Optimal. Leaf size=14 \[ -\frac {1}{3} (i-\sinh (x))^3 \]
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Rubi [A]
time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2746, 32}
\begin {gather*} -\frac {1}{3} (-\sinh (x)+i)^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cosh ^5(x)}{(i+\sinh (x))^2} \, dx &=\text {Subst}\left (\int (i-x)^2 \, dx,x,\sinh (x)\right )\\ &=-\frac {1}{3} (i-\sinh (x))^3\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.79 \begin {gather*} -\frac {1}{2} i \cosh (2 x)-\frac {5 \sinh (x)}{4}+\frac {1}{12} \sinh (3 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 69 vs. \(2 (11 ) = 22\).
time = 0.56, size = 70, normalized size = 5.00
method | result | size |
risch | \(\frac {{\mathrm e}^{3 x}}{24}-\frac {i {\mathrm e}^{2 x}}{4}-\frac {5 \,{\mathrm e}^{x}}{8}+\frac {5 \,{\mathrm e}^{-x}}{8}-\frac {i {\mathrm e}^{-2 x}}{4}-\frac {{\mathrm e}^{-3 x}}{24}\) | \(38\) |
default | \(\frac {\frac {1}{2}-i}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1+i}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1-i}{\tanh \left (\frac {x}{2}\right )-1}+\frac {-\frac {1}{2}-i}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}\) | \(70\) |
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. 39 vs. \(2 (8) = 16\).
time = 0.28, size = 39, normalized size = 2.79 \begin {gather*} -\frac {1}{24} \, {\left (6 i \, e^{\left (-x\right )} + 15 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )} + \frac {5}{8} \, e^{\left (-x\right )} - \frac {1}{4} i \, e^{\left (-2 \, x\right )} - \frac {1}{24} \, e^{\left (-3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 34 vs. \(2 (8) = 16\).
time = 0.37, size = 34, normalized size = 2.43 \begin {gather*} \frac {1}{24} \, {\left (e^{\left (6 \, x\right )} - 6 i \, e^{\left (5 \, x\right )} - 15 \, e^{\left (4 \, x\right )} + 15 \, e^{\left (2 \, x\right )} - 6 i \, e^{x} - 1\right )} e^{\left (-3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 44 vs. \(2 (8) = 16\).
time = 0.08, size = 44, normalized size = 3.14 \begin {gather*} \frac {e^{3 x}}{24} - \frac {i e^{2 x}}{4} - \frac {5 e^{x}}{8} + \frac {5 e^{- x}}{8} - \frac {i e^{- 2 x}}{4} - \frac {e^{- 3 x}}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 35 vs. \(2 (8) = 16\).
time = 0.42, size = 35, normalized size = 2.50 \begin {gather*} \frac {1}{24} \, {\left (15 \, e^{\left (2 \, x\right )} - 6 i \, e^{x} - 1\right )} e^{\left (-3 \, x\right )} + \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {1}{4} i \, e^{\left (2 \, x\right )} - \frac {5}{8} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 37, normalized size = 2.64 \begin {gather*} \frac {5\,{\mathrm {e}}^{-x}}{8}-\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{24}-\frac {5\,{\mathrm {e}}^x}{8}-\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{4}-\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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