Optimal. Leaf size=31 \[ -\frac {1}{5} i \tanh ^5(x)-\frac {1}{5} \text {sech}^5(x)+\frac {2 \text {sech}^3(x)}{3}-\text {sech}(x) \]
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Rubi [A] time = 0.08, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2706, 2607, 30, 2606, 194} \[ -\frac {1}{5} i \tanh ^5(x)-\frac {1}{5} \text {sech}^5(x)+\frac {2 \text {sech}^3(x)}{3}-\text {sech}(x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 194
Rule 2606
Rule 2607
Rule 2706
Rubi steps
\begin {align*} \int \frac {\tanh ^4(x)}{i+\sinh (x)} \, dx &=-\left (i \int \text {sech}^2(x) \tanh ^4(x) \, dx\right )+\int \text {sech}(x) \tanh ^5(x) \, dx\\ &=-\operatorname {Subst}\left (\int x^4 \, dx,x,i \tanh (x)\right )-\operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\text {sech}(x)\right )\\ &=-\frac {1}{5} i \tanh ^5(x)-\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\text {sech}(x)\right )\\ &=-\text {sech}(x)+\frac {2 \text {sech}^3(x)}{3}-\frac {\text {sech}^5(x)}{5}-\frac {1}{5} i \tanh ^5(x)\\ \end {align*}
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Mathematica [B] time = 0.14, size = 96, normalized size = 3.10 \[ -\frac {64 i \sinh (x)+178 i \sinh (2 x)-192 i \sinh (3 x)+89 i \sinh (4 x)-534 \cosh (x)+288 \cosh (2 x)-178 \cosh (3 x)+24 \cosh (4 x)+200}{960 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^5 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 88, normalized size = 2.84 \[ -\frac {30 \, e^{\left (7 \, x\right )} + 30 i \, e^{\left (6 \, x\right )} + 10 \, e^{\left (5 \, x\right )} + 50 i \, e^{\left (4 \, x\right )} + 26 \, e^{\left (3 \, x\right )} + 42 i \, e^{\left (2 \, x\right )} - 18 \, e^{x} + 6 i}{15 \, e^{\left (8 \, x\right )} + 30 i \, e^{\left (7 \, x\right )} + 30 \, e^{\left (6 \, x\right )} + 90 i \, e^{\left (5 \, x\right )} + 90 i \, e^{\left (3 \, x\right )} - 30 \, e^{\left (2 \, x\right )} + 30 i \, e^{x} - 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 53, normalized size = 1.71 \[ -\frac {15 \, e^{\left (2 \, x\right )} - 24 i \, e^{x} - 13}{24 \, {\left (e^{x} - i\right )}^{3}} - \frac {165 \, e^{\left (4 \, x\right )} + 480 i \, e^{\left (3 \, x\right )} - 650 \, e^{\left (2 \, x\right )} - 400 i \, e^{x} + 113}{120 \, {\left (e^{x} + i\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 93, normalized size = 3.00 \[ \frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {2 i}{5 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}-\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )-i\right )}+\frac {i}{6 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 413, normalized size = 13.32 \[ \frac {18 \, e^{\left (-x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {42 i \, e^{\left (-2 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} - \frac {26 \, e^{\left (-3 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {50 i \, e^{\left (-4 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} - \frac {10 \, e^{\left (-5 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {30 i \, e^{\left (-6 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} - \frac {30 \, e^{\left (-7 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {6 i}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 231, normalized size = 7.45 \[ -\frac {1}{6\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}\right )}-\frac {\frac {11\,{\mathrm {e}}^x}{40}+\frac {1}{8}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {\frac {11\,{\mathrm {e}}^{2\,x}}{40}-\frac {17}{120}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{4}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}+\frac {1{}\mathrm {i}}{4\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {5}{8\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}-\frac {11}{40\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}-\frac {\frac {{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}}{8}+\frac {11\,{\mathrm {e}}^{3\,x}}{40}-\frac {17\,{\mathrm {e}}^x}{40}-\frac {1}{8}{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}}-\frac {\frac {11\,{\mathrm {e}}^{4\,x}}{40}-\frac {17\,{\mathrm {e}}^{2\,x}}{20}+\frac {11}{40}+\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}}{{\mathrm {e}}^{5\,x}-10\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\,5{}\mathrm {i}-{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}+5\,{\mathrm {e}}^x+1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.29, size = 107, normalized size = 3.45 \[ \frac {30 e^{7 x} + 30 i e^{6 x} + 10 e^{5 x} + 50 i e^{4 x} + 26 e^{3 x} + 42 i e^{2 x} - 18 e^{x} + 6 i}{- 15 e^{8 x} - 30 i e^{7 x} - 30 e^{6 x} - 90 i e^{5 x} - 90 i e^{3 x} + 30 e^{2 x} - 30 i e^{x} + 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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