Optimal. Leaf size=50 \[ -\frac {5 i x}{16}+\frac {\cosh ^7(x)}{7}-\frac {1}{6} i \sinh (x) \cosh ^5(x)-\frac {5}{24} i \sinh (x) \cosh ^3(x)-\frac {5}{16} i \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.05, antiderivative size = 50, 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 = {2682, 2635, 8} \[ -\frac {5 i x}{16}+\frac {\cosh ^7(x)}{7}-\frac {1}{6} i \sinh (x) \cosh ^5(x)-\frac {5}{24} i \sinh (x) \cosh ^3(x)-\frac {5}{16} i \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2682
Rubi steps
\begin {align*} \int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx &=\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*}
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Mathematica [B] time = 0.16, size = 219, normalized size = 4.38 \[ \frac {\cosh ^9(x) \left (48 \sqrt {1+i \sinh (x)} \sinh ^7(x)-8 i \sqrt {1+i \sinh (x)} \sinh ^6(x)+200 \sqrt {1+i \sinh (x)} \sinh ^5(x)-38 i \sqrt {1+i \sinh (x)} \sinh ^4(x)+326 \sqrt {1+i \sinh (x)} \sinh ^3(x)-87 i \sqrt {1+i \sinh (x)} \sinh ^2(x)+279 \sqrt {1+i \sinh (x)} \sinh (x)+6 i \left (8 \sqrt {1+i \sinh (x)}+35 \sqrt {1-i \sinh (x)} \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )\right )\right )}{336 \sqrt {1+i \sinh (x)} (\sinh (x)-i)^4 (\sinh (x)+i)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 91, normalized size = 1.82 \[ \frac {1}{2688} \, {\left (-840 i \, x e^{\left (7 \, x\right )} + 3 \, e^{\left (14 \, x\right )} - 7 i \, e^{\left (13 \, x\right )} + 21 \, e^{\left (12 \, x\right )} - 63 i \, e^{\left (11 \, x\right )} + 63 \, e^{\left (10 \, x\right )} - 315 i \, e^{\left (9 \, x\right )} + 105 \, e^{\left (8 \, x\right )} + 105 \, e^{\left (6 \, x\right )} + 315 i \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 i \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 i \, e^{x} + 3\right )} e^{\left (-7 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 86, normalized size = 1.72 \[ \frac {1}{2688} \, {\left (105 \, e^{\left (6 \, x\right )} + 315 i \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 i \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 i \, e^{x} + 3\right )} e^{\left (-7 \, x\right )} - \frac {5}{16} i \, x + \frac {1}{896} \, e^{\left (7 \, x\right )} - \frac {1}{384} i \, e^{\left (6 \, x\right )} + \frac {1}{128} \, e^{\left (5 \, x\right )} - \frac {3}{128} i \, e^{\left (4 \, x\right )} + \frac {3}{128} \, e^{\left (3 \, x\right )} - \frac {15}{128} i \, e^{\left (2 \, x\right )} + \frac {5}{128} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 292, normalized size = 5.84 \[ \frac {5 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{16}-\frac {11 i}{16 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {5 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{16}-\frac {11 i}{16 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{7 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{7}}+\frac {1}{7 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{7}}-\frac {5}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {5}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{6}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{6}}-\frac {9}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {9}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {5}{16 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {11}{16 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {11}{16 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {5}{16 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {i}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{6}}-\frac {19 i}{16 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {7 i}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {19 i}{16 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {7 i}{6 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {i}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {i}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {i}{6 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 90, normalized size = 1.80 \[ -\frac {1}{5376} \, {\left (14 i \, e^{\left (-x\right )} - 42 \, e^{\left (-2 \, x\right )} + 126 i \, e^{\left (-3 \, x\right )} - 126 \, e^{\left (-4 \, x\right )} + 630 i \, e^{\left (-5 \, x\right )} - 210 \, e^{\left (-6 \, x\right )} - 6\right )} e^{\left (7 \, x\right )} - \frac {5}{16} i \, x + \frac {5}{128} \, e^{\left (-x\right )} + \frac {15}{128} i \, e^{\left (-2 \, x\right )} + \frac {3}{128} \, e^{\left (-3 \, x\right )} + \frac {3}{128} i \, e^{\left (-4 \, x\right )} + \frac {1}{128} \, e^{\left (-5 \, x\right )} + \frac {1}{384} i \, e^{\left (-6 \, x\right )} + \frac {1}{896} \, e^{\left (-7 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 93, normalized size = 1.86 \[ \frac {5\,{\mathrm {e}}^{-x}}{128}+\frac {3\,{\mathrm {e}}^{-3\,x}}{128}+\frac {3\,{\mathrm {e}}^{3\,x}}{128}+\frac {{\mathrm {e}}^{-5\,x}}{128}+\frac {{\mathrm {e}}^{5\,x}}{128}+\frac {{\mathrm {e}}^{-7\,x}}{896}+\frac {{\mathrm {e}}^{7\,x}}{896}+\frac {5\,{\mathrm {e}}^x}{128}-\frac {x\,5{}\mathrm {i}}{16}+\frac {{\mathrm {e}}^{-2\,x}\,15{}\mathrm {i}}{128}-\frac {{\mathrm {e}}^{2\,x}\,15{}\mathrm {i}}{128}+\frac {{\mathrm {e}}^{-4\,x}\,3{}\mathrm {i}}{128}-\frac {{\mathrm {e}}^{4\,x}\,3{}\mathrm {i}}{128}+\frac {{\mathrm {e}}^{-6\,x}\,1{}\mathrm {i}}{384}-\frac {{\mathrm {e}}^{6\,x}\,1{}\mathrm {i}}{384} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.30, size = 124, normalized size = 2.48 \[ - \frac {5 i x}{16} + \frac {e^{7 x}}{896} - \frac {i e^{6 x}}{384} + \frac {e^{5 x}}{128} - \frac {3 i e^{4 x}}{128} + \frac {3 e^{3 x}}{128} - \frac {15 i e^{2 x}}{128} + \frac {5 e^{x}}{128} + \frac {5 e^{- x}}{128} + \frac {15 i e^{- 2 x}}{128} + \frac {3 e^{- 3 x}}{128} + \frac {3 i e^{- 4 x}}{128} + \frac {e^{- 5 x}}{128} + \frac {i e^{- 6 x}}{384} + \frac {e^{- 7 x}}{896} \]
Verification of antiderivative is not currently implemented for this CAS.
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