Optimal. Leaf size=67 \[ \frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}-\frac {\sinh (a+b x) \cosh ^3(a+b x)}{24 b}-\frac {\sinh (a+b x) \cosh (a+b x)}{16 b}-\frac {x}{16} \]
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Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}-\frac {\sinh (a+b x) \cosh ^3(a+b x)}{24 b}-\frac {\sinh (a+b x) \cosh (a+b x)}{16 b}-\frac {x}{16} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2568
Rule 2635
Rubi steps
\begin {align*} \int \cosh ^4(a+b x) \sinh ^2(a+b x) \, dx &=\frac {\cosh ^5(a+b x) \sinh (a+b x)}{6 b}-\frac {1}{6} \int \cosh ^4(a+b x) \, dx\\ &=-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{24 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{6 b}-\frac {1}{8} \int \cosh ^2(a+b x) \, dx\\ &=-\frac {\cosh (a+b x) \sinh (a+b x)}{16 b}-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{24 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{6 b}-\frac {\int 1 \, dx}{16}\\ &=-\frac {x}{16}-\frac {\cosh (a+b x) \sinh (a+b x)}{16 b}-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{24 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 40, normalized size = 0.60 \[ \frac {-3 \sinh (2 (a+b x))+3 \sinh (4 (a+b x))+\sinh (6 (a+b x))-12 b x}{192 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 90, normalized size = 1.34 \[ \frac {3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 6 \, b x + 3 \, {\left (\cosh \left (b x + a\right )^{5} + 2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 88, normalized size = 1.31 \[ -\frac {1}{16} \, x + \frac {e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{128 \, b} - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{128 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{128 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{128 \, b} - \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 56, normalized size = 0.84 \[ \frac {\frac {\sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{6}-\frac {\left (\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )}{6}-\frac {b x}{16}-\frac {a}{16}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 88, normalized size = 1.31 \[ \frac {{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} - \frac {b x + a}{16 \, b} + \frac {3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.60, size = 42, normalized size = 0.63 \[ \frac {\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{64}-\frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}+\frac {\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{192}}{b}-\frac {x}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.87, size = 136, normalized size = 2.03 \[ \begin {cases} \frac {x \sinh ^{6}{\left (a + b x \right )}}{16} - \frac {3 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} + \frac {3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{16} - \frac {x \cosh ^{6}{\left (a + b x \right )}}{16} - \frac {\sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{16 b} + \frac {\sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{6 b} + \frac {\sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \sinh ^{2}{\relax (a )} \cosh ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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