Integrand size = 13, antiderivative size = 50 \[ \int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx=-\frac {5 i x}{16}+\frac {\cosh ^7(x)}{7}-\frac {5}{16} i \cosh (x) \sinh (x)-\frac {5}{24} i \cosh ^3(x) \sinh (x)-\frac {1}{6} i \cosh ^5(x) \sinh (x) \]
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Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2761, 2715, 8} \[ \int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx=-\frac {5 i x}{16}+\frac {\cosh ^7(x)}{7}-\frac {1}{6} i \sinh (x) \cosh ^5(x)-\frac {5}{24} i \sinh (x) \cosh ^3(x)-\frac {5}{16} i \sinh (x) \cosh (x) \]
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Rule 8
Rule 2715
Rule 2761
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh ^7(x)}{7}-i \int \cosh ^6(x) \, dx \\ & = \frac {\cosh ^7(x)}{7}-\frac {1}{6} i \cosh ^5(x) \sinh (x)-\frac {5}{6} i \int \cosh ^4(x) \, dx \\ & = \frac {\cosh ^7(x)}{7}-\frac {5}{24} i \cosh ^3(x) \sinh (x)-\frac {1}{6} i \cosh ^5(x) \sinh (x)-\frac {5}{8} i \int \cosh ^2(x) \, dx \\ & = \frac {\cosh ^7(x)}{7}-\frac {5}{16} i \cosh (x) \sinh (x)-\frac {5}{24} i \cosh ^3(x) \sinh (x)-\frac {1}{6} i \cosh ^5(x) \sinh (x)-\frac {5}{16} i \int 1 \, dx \\ & = -\frac {5 i x}{16}+\frac {\cosh ^7(x)}{7}-\frac {5}{16} i \cosh (x) \sinh (x)-\frac {5}{24} i \cosh ^3(x) \sinh (x)-\frac {1}{6} i \cosh ^5(x) \sinh (x) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(219\) vs. \(2(50)=100\).
Time = 0.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 4.38 \[ \int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx=\frac {\cosh ^9(x) \left (6 i \left (35 \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right ) \sqrt {1-i \sinh (x)}+8 \sqrt {1+i \sinh (x)}\right )+279 \sqrt {1+i \sinh (x)} \sinh (x)-87 i \sqrt {1+i \sinh (x)} \sinh ^2(x)+326 \sqrt {1+i \sinh (x)} \sinh ^3(x)-38 i \sqrt {1+i \sinh (x)} \sinh ^4(x)+200 \sqrt {1+i \sinh (x)} \sinh ^5(x)-8 i \sqrt {1+i \sinh (x)} \sinh ^6(x)+48 \sqrt {1+i \sinh (x)} \sinh ^7(x)\right )}{336 \sqrt {1+i \sinh (x)} (-i+\sinh (x))^4 (i+\sinh (x))^5} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (36 ) = 72\).
Time = 0.48 (sec) , antiderivative size = 186, normalized size of antiderivative = 3.72
\[\frac {-\frac {1}{2}+\frac {i}{6}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{6}}+\frac {-\frac {11}{16}-\frac {19 i}{16}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {-\frac {9}{8}-\frac {7 i}{6}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {-\frac {5}{4}+i}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {-\frac {1}{2}-\frac {i}{6}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{6}}+\frac {\frac {5}{16}-\frac {11 i}{16}}{\tanh \left (\frac {x}{2}\right )+1}-\frac {5 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{16}+\frac {1}{7 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{7}}+\frac {-\frac {5}{4}-i}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {\frac {9}{8}-\frac {7 i}{6}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {-1-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}+\frac {5 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{16}+\frac {-\frac {5}{16}-\frac {11 i}{16}}{\tanh \left (\frac {x}{2}\right )-1}+\frac {1-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {-\frac {11}{16}+\frac {19 i}{16}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{7 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{7}}\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (32) = 64\).
Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.82 \[ \int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx=\frac {1}{2688} \, {\left (-840 i \, x e^{\left (7 \, x\right )} + 3 \, e^{\left (14 \, x\right )} - 7 i \, e^{\left (13 \, x\right )} + 21 \, e^{\left (12 \, x\right )} - 63 i \, e^{\left (11 \, x\right )} + 63 \, e^{\left (10 \, x\right )} - 315 i \, e^{\left (9 \, x\right )} + 105 \, e^{\left (8 \, x\right )} + 105 \, e^{\left (6 \, x\right )} + 315 i \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 i \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 i \, e^{x} + 3\right )} e^{\left (-7 \, x\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (49) = 98\).
Time = 0.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.48 \[ \int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx=- \frac {5 i x}{16} + \frac {e^{7 x}}{896} - \frac {i e^{6 x}}{384} + \frac {e^{5 x}}{128} - \frac {3 i e^{4 x}}{128} + \frac {3 e^{3 x}}{128} - \frac {15 i e^{2 x}}{128} + \frac {5 e^{x}}{128} + \frac {5 e^{- x}}{128} + \frac {15 i e^{- 2 x}}{128} + \frac {3 e^{- 3 x}}{128} + \frac {3 i e^{- 4 x}}{128} + \frac {e^{- 5 x}}{128} + \frac {i e^{- 6 x}}{384} + \frac {e^{- 7 x}}{896} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (32) = 64\).
Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.80 \[ \int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx=-\frac {1}{2688} \, {\left (7 i \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 63 i \, e^{\left (-3 \, x\right )} - 63 \, e^{\left (-4 \, x\right )} + 315 i \, e^{\left (-5 \, x\right )} - 105 \, e^{\left (-6 \, x\right )} - 3\right )} e^{\left (7 \, x\right )} - \frac {5}{16} i \, x + \frac {5}{128} \, e^{\left (-x\right )} + \frac {15}{128} i \, e^{\left (-2 \, x\right )} + \frac {3}{128} \, e^{\left (-3 \, x\right )} + \frac {3}{128} i \, e^{\left (-4 \, x\right )} + \frac {1}{128} \, e^{\left (-5 \, x\right )} + \frac {1}{384} i \, e^{\left (-6 \, x\right )} + \frac {1}{896} \, e^{\left (-7 \, x\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (32) = 64\).
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.72 \[ \int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx=\frac {1}{2688} \, {\left (105 \, e^{\left (6 \, x\right )} + 315 i \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 i \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 i \, e^{x} + 3\right )} e^{\left (-7 \, x\right )} - \frac {5}{16} i \, x + \frac {1}{896} \, e^{\left (7 \, x\right )} - \frac {1}{384} i \, e^{\left (6 \, x\right )} + \frac {1}{128} \, e^{\left (5 \, x\right )} - \frac {3}{128} i \, e^{\left (4 \, x\right )} + \frac {3}{128} \, e^{\left (3 \, x\right )} - \frac {15}{128} i \, e^{\left (2 \, x\right )} + \frac {5}{128} \, e^{x} \]
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Time = 1.51 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.86 \[ \int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx=\frac {5\,{\mathrm {e}}^{-x}}{128}+\frac {3\,{\mathrm {e}}^{-3\,x}}{128}+\frac {3\,{\mathrm {e}}^{3\,x}}{128}+\frac {{\mathrm {e}}^{-5\,x}}{128}+\frac {{\mathrm {e}}^{5\,x}}{128}+\frac {{\mathrm {e}}^{-7\,x}}{896}+\frac {{\mathrm {e}}^{7\,x}}{896}+\frac {5\,{\mathrm {e}}^x}{128}-\frac {x\,5{}\mathrm {i}}{16}+\frac {{\mathrm {e}}^{-2\,x}\,15{}\mathrm {i}}{128}-\frac {{\mathrm {e}}^{2\,x}\,15{}\mathrm {i}}{128}+\frac {{\mathrm {e}}^{-4\,x}\,3{}\mathrm {i}}{128}-\frac {{\mathrm {e}}^{4\,x}\,3{}\mathrm {i}}{128}+\frac {{\mathrm {e}}^{-6\,x}\,1{}\mathrm {i}}{384}-\frac {{\mathrm {e}}^{6\,x}\,1{}\mathrm {i}}{384} \]
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