Optimal. Leaf size=36 \[ -\frac {1}{4} i \tanh ^4(x)+\frac {3}{8} \tan ^{-1}(\sinh (x))-\frac {1}{4} \tanh ^3(x) \text {sech}(x)-\frac {3}{8} \tanh (x) \text {sech}(x) \]
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Rubi [A] time = 0.09, antiderivative size = 36, 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, 2611, 3770} \[ -\frac {1}{4} i \tanh ^4(x)+\frac {3}{8} \tan ^{-1}(\sinh (x))-\frac {1}{4} \tanh ^3(x) \text {sech}(x)-\frac {3}{8} \tanh (x) \text {sech}(x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 2611
Rule 2706
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tanh ^3(x)}{i+\sinh (x)} \, dx &=-\left (i \int \text {sech}^2(x) \tanh ^3(x) \, dx\right )+\int \text {sech}(x) \tanh ^4(x) \, dx\\ &=-\frac {1}{4} \text {sech}(x) \tanh ^3(x)-i \operatorname {Subst}\left (\int x^3 \, dx,x,i \tanh (x)\right )+\frac {3}{4} \int \text {sech}(x) \tanh ^2(x) \, dx\\ &=-\frac {3}{8} \text {sech}(x) \tanh (x)-\frac {1}{4} \text {sech}(x) \tanh ^3(x)-\frac {1}{4} i \tanh ^4(x)+\frac {3}{8} \int \text {sech}(x) \, dx\\ &=\frac {3}{8} \tan ^{-1}(\sinh (x))-\frac {3}{8} \text {sech}(x) \tanh (x)-\frac {1}{4} \text {sech}(x) \tanh ^3(x)-\frac {1}{4} i \tanh ^4(x)\\ \end {align*}
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Mathematica [A] time = 0.08, size = 42, normalized size = 1.17 \[ \frac {1}{8} \left (3 \tan ^{-1}(\sinh (x))-\frac {5 \sinh ^2(x)+i \sinh (x)+2}{(\sinh (x)-i) (\sinh (x)+i)^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 152, normalized size = 4.22 \[ \frac {{\left (3 i \, e^{\left (6 \, x\right )} - 6 \, e^{\left (5 \, x\right )} + 3 i \, e^{\left (4 \, x\right )} - 12 \, e^{\left (3 \, x\right )} - 3 i \, e^{\left (2 \, x\right )} - 6 \, e^{x} - 3 i\right )} \log \left (e^{x} + i\right ) + {\left (-3 i \, e^{\left (6 \, x\right )} + 6 \, e^{\left (5 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} + 12 \, e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} + 6 \, e^{x} + 3 i\right )} \log \left (e^{x} - i\right ) - 10 \, e^{\left (5 \, x\right )} - 4 i \, e^{\left (4 \, x\right )} + 4 \, e^{\left (3 \, x\right )} + 4 i \, e^{\left (2 \, x\right )} - 10 \, e^{x}}{8 \, e^{\left (6 \, x\right )} + 16 i \, e^{\left (5 \, x\right )} + 8 \, e^{\left (4 \, x\right )} + 32 i \, e^{\left (3 \, x\right )} - 8 \, e^{\left (2 \, x\right )} + 16 i \, e^{x} - 8} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 92, normalized size = 2.56 \[ \frac {3 i \, e^{\left (-x\right )} - 3 i \, e^{x} - 2}{16 \, {\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}} - \frac {9 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, e^{\left (-x\right )} - 4 \, e^{x} + 12 i}{32 \, {\left (e^{\left (-x\right )} - e^{x} - 2 i\right )}^{2}} + \frac {3}{16} i \, \log \left (-e^{\left (-x\right )} + e^{x} + 2 i\right ) - \frac {3}{16} i \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 79, normalized size = 2.19 \[ -\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}+\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{8}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2 i}-\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )-i\right )}{8}+\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}+\frac {1}{4 \tanh \left (\frac {x}{2}\right )-4 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 95, normalized size = 2.64 \[ \frac {5 \, e^{\left (-x\right )} + 2 i \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} - 2 i \, e^{\left (-4 \, x\right )} + 5 \, e^{\left (-5 \, x\right )}}{-8 i \, e^{\left (-x\right )} - 4 \, e^{\left (-2 \, x\right )} - 16 i \, e^{\left (-3 \, x\right )} + 4 \, e^{\left (-4 \, x\right )} - 8 i \, e^{\left (-5 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - 4} + \frac {3}{8} i \, \log \left (i \, e^{\left (-x\right )} + 1\right ) - \frac {3}{8} i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 113, normalized size = 3.14 \[ \frac {3\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{4}+\frac {3{}\mathrm {i}}{2\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {1{}\mathrm {i}}{2\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}\right )}+\frac {1{}\mathrm {i}}{4\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {1}{4\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}-\frac {1}{{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {1}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.26, size = 100, normalized size = 2.78 \[ \frac {5 e^{5 x} + 2 i e^{4 x} - 2 e^{3 x} - 2 i e^{2 x} + 5 e^{x}}{- 4 e^{6 x} - 8 i e^{5 x} - 4 e^{4 x} - 16 i e^{3 x} + 4 e^{2 x} - 8 i e^{x} + 4} + \frac {3 \log {\left (e^{x} - i \right )}}{8} - \frac {3 \log {\left (e^{x} + i \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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