Integrand size = 13, antiderivative size = 49 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {i \text {sech}^3(x)}{7 (i+\sinh (x))^2}-\frac {\text {sech}^3(x)}{7 (i+\sinh (x))}-\frac {4 \tanh (x)}{7}+\frac {4 \tanh ^3(x)}{21} \]
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Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2751, 3852} \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=\frac {4 \tanh ^3(x)}{21}-\frac {4 \tanh (x)}{7}-\frac {\text {sech}^3(x)}{7 (\sinh (x)+i)}-\frac {i \text {sech}^3(x)}{7 (\sinh (x)+i)^2} \]
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Rule 2751
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {sech}^3(x)}{7 (i+\sinh (x))^2}-\frac {5}{7} i \int \frac {\text {sech}^4(x)}{i+\sinh (x)} \, dx \\ & = -\frac {i \text {sech}^3(x)}{7 (i+\sinh (x))^2}-\frac {\text {sech}^3(x)}{7 (i+\sinh (x))}-\frac {4}{7} \int \text {sech}^4(x) \, dx \\ & = -\frac {i \text {sech}^3(x)}{7 (i+\sinh (x))^2}-\frac {\text {sech}^3(x)}{7 (i+\sinh (x))}-\frac {4}{7} i \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (x)\right ) \\ & = -\frac {i \text {sech}^3(x)}{7 (i+\sinh (x))^2}-\frac {\text {sech}^3(x)}{7 (i+\sinh (x))}-\frac {4 \tanh (x)}{7}+\frac {4 \tanh ^3(x)}{21} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.96 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {\text {sech}^3(x) (8 i \cosh (2 x)+4 i \cosh (4 x)-14 \sinh (x)-3 \sinh (3 x)+\sinh (5 x))}{42 (i+\sinh (x))^2} \]
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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. \(2 (38 ) = 76\).
Time = 0.41 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.37
\[-\frac {i}{8 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{2}}+\frac {1}{12 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{3}}-\frac {3}{8 \left (-i+\tanh \left (\frac {x}{2}\right )\right )}+\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{6}}-\frac {5 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}+\frac {23 i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {4}{7 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{7}}-\frac {4}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}+\frac {55}{12 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {13}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (35) = 70\).
Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.63 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {16 \, {\left (14 \, e^{\left (4 \, x\right )} + 8 i \, e^{\left (3 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 4 i \, e^{x} - 1\right )}}{21 \, {\left (e^{\left (10 \, x\right )} + 4 i \, e^{\left (9 \, x\right )} - 3 \, e^{\left (8 \, x\right )} + 8 i \, e^{\left (7 \, x\right )} - 14 \, e^{\left (6 \, x\right )} - 14 \, e^{\left (4 \, x\right )} - 8 i \, e^{\left (3 \, x\right )} - 3 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} + 1\right )}} \]
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\[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=\int \frac {\operatorname {sech}^{4}{\left (x \right )}}{\left (\sinh {\left (x \right )} + i\right )^{2}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (35) = 70\).
Time = 0.21 (sec) , antiderivative size = 317, normalized size of antiderivative = 6.47 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {64 i \, e^{\left (-x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} + \frac {48 \, e^{\left (-2 \, x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} - \frac {128 i \, e^{\left (-3 \, x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} + \frac {224 \, e^{\left (-4 \, x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} - \frac {16}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} \]
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Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.33 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {6 i \, e^{\left (2 \, x\right )} + 15 \, e^{x} - 7 i}{24 \, {\left (e^{x} - i\right )}^{3}} - \frac {-42 i \, e^{\left (6 \, x\right )} + 315 \, e^{\left (5 \, x\right )} + 1015 i \, e^{\left (4 \, x\right )} - 1750 \, e^{\left (3 \, x\right )} - 1344 i \, e^{\left (2 \, x\right )} + 511 \, e^{x} + 79 i}{168 \, {\left (e^{x} + i\right )}^{7}} \]
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Time = 0.53 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.84 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=\frac {\left (4\,{\mathrm {e}}^{3\,x}-4\,{\mathrm {e}}^x\right )\,\left (\frac {16\,{\mathrm {e}}^{2\,x}}{7}+\frac {32\,{\mathrm {e}}^{4\,x}}{3}-\frac {16}{21}\right )\,1{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^7}-\frac {\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1\right )\,\left (\frac {16\,{\mathrm {e}}^{2\,x}}{7}+\frac {32\,{\mathrm {e}}^{4\,x}}{3}-\frac {16}{21}\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^7}-\frac {\left (4\,{\mathrm {e}}^{3\,x}-4\,{\mathrm {e}}^x\right )\,\left (\frac {128\,{\mathrm {e}}^{3\,x}}{21}+\frac {64\,{\mathrm {e}}^x}{21}\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^7}-\frac {\left (\frac {128\,{\mathrm {e}}^{3\,x}}{21}+\frac {64\,{\mathrm {e}}^x}{21}\right )\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1\right )\,1{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^7} \]
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