Optimal. Leaf size=31 \[ -\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {2 i \cosh (x)}{3 (i+\sinh (x))} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2829, 2727}
\begin {gather*} -\frac {2 i \cosh (x)}{3 (\sinh (x)+i)}-\frac {\cosh (x)}{3 (\sinh (x)+i)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2829
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{(i+\sinh (x))^2} \, dx &=-\frac {\cosh (x)}{3 (i+\sinh (x))^2}+\frac {2}{3} \int \frac {1}{i+\sinh (x)} \, dx\\ &=-\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {2 i \cosh (x)}{3 (i+\sinh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.71 \begin {gather*} \frac {\cosh (x) (1-2 i \sinh (x))}{3 (i+\sinh (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 25, normalized size = 0.81
method | result | size |
risch | \(-\frac {2 \left (3 i {\mathrm e}^{x}+3 \,{\mathrm e}^{2 x}-2\right )}{3 \left ({\mathrm e}^{x}+i\right )^{3}}\) | \(23\) |
default | \(-\frac {4 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 81 vs. \(2 (21) = 42\).
time = 0.30, size = 81, normalized size = 2.61 \begin {gather*} -\frac {2 i \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i} + \frac {2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i} - \frac {4}{3 \, {\left (3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 32, normalized size = 1.03 \begin {gather*} -\frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} + 3 i \, e^{x} - 2\right )}}{3 \, {\left (e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 37, normalized size = 1.19 \begin {gather*} \frac {- 6 e^{2 x} - 6 i e^{x} + 4}{3 e^{3 x} + 9 i e^{2 x} - 9 e^{x} - 3 i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 20, normalized size = 0.65 \begin {gather*} -\frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} + 3 i \, e^{x} - 2\right )}}{3 \, {\left (e^{x} + i\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 25, normalized size = 0.81 \begin {gather*} -\frac {2\,\left (3\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+2{}\mathrm {i}\right )}{3\,{\left (-1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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