Optimal. Leaf size=15 \[ \frac {\sinh ^2(x)}{2}-i \sinh (x) \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2667} \[ \frac {\sinh ^2(x)}{2}-i \sinh (x) \]
Antiderivative was successfully verified.
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Rule 2667
Rubi steps
\begin {align*} \int \frac {\cosh ^3(x)}{i+\sinh (x)} \, dx &=-\operatorname {Subst}(\int (i-x) \, dx,x,\sinh (x))\\ &=-i \sinh (x)+\frac {\sinh ^2(x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 0.80 \[ \frac {1}{2} \sinh (x) (\sinh (x)-2 i) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 22, normalized size = 1.47 \[ \frac {1}{8} \, {\left (e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} + 4 i \, e^{x} + 1\right )} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 23, normalized size = 1.53 \[ -\frac {1}{8} \, {\left (-4 i \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} + \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{2} i \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 13, normalized size = 0.87 \[ -i \sinh \relax (x )+\frac {\left (\sinh ^{2}\relax (x )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 27, normalized size = 1.80 \[ \frac {1}{8} \, {\left (-4 i \, e^{\left (-x\right )} + 1\right )} e^{\left (2 \, x\right )} + \frac {1}{2} i \, e^{\left (-x\right )} + \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 31, normalized size = 2.07 \[ \frac {{\mathrm {e}}^{-2\,x}\,\left ({\mathrm {e}}^{4\,x}+1\right )}{8}-\frac {{\mathrm {e}}^{-2\,x}\,\left (4\,{\mathrm {e}}^{3\,x}-4\,{\mathrm {e}}^x\right )\,1{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 27, normalized size = 1.80 \[ \frac {e^{2 x}}{8} - \frac {i e^{x}}{2} + \frac {i e^{- x}}{2} + \frac {e^{- 2 x}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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