Optimal. Leaf size=113 \[ \frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{16 b}+\frac {\sinh (a+b x) \cosh ^5(a+b x)}{32 b}-\frac {\sinh (a+b x) \cosh ^3(a+b x)}{128 b}-\frac {3 \sinh (a+b x) \cosh (a+b x)}{256 b}-\frac {3 x}{256} \]
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Rubi [A] time = 0.12, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{16 b}+\frac {\sinh (a+b x) \cosh ^5(a+b x)}{32 b}-\frac {\sinh (a+b x) \cosh ^3(a+b x)}{128 b}-\frac {3 \sinh (a+b x) \cosh (a+b x)}{256 b}-\frac {3 x}{256} \]
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 ^6(a+b x) \, dx &=\frac {\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}-\frac {1}{2} \int \cosh ^4(a+b x) \sinh ^4(a+b x) \, dx\\ &=-\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}+\frac {3}{16} \int \cosh ^4(a+b x) \sinh ^2(a+b x) \, dx\\ &=\frac {\cosh ^5(a+b x) \sinh (a+b x)}{32 b}-\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}-\frac {1}{32} \int \cosh ^4(a+b x) \, dx\\ &=-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{128 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{32 b}-\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}-\frac {3}{128} \int \cosh ^2(a+b x) \, dx\\ &=-\frac {3 \cosh (a+b x) \sinh (a+b x)}{256 b}-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{128 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{32 b}-\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}-\frac {3 \int 1 \, dx}{256}\\ &=-\frac {3 x}{256}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{256 b}-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{128 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{32 b}-\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 62, normalized size = 0.55 \[ \frac {20 \sinh (2 (a+b x))+40 \sinh (4 (a+b x))-10 \sinh (6 (a+b x))-5 \sinh (8 (a+b x))+2 \sinh (10 (a+b x))-120 b x}{10240 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 197, normalized size = 1.74 \[ \frac {5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{9} + 10 \, {\left (6 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{7} + {\left (126 \, \cosh \left (b x + a\right )^{5} - 70 \, \cosh \left (b x + a\right )^{3} - 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 10 \, {\left (6 \, \cosh \left (b x + a\right )^{7} - 7 \, \cosh \left (b x + a\right )^{5} - 5 \, \cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 30 \, b x + 5 \, {\left (\cosh \left (b x + a\right )^{9} - 2 \, \cosh \left (b x + a\right )^{7} - 3 \, \cosh \left (b x + a\right )^{5} + 8 \, \cosh \left (b x + a\right )^{3} + 2 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{2560 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 144, normalized size = 1.27 \[ -\frac {3}{256} \, x + \frac {e^{\left (10 \, b x + 10 \, a\right )}}{10240 \, b} - \frac {e^{\left (8 \, b x + 8 \, a\right )}}{4096 \, b} - \frac {e^{\left (6 \, b x + 6 \, a\right )}}{2048 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{512 \, b} + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{1024 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{1024 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b} + \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{2048 \, b} + \frac {e^{\left (-8 \, b x - 8 \, a\right )}}{4096 \, b} - \frac {e^{\left (-10 \, b x - 10 \, a\right )}}{10240 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 92, normalized size = 0.81 \[ \frac {\frac {\left (\sinh ^{5}\left (b x +a \right )\right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{10}-\frac {\left (\sinh ^{3}\left (b x +a \right )\right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{16}+\frac {\sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{32}-\frac {\left (\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )}{32}-\frac {3 b x}{256}-\frac {3 a}{256}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 132, normalized size = 1.17 \[ -\frac {{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} - 40 \, e^{\left (-6 \, b x - 6 \, a\right )} - 20 \, e^{\left (-8 \, b x - 8 \, a\right )} - 2\right )} e^{\left (10 \, b x + 10 \, a\right )}}{20480 \, b} - \frac {3 \, {\left (b x + a\right )}}{256 \, b} - \frac {20 \, e^{\left (-2 \, b x - 2 \, a\right )} + 40 \, e^{\left (-4 \, b x - 4 \, a\right )} - 10 \, e^{\left (-6 \, b x - 6 \, a\right )} - 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + 2 \, e^{\left (-10 \, b x - 10 \, a\right )}}{20480 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.74, size = 65, normalized size = 0.58 \[ \frac {20\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+40\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )-10\,\mathrm {sinh}\left (6\,a+6\,b\,x\right )-5\,\mathrm {sinh}\left (8\,a+8\,b\,x\right )+2\,\mathrm {sinh}\left (10\,a+10\,b\,x\right )-120\,b\,x}{10240\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.04, size = 231, normalized size = 2.04 \[ \begin {cases} \frac {3 x \sinh ^{10}{\left (a + b x \right )}}{256} - \frac {15 x \sinh ^{8}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{256} + \frac {15 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{128} - \frac {15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{128} + \frac {15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{8}{\left (a + b x \right )}}{256} - \frac {3 x \cosh ^{10}{\left (a + b x \right )}}{256} - \frac {3 \sinh ^{9}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{256 b} + \frac {7 \sinh ^{7}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{128 b} + \frac {\sinh ^{5}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{10 b} - \frac {7 \sinh ^{3}{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} + \frac {3 \sinh {\left (a + b x \right )} \cosh ^{9}{\left (a + b x \right )}}{256 b} & \text {for}\: b \neq 0 \\x \sinh ^{6}{\relax (a )} \cosh ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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