Integrand size = 13, antiderivative size = 38 \[ \int \frac {\cosh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {i x}{8}+\frac {\cosh ^3(x)}{3}+\frac {1}{8} i \cosh (x) \sinh (x)-\frac {1}{4} i \cosh ^3(x) \sinh (x) \]
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Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3957, 2918, 2645, 30, 2648, 2715, 8} \[ \int \frac {\cosh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {i x}{8}+\frac {\cosh ^3(x)}{3}-\frac {1}{4} i \sinh (x) \cosh ^3(x)+\frac {1}{8} i \sinh (x) \cosh (x) \]
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Rule 8
Rule 30
Rule 2645
Rule 2648
Rule 2715
Rule 2918
Rule 3957
Rubi steps \begin{align*} \text {integral}& = i \int \frac {\cosh ^4(x) \sinh (x)}{i-\sinh (x)} \, dx \\ & = -\left (i \int \cosh ^2(x) \sinh ^2(x) \, dx\right )+\int \cosh ^2(x) \sinh (x) \, dx \\ & = -\frac {1}{4} i \cosh ^3(x) \sinh (x)+\frac {1}{4} i \int \cosh ^2(x) \, dx+\text {Subst}\left (\int x^2 \, dx,x,\cosh (x)\right ) \\ & = \frac {\cosh ^3(x)}{3}+\frac {1}{8} i \cosh (x) \sinh (x)-\frac {1}{4} i \cosh ^3(x) \sinh (x)+\frac {1}{8} i \int 1 \, dx \\ & = \frac {i x}{8}+\frac {\cosh ^3(x)}{3}+\frac {1}{8} i \cosh (x) \sinh (x)-\frac {1}{4} i \cosh ^3(x) \sinh (x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \frac {\cosh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {i x}{8}+\frac {\cosh (x)}{4}+\frac {1}{12} \cosh (3 x)-\frac {1}{32} i \sinh (4 x) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (27 ) = 54\).
Time = 0.15 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.05
\[-\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}-\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {-\frac {1}{3}-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {-\frac {1}{2}-\frac {3 i}{8}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {-\frac {1}{2}-\frac {i}{8}}{\tanh \left (\frac {x}{2}\right )-1}+\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8}+\frac {\frac {1}{3}-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\frac {1}{2}-\frac {i}{8}}{\tanh \left (\frac {x}{2}\right )+1}+\frac {-\frac {1}{2}+\frac {3 i}{8}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}\]
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.13 \[ \int \frac {\cosh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {1}{192} \, {\left (24 i \, x e^{\left (4 \, x\right )} - 3 i \, e^{\left (8 \, x\right )} + 8 \, e^{\left (7 \, x\right )} + 24 \, e^{\left (5 \, x\right )} + 24 \, e^{\left (3 \, x\right )} + 8 \, e^{x} + 3 i\right )} e^{\left (-4 \, x\right )} \]
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\[ \int \frac {\cosh ^4(x)}{i+\text {csch}(x)} \, dx=\int \frac {\cosh ^{4}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
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Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {1}{192} \, {\left (8 \, e^{\left (-x\right )} + 24 \, e^{\left (-3 \, x\right )} - 3 i\right )} e^{\left (4 \, x\right )} + \frac {1}{8} i \, x + \frac {1}{8} \, e^{\left (-x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {1}{64} i \, e^{\left (-4 \, x\right )} \]
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Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {1}{192} \, {\left (24 \, e^{\left (3 \, x\right )} + 8 \, e^{x} + 3 i\right )} e^{\left (-4 \, x\right )} + \frac {1}{8} i \, x - \frac {1}{64} i \, e^{\left (4 \, x\right )} + \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{x} \]
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Time = 2.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int \frac {\cosh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{-x}}{8}+\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{24}+\frac {{\mathrm {e}}^x}{8}+\frac {x\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{-4\,x}\,1{}\mathrm {i}}{64}-\frac {{\mathrm {e}}^{4\,x}\,1{}\mathrm {i}}{64} \]
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