Optimal. Leaf size=38 \[ -\frac {3 i x}{8}+\frac {\cosh ^5(x)}{5}-\frac {1}{4} i \sinh (x) \cosh ^3(x)-\frac {3}{8} i \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.05, antiderivative size = 38, 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} \[ -\frac {3 i x}{8}+\frac {\cosh ^5(x)}{5}-\frac {1}{4} i \sinh (x) \cosh ^3(x)-\frac {3}{8} i \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2682
Rubi steps
\begin {align*} \int \frac {\cosh ^6(x)}{i+\sinh (x)} \, dx &=\frac {\cosh ^5(x)}{5}-i \int \cosh ^4(x) \, dx\\ &=\frac {\cosh ^5(x)}{5}-\frac {1}{4} i \cosh ^3(x) \sinh (x)-\frac {3}{4} i \int \cosh ^2(x) \, dx\\ &=\frac {\cosh ^5(x)}{5}-\frac {3}{8} i \cosh (x) \sinh (x)-\frac {1}{4} i \cosh ^3(x) \sinh (x)-\frac {3}{8} i \int 1 \, dx\\ &=-\frac {3 i x}{8}+\frac {\cosh ^5(x)}{5}-\frac {3}{8} i \cosh (x) \sinh (x)-\frac {1}{4} i \cosh ^3(x) \sinh (x)\\ \end {align*}
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Mathematica [B] time = 0.24, size = 131, normalized size = 3.45 \[ -\frac {i \cosh ^7(x) \left (8 \sinh ^5(x)-2 i \sinh ^4(x)+26 \sinh ^3(x)-9 i \sinh ^2(x)+33 \sinh (x)+\frac {30 i \sqrt {1-i \sinh (x)} \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {1+i \sinh (x)}}+8 i\right )}{40 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^8 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 67, normalized size = 1.76 \[ \frac {1}{320} \, {\left (-120 i \, x e^{\left (5 \, x\right )} + 2 \, e^{\left (10 \, x\right )} - 5 i \, e^{\left (9 \, x\right )} + 10 \, e^{\left (8 \, x\right )} - 40 i \, e^{\left (7 \, x\right )} + 20 \, e^{\left (6 \, x\right )} + 20 \, e^{\left (4 \, x\right )} + 40 i \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} + 5 i \, e^{x} + 2\right )} e^{\left (-5 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 62, normalized size = 1.63 \[ \frac {1}{320} \, {\left (20 \, e^{\left (4 \, x\right )} + 40 i \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} + 5 i \, e^{x} + 2\right )} e^{\left (-5 \, x\right )} - \frac {3}{8} i \, x + \frac {1}{160} \, e^{\left (5 \, x\right )} - \frac {1}{64} i \, e^{\left (4 \, x\right )} + \frac {1}{32} \, e^{\left (3 \, x\right )} - \frac {1}{8} i \, e^{\left (2 \, x\right )} + \frac {1}{16} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 210, normalized size = 5.53 \[ -\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8}-\frac {5}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {7 i}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {3}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {5 i}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{5 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}+\frac {7 i}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}+\frac {3}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {5}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {5 i}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {1}{5 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 66, normalized size = 1.74 \[ -\frac {1}{320} \, {\left (5 i \, e^{\left (-x\right )} - 10 \, e^{\left (-2 \, x\right )} + 40 i \, e^{\left (-3 \, x\right )} - 20 \, e^{\left (-4 \, x\right )} - 2\right )} e^{\left (5 \, x\right )} - \frac {3}{8} i \, x + \frac {1}{16} \, e^{\left (-x\right )} + \frac {1}{8} i \, e^{\left (-2 \, x\right )} + \frac {1}{32} \, e^{\left (-3 \, x\right )} + \frac {1}{64} i \, e^{\left (-4 \, x\right )} + \frac {1}{160} \, e^{\left (-5 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 67, normalized size = 1.76 \[ \frac {{\mathrm {e}}^{-x}}{16}+\frac {{\mathrm {e}}^{-3\,x}}{32}+\frac {{\mathrm {e}}^{3\,x}}{32}+\frac {{\mathrm {e}}^{-5\,x}}{160}+\frac {{\mathrm {e}}^{5\,x}}{160}+\frac {{\mathrm {e}}^x}{16}-\frac {x\,3{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{-4\,x}\,1{}\mathrm {i}}{64}-\frac {{\mathrm {e}}^{4\,x}\,1{}\mathrm {i}}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.23, size = 82, normalized size = 2.16 \[ - \frac {3 i x}{8} + \frac {e^{5 x}}{160} - \frac {i e^{4 x}}{64} + \frac {e^{3 x}}{32} - \frac {i e^{2 x}}{8} + \frac {e^{x}}{16} + \frac {e^{- x}}{16} + \frac {i e^{- 2 x}}{8} + \frac {e^{- 3 x}}{32} + \frac {i e^{- 4 x}}{64} + \frac {e^{- 5 x}}{160} \]
Verification of antiderivative is not currently implemented for this CAS.
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