 45 \, 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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