Optimal. Leaf size=44 \[ -\frac {3 x}{8 a}+\frac {\sinh ^5(x)}{5 a}-\frac {\sinh ^3(x) \cosh (x)}{4 a}+\frac {3 \sinh (x) \cosh (x)}{8 a} \]
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Rubi [A] time = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2682, 2635, 8} \[ -\frac {3 x}{8 a}+\frac {\sinh ^5(x)}{5 a}-\frac {\sinh ^3(x) \cosh (x)}{4 a}+\frac {3 \sinh (x) \cosh (x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2682
Rubi steps
\begin {align*} \int \frac {\sinh ^6(x)}{a+a \cosh (x)} \, dx &=\frac {\sinh ^5(x)}{5 a}-\frac {\int \sinh ^4(x) \, dx}{a}\\ &=-\frac {\cosh (x) \sinh ^3(x)}{4 a}+\frac {\sinh ^5(x)}{5 a}+\frac {3 \int \sinh ^2(x) \, dx}{4 a}\\ &=\frac {3 \cosh (x) \sinh (x)}{8 a}-\frac {\cosh (x) \sinh ^3(x)}{4 a}+\frac {\sinh ^5(x)}{5 a}-\frac {3 \int 1 \, dx}{8 a}\\ &=-\frac {3 x}{8 a}+\frac {3 \cosh (x) \sinh (x)}{8 a}-\frac {\cosh (x) \sinh ^3(x)}{4 a}+\frac {\sinh ^5(x)}{5 a}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 39, normalized size = 0.89 \[ \frac {-60 x+20 \sinh (x)+40 \sinh (2 x)-10 \sinh (3 x)-5 \sinh (4 x)+2 \sinh (5 x)}{160 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 57, normalized size = 1.30 \[ \frac {\sinh \relax (x)^{5} + 5 \, {\left (2 \, \cosh \relax (x)^{2} - 2 \, \cosh \relax (x) - 1\right )} \sinh \relax (x)^{3} + 5 \, {\left (\cosh \relax (x)^{4} - 2 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)^{2} + 8 \, \cosh \relax (x) + 2\right )} \sinh \relax (x) - 30 \, x}{80 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 66, normalized size = 1.50 \[ -\frac {{\left (20 \, e^{\left (4 \, x\right )} + 40 \, e^{\left (3 \, x\right )} - 10 \, e^{\left (2 \, x\right )} - 5 \, e^{x} + 2\right )} e^{\left (-5 \, x\right )} + 120 \, x - 2 \, e^{\left (5 \, x\right )} + 5 \, e^{\left (4 \, x\right )} + 10 \, e^{\left (3 \, x\right )} - 40 \, e^{\left (2 \, x\right )} - 20 \, e^{x}}{320 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 156, normalized size = 3.55 \[ -\frac {1}{5 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {3}{4 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {3}{4 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{8 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 a}-\frac {1}{5 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {3}{4 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {3}{4 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{8 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 78, normalized size = 1.77 \[ -\frac {{\left (5 \, e^{\left (-x\right )} + 10 \, e^{\left (-2 \, x\right )} - 40 \, e^{\left (-3 \, x\right )} - 20 \, e^{\left (-4 \, x\right )} - 2\right )} e^{\left (5 \, x\right )}}{320 \, a} - \frac {3 \, x}{8 \, a} - \frac {20 \, e^{\left (-x\right )} + 40 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-3 \, x\right )} - 5 \, e^{\left (-4 \, x\right )} + 2 \, e^{\left (-5 \, x\right )}}{320 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 95, normalized size = 2.16 \[ \frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-x}}{16\,a}+\frac {{\mathrm {e}}^{-3\,x}}{32\,a}-\frac {{\mathrm {e}}^{3\,x}}{32\,a}+\frac {{\mathrm {e}}^{-4\,x}}{64\,a}-\frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {{\mathrm {e}}^{-5\,x}}{160\,a}+\frac {{\mathrm {e}}^{5\,x}}{160\,a}-\frac {3\,x}{8\,a}+\frac {{\mathrm {e}}^x}{16\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.71, size = 692, normalized size = 15.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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