Optimal. Leaf size=52 \[ -i x+\frac {1}{5} \tanh ^5(x) (-\text {csch}(x)+i)+\frac {1}{15} \tanh ^3(x) (-4 \text {csch}(x)+5 i)+\frac {1}{15} \tanh (x) (-8 \text {csch}(x)+15 i) \]
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Rubi [A] time = 0.09, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3888, 3882, 8} \[ -i x+\frac {1}{5} \tanh ^5(x) (-\text {csch}(x)+i)+\frac {1}{15} \tanh ^3(x) (-4 \text {csch}(x)+5 i)+\frac {1}{15} \tanh (x) (-8 \text {csch}(x)+15 i) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3882
Rule 3888
Rubi steps
\begin {align*} \int \frac {\tanh ^4(x)}{i+\text {csch}(x)} \, dx &=\int (-i+\text {csch}(x)) \tanh ^6(x) \, dx\\ &=\frac {1}{5} (i-\text {csch}(x)) \tanh ^5(x)-\frac {1}{5} \int (5 i-4 \text {csch}(x)) \tanh ^4(x) \, dx\\ &=\frac {1}{15} (5 i-4 \text {csch}(x)) \tanh ^3(x)+\frac {1}{5} (i-\text {csch}(x)) \tanh ^5(x)+\frac {1}{15} \int (-15 i+8 \text {csch}(x)) \tanh ^2(x) \, dx\\ &=\frac {1}{15} (15 i-8 \text {csch}(x)) \tanh (x)+\frac {1}{15} (5 i-4 \text {csch}(x)) \tanh ^3(x)+\frac {1}{5} (i-\text {csch}(x)) \tanh ^5(x)-\frac {1}{15} \int 15 i \, dx\\ &=-i x+\frac {1}{15} (15 i-8 \text {csch}(x)) \tanh (x)+\frac {1}{15} (5 i-4 \text {csch}(x)) \tanh ^3(x)+\frac {1}{5} (i-\text {csch}(x)) \tanh ^5(x)\\ \end {align*}
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Mathematica [B] time = 0.14, size = 126, normalized size = 2.42 \[ \frac {64 i \sinh (x)+240 x \sinh (2 x)+178 i \sinh (2 x)+128 i \sinh (3 x)+120 x \sinh (4 x)+89 i \sinh (4 x)+6 (89-120 i x) \cosh (x)-128 \cosh (2 x)-240 i x \cosh (3 x)+178 \cosh (3 x)-184 \cosh (4 x)-200}{960 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^3 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 123, normalized size = 2.37 \[ \frac {-15 i \, x e^{\left (8 \, x\right )} - 30 \, {\left (x + 1\right )} e^{\left (7 \, x\right )} + {\left (-30 i \, x - 30 i\right )} e^{\left (6 \, x\right )} - 10 \, {\left (9 \, x + 13\right )} e^{\left (5 \, x\right )} - 2 \, {\left (45 \, x + 73\right )} e^{\left (3 \, x\right )} + {\left (30 i \, x + 62 i\right )} e^{\left (2 \, x\right )} - 2 \, {\left (15 \, x + 31\right )} e^{x} + 15 i \, x + 50 i \, e^{\left (4 \, x\right )} + 46 i}{15 \, e^{\left (8 \, x\right )} - 30 i \, e^{\left (7 \, x\right )} + 30 \, e^{\left (6 \, x\right )} - 90 i \, e^{\left (5 \, x\right )} - 90 i \, e^{\left (3 \, x\right )} - 30 \, e^{\left (2 \, x\right )} - 30 i \, e^{x} - 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 62, normalized size = 1.19 \[ -\frac {21 i \, e^{\left (2 \, x\right )} - 36 \, e^{x} - 19 i}{24 \, {\left (i \, e^{x} - 1\right )}^{3}} - \frac {115 \, e^{\left (4 \, x\right )} - 380 i \, e^{\left (3 \, x\right )} - 530 \, e^{\left (2 \, x\right )} + 340 i \, e^{x} + 91}{40 \, {\left (e^{x} - i\right )}^{5}} - i \, \log \left (i \, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.25, size = 99, normalized size = 1.90 \[ i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {5 i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}+\frac {i}{6 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {11 i}{8 \left (\tanh \left (\frac {x}{2}\right )-i\right )}+\frac {2 i}{5 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{5}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{4}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 96, normalized size = 1.85 \[ -i \, x - \frac {62 \, e^{\left (-x\right )} + 62 i \, e^{\left (-2 \, x\right )} + 146 \, e^{\left (-3 \, x\right )} + 50 i \, e^{\left (-4 \, x\right )} + 130 \, e^{\left (-5 \, x\right )} - 30 i \, e^{\left (-6 \, x\right )} + 30 \, e^{\left (-7 \, x\right )} + 46 i}{30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} + 90 i \, e^{\left (-3 \, x\right )} + 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} + 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 237, normalized size = 4.56 \[ -x\,1{}\mathrm {i}-\frac {1{}\mathrm {i}}{4\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}+\frac {\frac {23\,{\mathrm {e}}^x}{40}-\frac {3}{8}{}\mathrm {i}}{1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {23}{40\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}+\frac {7}{8\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}+\frac {\frac {{\mathrm {e}}^{2\,x}\,9{}\mathrm {i}}{8}-\frac {23\,{\mathrm {e}}^{3\,x}}{40}+\frac {9\,{\mathrm {e}}^x}{8}-\frac {3}{8}{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1-{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}+{\mathrm {e}}^x\,4{}\mathrm {i}}-\frac {\frac {3}{8}-\frac {23\,{\mathrm {e}}^{2\,x}}{40}+\frac {{\mathrm {e}}^x\,3{}\mathrm {i}}{4}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {1}{6\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}\right )}-\frac {\frac {23\,{\mathrm {e}}^{4\,x}}{40}-\frac {9\,{\mathrm {e}}^{2\,x}}{4}+\frac {23}{40}-\frac {{\mathrm {e}}^{3\,x}\,3{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^x\,3{}\mathrm {i}}{2}}{{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}-10\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{4\,x}\,5{}\mathrm {i}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x-\mathrm {i}} \]
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 \[ \int \frac {\tanh ^{4}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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