Optimal. Leaf size=134 \[ \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{24 b}+\frac {\sinh (a+b x) \cosh ^7(a+b x)}{64 b}-\frac {\sinh (a+b x) \cosh ^5(a+b x)}{384 b}-\frac {5 \sinh (a+b x) \cosh ^3(a+b x)}{1536 b}-\frac {5 \sinh (a+b x) \cosh (a+b x)}{1024 b}-\frac {5 x}{1024} \]
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Rubi [A] time = 0.13, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{24 b}+\frac {\sinh (a+b x) \cosh ^7(a+b x)}{64 b}-\frac {\sinh (a+b x) \cosh ^5(a+b x)}{384 b}-\frac {5 \sinh (a+b x) \cosh ^3(a+b x)}{1536 b}-\frac {5 \sinh (a+b x) \cosh (a+b x)}{1024 b}-\frac {5 x}{1024} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2568
Rule 2635
Rubi steps
\begin {align*} \int \cosh ^6(a+b x) \sinh ^6(a+b x) \, dx &=\frac {\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}-\frac {5}{12} \int \cosh ^6(a+b x) \sinh ^4(a+b x) \, dx\\ &=-\frac {\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}+\frac {1}{8} \int \cosh ^6(a+b x) \sinh ^2(a+b x) \, dx\\ &=\frac {\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac {\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}-\frac {1}{64} \int \cosh ^6(a+b x) \, dx\\ &=-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{384 b}+\frac {\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac {\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}-\frac {5}{384} \int \cosh ^4(a+b x) \, dx\\ &=-\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{1536 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{384 b}+\frac {\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac {\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}-\frac {5}{512} \int \cosh ^2(a+b x) \, dx\\ &=-\frac {5 \cosh (a+b x) \sinh (a+b x)}{1024 b}-\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{1536 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{384 b}+\frac {\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac {\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}-\frac {5 \int 1 \, dx}{1024}\\ &=-\frac {5 x}{1024}-\frac {5 \cosh (a+b x) \sinh (a+b x)}{1024 b}-\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{1536 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{384 b}+\frac {\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac {\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 43, normalized size = 0.32 \[ \frac {45 \sinh (4 (a+b x))-9 \sinh (8 (a+b x))+\sinh (12 (a+b x))-120 a-120 b x}{24576 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 179, normalized size = 1.34 \[ \frac {55 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{9} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{11} + 18 \, {\left (11 \, \cosh \left (b x + a\right )^{5} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{7} + 18 \, {\left (11 \, \cosh \left (b x + a\right )^{7} - 7 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{5} + {\left (55 \, \cosh \left (b x + a\right )^{9} - 126 \, \cosh \left (b x + a\right )^{5} + 45 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 30 \, b x + 3 \, {\left (\cosh \left (b x + a\right )^{11} - 6 \, \cosh \left (b x + a\right )^{7} + 15 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )}{6144 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 88, normalized size = 0.66 \[ -\frac {5}{1024} \, x + \frac {e^{\left (12 \, b x + 12 \, a\right )}}{49152 \, b} - \frac {3 \, e^{\left (8 \, b x + 8 \, a\right )}}{16384 \, b} + \frac {15 \, e^{\left (4 \, b x + 4 \, a\right )}}{16384 \, b} - \frac {15 \, e^{\left (-4 \, b x - 4 \, a\right )}}{16384 \, b} + \frac {3 \, e^{\left (-8 \, b x - 8 \, a\right )}}{16384 \, b} - \frac {e^{\left (-12 \, b x - 12 \, a\right )}}{49152 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 102, normalized size = 0.76 \[ \frac {\frac {\left (\sinh ^{5}\left (b x +a \right )\right ) \left (\cosh ^{7}\left (b x +a \right )\right )}{12}-\frac {\left (\sinh ^{3}\left (b x +a \right )\right ) \left (\cosh ^{7}\left (b x +a \right )\right )}{24}+\frac {\sinh \left (b x +a \right ) \left (\cosh ^{7}\left (b x +a \right )\right )}{64}-\frac {\left (\frac {\left (\cosh ^{5}\left (b x +a \right )\right )}{6}+\frac {5 \left (\cosh ^{3}\left (b x +a \right )\right )}{24}+\frac {5 \cosh \left (b x +a \right )}{16}\right ) \sinh \left (b x +a \right )}{64}-\frac {5 b x}{1024}-\frac {5 a}{1024}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 86, normalized size = 0.64 \[ -\frac {{\left (9 \, e^{\left (-4 \, b x - 4 \, a\right )} - 45 \, e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )} e^{\left (12 \, b x + 12 \, a\right )}}{49152 \, b} - \frac {5 \, {\left (b x + a\right )}}{1024 \, b} - \frac {45 \, e^{\left (-4 \, b x - 4 \, a\right )} - 9 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )}}{49152 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.82, size = 42, normalized size = 0.31 \[ \frac {\frac {15\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{8192}-\frac {3\,\mathrm {sinh}\left (8\,a+8\,b\,x\right )}{8192}+\frac {\mathrm {sinh}\left (12\,a+12\,b\,x\right )}{24576}}{b}-\frac {5\,x}{1024} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 47.86, size = 277, normalized size = 2.07 \[ \begin {cases} - \frac {5 x \sinh ^{12}{\left (a + b x \right )}}{1024} + \frac {15 x \sinh ^{10}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{512} - \frac {75 x \sinh ^{8}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{1024} + \frac {25 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{256} - \frac {75 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{8}{\left (a + b x \right )}}{1024} + \frac {15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{10}{\left (a + b x \right )}}{512} - \frac {5 x \cosh ^{12}{\left (a + b x \right )}}{1024} + \frac {5 \sinh ^{11}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{1024 b} - \frac {85 \sinh ^{9}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3072 b} + \frac {33 \sinh ^{7}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{512 b} + \frac {33 \sinh ^{5}{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{512 b} - \frac {85 \sinh ^{3}{\left (a + b x \right )} \cosh ^{9}{\left (a + b x \right )}}{3072 b} + \frac {5 \sinh {\left (a + b x \right )} \cosh ^{11}{\left (a + b x \right )}}{1024 b} & \text {for}\: b \neq 0 \\x \sinh ^{6}{\relax (a )} \cosh ^{6}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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