Optimal. Leaf size=29 \[ \frac {1}{5} i \tanh ^5(x)-\frac {1}{3} i \tanh ^3(x)-\frac {1}{5} \text {sech}^5(x) \]
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Rubi [A] time = 0.12, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3872, 2839, 2606, 30, 2607, 14} \[ \frac {1}{5} i \tanh ^5(x)-\frac {1}{3} i \tanh ^3(x)-\frac {1}{5} \text {sech}^5(x) \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2606
Rule 2607
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {\text {sech}^4(x)}{i+\text {csch}(x)} \, dx &=i \int \frac {\text {sech}^3(x) \tanh (x)}{i-\sinh (x)} \, dx\\ &=-\left (i \int \text {sech}^4(x) \tanh ^2(x) \, dx\right )+\int \text {sech}^5(x) \tanh (x) \, dx\\ &=-\operatorname {Subst}\left (\int x^4 \, dx,x,\text {sech}(x)\right )+\operatorname {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,i \tanh (x)\right )\\ &=-\frac {1}{5} \text {sech}^5(x)+\operatorname {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,i \tanh (x)\right )\\ &=-\frac {1}{5} \text {sech}^5(x)-\frac {1}{3} i \tanh ^3(x)+\frac {1}{5} i \tanh ^5(x)\\ \end {align*}
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Mathematica [B] time = 0.11, size = 96, normalized size = 3.31 \[ \frac {-96 i \sinh (x)+18 i \sinh (2 x)-32 i \sinh (3 x)+9 i \sinh (4 x)+54 \cosh (x)+32 \cosh (2 x)+18 \cosh (3 x)+16 \cosh (4 x)-240}{960 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^3 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 69, normalized size = 2.38 \[ \frac {60 i \, e^{\left (4 \, x\right )} + 24 \, e^{\left (3 \, x\right )} - 8 i \, e^{\left (2 \, x\right )} + 8 \, e^{x} - 4 i}{15 \, e^{\left (8 \, x\right )} - 30 i \, e^{\left (7 \, x\right )} + 30 \, e^{\left (6 \, x\right )} - 90 i \, e^{\left (5 \, x\right )} - 90 i \, e^{\left (3 \, x\right )} - 30 \, e^{\left (2 \, x\right )} - 30 i \, e^{x} - 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 55, normalized size = 1.90 \[ -\frac {-3 i \, e^{\left (2 \, x\right )} + 12 \, e^{x} + 5 i}{24 \, {\left (i \, e^{x} - 1\right )}^{3}} + \frac {15 \, e^{\left (4 \, x\right )} - 60 i \, e^{\left (3 \, x\right )} - 10 \, e^{\left (2 \, x\right )} + 20 i \, e^{x} + 7}{120 \, {\left (e^{x} - i\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 93, normalized size = 3.21 \[ \frac {i}{6 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {4 i}{3 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}+\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )-i\right )}+\frac {2 i}{5 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{5}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{4}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 257, normalized size = 8.86 \[ \frac {32 \, e^{\left (-x\right )}}{120 i \, e^{\left (-x\right )} - 120 \, e^{\left (-2 \, x\right )} + 360 i \, e^{\left (-3 \, x\right )} + 360 i \, e^{\left (-5 \, x\right )} + 120 \, e^{\left (-6 \, x\right )} + 120 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} - 60} + \frac {32 i \, e^{\left (-2 \, x\right )}}{120 i \, e^{\left (-x\right )} - 120 \, e^{\left (-2 \, x\right )} + 360 i \, e^{\left (-3 \, x\right )} + 360 i \, e^{\left (-5 \, x\right )} + 120 \, e^{\left (-6 \, x\right )} + 120 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} - 60} + \frac {96 \, e^{\left (-3 \, x\right )}}{120 i \, e^{\left (-x\right )} - 120 \, e^{\left (-2 \, x\right )} + 360 i \, e^{\left (-3 \, x\right )} + 360 i \, e^{\left (-5 \, x\right )} + 120 \, e^{\left (-6 \, x\right )} + 120 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} - 60} - \frac {240 i \, e^{\left (-4 \, x\right )}}{120 i \, e^{\left (-x\right )} - 120 \, e^{\left (-2 \, x\right )} + 360 i \, e^{\left (-3 \, x\right )} + 360 i \, e^{\left (-5 \, x\right )} + 120 \, e^{\left (-6 \, x\right )} + 120 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} - 60} + \frac {16 i}{120 i \, e^{\left (-x\right )} - 120 \, e^{\left (-2 \, x\right )} + 360 i \, e^{\left (-3 \, x\right )} + 360 i \, e^{\left (-5 \, x\right )} + 120 \, e^{\left (-6 \, x\right )} + 120 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} - 60} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.92, size = 207, normalized size = 7.14 \[ -\frac {1{}\mathrm {i}}{4\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}+\frac {1}{20\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}-\frac {1}{8\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}+\frac {\frac {{\mathrm {e}}^{3\,x}}{40}-\frac {{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}}{40}+\frac {{\mathrm {e}}^x}{8}+\frac {1}{40}{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1-{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}+{\mathrm {e}}^x\,4{}\mathrm {i}}-\frac {\frac {{\mathrm {e}}^{2\,x}}{40}+\frac {1}{24}-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{20}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {1}{6\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}\right )}+\frac {\frac {{\mathrm {e}}^{2\,x}}{4}+\frac {{\mathrm {e}}^{4\,x}}{40}+\frac {1}{40}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{10}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{10}}{{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}-10\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{4\,x}\,5{}\mathrm {i}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x-\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{4}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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