Optimal. Leaf size=36 \[ \frac {3 i x}{2}+2 \cosh (x)-\frac {3}{2} i \sinh (x) \cosh (x)-\frac {\sinh (x) \cosh (x)}{\text {csch}(x)+i} \]
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Rubi [A] time = 0.06, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3819, 3787, 2635, 8, 2638} \[ \frac {3 i x}{2}+2 \cosh (x)-\frac {3}{2} i \sinh (x) \cosh (x)-\frac {\sinh (x) \cosh (x)}{\text {csch}(x)+i} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 3787
Rule 3819
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{i+\text {csch}(x)} \, dx &=-\frac {\cosh (x) \sinh (x)}{i+\text {csch}(x)}+\int (-3 i+2 \text {csch}(x)) \sinh ^2(x) \, dx\\ &=-\frac {\cosh (x) \sinh (x)}{i+\text {csch}(x)}-3 i \int \sinh ^2(x) \, dx+2 \int \sinh (x) \, dx\\ &=2 \cosh (x)-\frac {3}{2} i \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh (x)}{i+\text {csch}(x)}+\frac {3}{2} i \int 1 \, dx\\ &=\frac {3 i x}{2}+2 \cosh (x)-\frac {3}{2} i \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh (x)}{i+\text {csch}(x)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 46, normalized size = 1.28 \[ \cosh (x)+\frac {1}{4} i \left (6 x-\sinh (2 x)-\frac {8 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 55, normalized size = 1.53 \[ \frac {{\left (12 i \, x - 4 i\right )} e^{\left (3 \, x\right )} + 4 \, {\left (3 \, x + 5\right )} e^{\left (2 \, x\right )} - i \, e^{\left (5 \, x\right )} + 3 \, e^{\left (4 \, x\right )} - 3 i \, e^{x} + 1}{8 \, e^{\left (3 \, x\right )} - 8 i \, e^{\left (2 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 40, normalized size = 1.11 \[ \frac {3}{2} i \, x + \frac {{\left (-20 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} e^{\left (-2 \, x\right )}}{8 \, {\left (-i \, e^{x} - 1\right )}} - \frac {1}{8} i \, e^{\left (2 \, x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 96, normalized size = 2.67 \[ -\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {2 i}{\tanh \left (\frac {x}{2}\right )-i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 47, normalized size = 1.31 \[ \frac {3}{2} i \, x + \frac {3 i \, e^{\left (-x\right )} + 20 \, e^{\left (-2 \, x\right )} + 1}{4 \, {\left (2 i \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )}\right )}} + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{8} i \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 38, normalized size = 1.06 \[ \frac {x\,3{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^x}{2}+\frac {2}{{\mathrm {e}}^x-\mathrm {i}} \]
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 \[ \int \frac {\sinh ^{2}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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