Optimal. Leaf size=20 \[ -x+\frac {2 i \cosh (x)}{1+i \sinh (x)} \]
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Rubi [A]
time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4476, 2749,
2759, 8} \begin {gather*} -x+\frac {2 i \cosh (x)}{1+i \sinh (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2749
Rule 2759
Rule 4476
Rubi steps
\begin {align*} \int (\text {sech}(x)-i \tanh (x))^2 \, dx &=\int \text {sech}^2(x) (1-i \sinh (x))^2 \, dx\\ &=\int \frac {\cosh ^2(x)}{(1+i \sinh (x))^2} \, dx\\ &=\frac {2 i \cosh (x)}{1+i \sinh (x)}-\int 1 \, dx\\ &=-x+\frac {2 i \cosh (x)}{1+i \sinh (x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 0.70 \begin {gather*} -x+2 i \text {sech}(x)+2 \tanh (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.00, size = 15, normalized size = 0.75
method | result | size |
risch | \(-x +\frac {4 i}{{\mathrm e}^{x}-i}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 25, normalized size = 1.25 \begin {gather*} -x + \frac {4 i}{e^{\left (-x\right )} + e^{x}} + \frac {4}{e^{\left (-2 \, x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 17, normalized size = 0.85 \begin {gather*} -\frac {x e^{x} - i \, x - 4 i}{e^{x} - i} \end {gather*}
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 \begin {gather*} \int \left (- i \tanh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 12, normalized size = 0.60 \begin {gather*} -x + \frac {4 i}{e^{x} - i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.56, size = 14, normalized size = 0.70 \begin {gather*} -x+\frac {4{}\mathrm {i}}{{\mathrm {e}}^x-\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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