Integrand size = 13, antiderivative size = 60 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{4} \arctan (\sinh (x))+\frac {1}{16 (i-\sinh (x))}+\frac {1}{12 (i+\sinh (x))^3}-\frac {i}{8 (i+\sinh (x))^2}-\frac {3}{16 (i+\sinh (x))} \]
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Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2746, 46, 209} \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{4} \arctan (\sinh (x))+\frac {1}{16 (-\sinh (x)+i)}-\frac {3}{16 (\sinh (x)+i)}-\frac {i}{8 (\sinh (x)+i)^2}+\frac {1}{12 (\sinh (x)+i)^3} \]
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Rule 46
Rule 209
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{(i-x)^2 (i+x)^4} \, dx,x,\sinh (x)\right ) \\ & = \text {Subst}\left (\int \left (\frac {1}{16 (-i+x)^2}-\frac {1}{4 (i+x)^4}+\frac {i}{4 (i+x)^3}+\frac {3}{16 (i+x)^2}-\frac {1}{4 \left (1+x^2\right )}\right ) \, dx,x,\sinh (x)\right ) \\ & = \frac {1}{16 (i-\sinh (x))}+\frac {1}{12 (i+\sinh (x))^3}-\frac {i}{8 (i+\sinh (x))^2}-\frac {3}{16 (i+\sinh (x))}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (x)\right ) \\ & = -\frac {1}{4} \arctan (\sinh (x))+\frac {1}{16 (i-\sinh (x))}+\frac {1}{12 (i+\sinh (x))^3}-\frac {i}{8 (i+\sinh (x))^2}-\frac {3}{16 (i+\sinh (x))} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.13 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {\text {sech}^2(x) \left (4 i-3 \arctan (\sinh (x))+(-1+6 i \arctan (\sinh (x))) \sinh (x)+6 i \sinh ^2(x)+(3+6 i \arctan (\sinh (x))) \sinh ^3(x)+3 \arctan (\sinh (x)) \sinh ^4(x)\right )}{12 (i+\sinh (x))^2} \]
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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. \(2 (45 ) = 90\).
Time = 0.62 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.93
\[\frac {i}{8 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{2}}+\frac {i \ln \left (-i+\tanh \left (\frac {x}{2}\right )\right )}{4}+\frac {1}{-8 i+8 \tanh \left (\frac {x}{2}\right )}+\frac {7 i}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}-\frac {2 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{6}}-\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{4}-\frac {23 i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}-\frac {11}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {11}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (38) = 76\).
Time = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 3.35 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {3 \, {\left (i \, e^{\left (8 \, x\right )} - 4 \, e^{\left (7 \, x\right )} - 4 i \, e^{\left (6 \, x\right )} - 4 \, e^{\left (5 \, x\right )} - 10 i \, e^{\left (4 \, x\right )} + 4 \, e^{\left (3 \, x\right )} - 4 i \, e^{\left (2 \, x\right )} + 4 \, e^{x} + i\right )} \log \left (e^{x} + i\right ) + 3 \, {\left (-i \, e^{\left (8 \, x\right )} + 4 \, e^{\left (7 \, x\right )} + 4 i \, e^{\left (6 \, x\right )} + 4 \, e^{\left (5 \, x\right )} + 10 i \, e^{\left (4 \, x\right )} - 4 \, e^{\left (3 \, x\right )} + 4 i \, e^{\left (2 \, x\right )} - 4 \, e^{x} - i\right )} \log \left (e^{x} - i\right ) + 6 \, e^{\left (7 \, x\right )} + 24 i \, e^{\left (6 \, x\right )} - 26 \, e^{\left (5 \, x\right )} + 16 i \, e^{\left (4 \, x\right )} + 26 \, e^{\left (3 \, x\right )} + 24 i \, e^{\left (2 \, x\right )} - 6 \, e^{x}}{12 \, {\left (e^{\left (8 \, x\right )} + 4 i \, e^{\left (7 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 4 i \, e^{\left (5 \, x\right )} - 10 \, e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - 4 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} + 1\right )}} \]
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\[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=\int \frac {\operatorname {sech}^{3}{\left (x \right )}}{\left (\sinh {\left (x \right )} + i\right )^{2}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (38) = 76\).
Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.00 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {8 \, {\left (3 \, e^{\left (-x\right )} + 12 i \, e^{\left (-2 \, x\right )} - 13 \, e^{\left (-3 \, x\right )} + 8 i \, e^{\left (-4 \, x\right )} + 13 \, e^{\left (-5 \, x\right )} + 12 i \, e^{\left (-6 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}\right )}}{192 i \, e^{\left (-x\right )} - 192 \, e^{\left (-2 \, x\right )} + 192 i \, e^{\left (-3 \, x\right )} - 480 \, e^{\left (-4 \, x\right )} - 192 i \, e^{\left (-5 \, x\right )} - 192 \, e^{\left (-6 \, x\right )} - 192 i \, e^{\left (-7 \, x\right )} + 48 \, e^{\left (-8 \, x\right )} + 48} - \frac {1}{4} i \, \log \left (i \, e^{\left (-x\right )} + 1\right ) + \frac {1}{4} i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (38) = 76\).
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.75 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=\frac {-i \, e^{\left (-x\right )} + i \, e^{x} + 3}{8 \, {\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}} + \frac {11 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 84 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 228 i \, e^{\left (-x\right )} + 228 i \, e^{x} - 240}{48 \, {\left (e^{\left (-x\right )} - e^{x} - 2 i\right )}^{3}} - \frac {1}{8} i \, \log \left (-e^{\left (-x\right )} + e^{x} + 2 i\right ) + \frac {1}{8} i \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \]
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Time = 1.85 (sec) , antiderivative size = 198, normalized size of antiderivative = 3.30 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{2}-\frac {2}{{\mathrm {e}}^{5\,x}-10\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\,5{}\mathrm {i}-{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}+5\,{\mathrm {e}}^x+1{}\mathrm {i}}-\frac {1{}\mathrm {i}}{8\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {3{}\mathrm {i}}{2\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}\right )}+\frac {1{}\mathrm {i}}{8\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {1}{8\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}-\frac {3}{8\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}+\frac {2{}\mathrm {i}}{3\,\left (15\,{\mathrm {e}}^{2\,x}-15\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1-{\mathrm {e}}^{3\,x}\,20{}\mathrm {i}+{\mathrm {e}}^{5\,x}\,6{}\mathrm {i}+{\mathrm {e}}^x\,6{}\mathrm {i}\right )}-\frac {1}{3\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}\right )} \]
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