Optimal. Leaf size=57 \[ \frac {5 x}{16 a}-\frac {5 \cosh (x) \sinh (x)}{16 a}+\frac {5 \cosh (x) \sinh ^3(x)}{24 a}-\frac {\cosh (x) \sinh ^5(x)}{6 a}+\frac {\sinh ^7(x)}{7 a} \]
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Rubi [A]
time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2761, 2715, 8}
\begin {gather*} \frac {5 x}{16 a}+\frac {\sinh ^7(x)}{7 a}-\frac {\sinh ^5(x) \cosh (x)}{6 a}+\frac {5 \sinh ^3(x) \cosh (x)}{24 a}-\frac {5 \sinh (x) \cosh (x)}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2761
Rubi steps
\begin {align*} \int \frac {\sinh ^8(x)}{a+a \cosh (x)} \, dx &=\frac {\sinh ^7(x)}{7 a}-\frac {\int \sinh ^6(x) \, dx}{a}\\ &=-\frac {\cosh (x) \sinh ^5(x)}{6 a}+\frac {\sinh ^7(x)}{7 a}+\frac {5 \int \sinh ^4(x) \, dx}{6 a}\\ &=\frac {5 \cosh (x) \sinh ^3(x)}{24 a}-\frac {\cosh (x) \sinh ^5(x)}{6 a}+\frac {\sinh ^7(x)}{7 a}-\frac {5 \int \sinh ^2(x) \, dx}{8 a}\\ &=-\frac {5 \cosh (x) \sinh (x)}{16 a}+\frac {5 \cosh (x) \sinh ^3(x)}{24 a}-\frac {\cosh (x) \sinh ^5(x)}{6 a}+\frac {\sinh ^7(x)}{7 a}+\frac {5 \int 1 \, dx}{16 a}\\ &=\frac {5 x}{16 a}-\frac {5 \cosh (x) \sinh (x)}{16 a}+\frac {5 \cosh (x) \sinh ^3(x)}{24 a}-\frac {\cosh (x) \sinh ^5(x)}{6 a}+\frac {\sinh ^7(x)}{7 a}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 51, normalized size = 0.89 \begin {gather*} \frac {420 x-105 \sinh (x)-315 \sinh (2 x)+63 \sinh (3 x)+63 \sinh (4 x)-21 \sinh (5 x)-7 \sinh (6 x)+3 \sinh (7 x)}{1344 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(164\) vs.
\(2(47)=94\).
time = 0.52, size = 165, normalized size = 2.89
method | result | size |
risch | \(\frac {5 x}{16 a}+\frac {{\mathrm e}^{7 x}}{896 a}-\frac {{\mathrm e}^{6 x}}{384 a}-\frac {{\mathrm e}^{5 x}}{128 a}+\frac {3 \,{\mathrm e}^{4 x}}{128 a}+\frac {3 \,{\mathrm e}^{3 x}}{128 a}-\frac {15 \,{\mathrm e}^{2 x}}{128 a}-\frac {5 \,{\mathrm e}^{x}}{128 a}+\frac {5 \,{\mathrm e}^{-x}}{128 a}+\frac {15 \,{\mathrm e}^{-2 x}}{128 a}-\frac {3 \,{\mathrm e}^{-3 x}}{128 a}-\frac {3 \,{\mathrm e}^{-4 x}}{128 a}+\frac {{\mathrm e}^{-5 x}}{128 a}+\frac {{\mathrm e}^{-6 x}}{384 a}-\frac {{\mathrm e}^{-7 x}}{896 a}\) | \(132\) |
default | \(\frac {-\frac {1}{7 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{7}}+\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{6}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {11}{24 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {5}{16 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{16}-\frac {1}{7 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{7}}-\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{6}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {11}{24 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {5}{16 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{16}}{a}\) | \(165\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (47) = 94\).
time = 0.26, size = 102, normalized size = 1.79 \begin {gather*} -\frac {{\left (7 \, e^{\left (-x\right )} + 21 \, e^{\left (-2 \, x\right )} - 63 \, e^{\left (-3 \, x\right )} - 63 \, e^{\left (-4 \, x\right )} + 315 \, e^{\left (-5 \, x\right )} + 105 \, e^{\left (-6 \, x\right )} - 3\right )} e^{\left (7 \, x\right )}}{2688 \, a} + \frac {5 \, x}{16 \, a} + \frac {105 \, e^{\left (-x\right )} + 315 \, e^{\left (-2 \, x\right )} - 63 \, e^{\left (-3 \, x\right )} - 63 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 \, e^{\left (-6 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{2688 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs.
\(2 (47) = 94\).
time = 0.41, size = 101, normalized size = 1.77 \begin {gather*} \frac {3 \, \sinh \left (x\right )^{7} + 21 \, {\left (3 \, \cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{5} + 7 \, {\left (15 \, \cosh \left (x\right )^{4} - 20 \, \cosh \left (x\right )^{3} - 30 \, \cosh \left (x\right )^{2} + 36 \, \cosh \left (x\right ) + 9\right )} \sinh \left (x\right )^{3} + 21 \, {\left (\cosh \left (x\right )^{6} - 2 \, \cosh \left (x\right )^{5} - 5 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )^{2} - 30 \, \cosh \left (x\right ) - 5\right )} \sinh \left (x\right ) + 420 \, x}{1344 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1253 vs.
\(2 (51) = 102\).
time = 3.54, size = 1253, normalized size = 21.98 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 90, normalized size = 1.58 \begin {gather*} \frac {{\left (105 \, e^{\left (6 \, x\right )} + 315 \, e^{\left (5 \, x\right )} - 63 \, e^{\left (4 \, x\right )} - 63 \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 \, e^{x} - 3\right )} e^{\left (-7 \, x\right )} + 840 \, x + 3 \, e^{\left (7 \, x\right )} - 7 \, e^{\left (6 \, x\right )} - 21 \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 \, e^{\left (3 \, x\right )} - 315 \, e^{\left (2 \, x\right )} - 105 \, e^{x}}{2688 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.26, size = 131, normalized size = 2.30 \begin {gather*} \frac {5\,{\mathrm {e}}^{-x}}{128\,a}+\frac {15\,{\mathrm {e}}^{-2\,x}}{128\,a}-\frac {15\,{\mathrm {e}}^{2\,x}}{128\,a}-\frac {3\,{\mathrm {e}}^{-3\,x}}{128\,a}+\frac {3\,{\mathrm {e}}^{3\,x}}{128\,a}-\frac {3\,{\mathrm {e}}^{-4\,x}}{128\,a}+\frac {3\,{\mathrm {e}}^{4\,x}}{128\,a}+\frac {{\mathrm {e}}^{-5\,x}}{128\,a}-\frac {{\mathrm {e}}^{5\,x}}{128\,a}+\frac {{\mathrm {e}}^{-6\,x}}{384\,a}-\frac {{\mathrm {e}}^{6\,x}}{384\,a}-\frac {{\mathrm {e}}^{-7\,x}}{896\,a}+\frac {{\mathrm {e}}^{7\,x}}{896\,a}+\frac {5\,x}{16\,a}-\frac {5\,{\mathrm {e}}^x}{128\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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