Integrand size = 13, antiderivative size = 14 \[ \int \frac {\cosh ^5(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{3} (i-\sinh (x))^3 \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2746, 32} \[ \int \frac {\cosh ^5(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{3} (-\sinh (x)+i)^3 \]
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Rule 32
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int (i-x)^2 \, dx,x,\sinh (x)\right ) \\ & = -\frac {1}{3} (i-\sinh (x))^3 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {\cosh ^5(x)}{(i+\sinh (x))^2} \, dx=\frac {1}{6} (-7+\cosh (2 x)-6 i \sinh (x)) \sinh (x) \]
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Time = 0.39 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86
\[-\frac {\left (i-\sinh \left (x \right )\right )^{3}}{3}\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (8) = 16\).
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.43 \[ \int \frac {\cosh ^5(x)}{(i+\sinh (x))^2} \, dx=\frac {1}{24} \, {\left (e^{\left (6 \, x\right )} - 6 i \, e^{\left (5 \, x\right )} - 15 \, e^{\left (4 \, x\right )} + 15 \, e^{\left (2 \, x\right )} - 6 i \, e^{x} - 1\right )} e^{\left (-3 \, x\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (8) = 16\).
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 3.14 \[ \int \frac {\cosh ^5(x)}{(i+\sinh (x))^2} \, dx=\frac {e^{3 x}}{24} - \frac {i e^{2 x}}{4} - \frac {5 e^{x}}{8} + \frac {5 e^{- x}}{8} - \frac {i e^{- 2 x}}{4} - \frac {e^{- 3 x}}{24} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (8) = 16\).
Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.79 \[ \int \frac {\cosh ^5(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{24} \, {\left (6 i \, e^{\left (-x\right )} + 15 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )} + \frac {5}{8} \, e^{\left (-x\right )} - \frac {1}{4} i \, e^{\left (-2 \, x\right )} - \frac {1}{24} \, e^{\left (-3 \, x\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (8) = 16\).
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.50 \[ \int \frac {\cosh ^5(x)}{(i+\sinh (x))^2} \, dx=\frac {1}{24} \, {\left (15 \, e^{\left (2 \, x\right )} - 6 i \, e^{x} - 1\right )} e^{\left (-3 \, x\right )} + \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {1}{4} i \, e^{\left (2 \, x\right )} - \frac {5}{8} \, e^{x} \]
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Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.64 \[ \int \frac {\cosh ^5(x)}{(i+\sinh (x))^2} \, dx=\frac {5\,{\mathrm {e}}^{-x}}{8}-\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{24}-\frac {5\,{\mathrm {e}}^x}{8}-\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{4}-\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{4} \]
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