Optimal. Leaf size=58 \[ -\frac {7 x}{2}-\frac {16}{3} i \cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{3 (\sinh (x)+i)^2}-\frac {8 \sinh ^2(x) \cosh (x)}{3 (\sinh (x)+i)}+\frac {7}{2} \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.10, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2765, 2977, 2734} \[ -\frac {7 x}{2}-\frac {16}{3} i \cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{3 (\sinh (x)+i)^2}-\frac {8 \sinh ^2(x) \cosh (x)}{3 (\sinh (x)+i)}+\frac {7}{2} \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2765
Rule 2977
Rubi steps
\begin {align*} \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx &=-\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}+\frac {1}{3} \int \frac {\sinh ^2(x) (-3 i+5 \sinh (x))}{i+\sinh (x)} \, dx\\ &=-\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}-\frac {8 \cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))}-\frac {1}{3} i \int (16+21 i \sinh (x)) \sinh (x) \, dx\\ &=-\frac {7 x}{2}-\frac {16}{3} i \cosh (x)+\frac {7}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}-\frac {8 \cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))}\\ \end {align*}
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Mathematica [B] time = 0.19, size = 147, normalized size = 2.53 \[ -\frac {\sinh ^3(x) \cosh (x)}{2 (1-i \sinh (x))^2}-\frac {i \sqrt {2} \sqrt {1+\frac {1}{2} (-1+i \sinh (x))} \cosh (x)}{\sqrt {1+i \sinh (x)}}-\frac {31 i \cosh (x)}{6 (1-i \sinh (x))}+\frac {5 i \cosh (x)}{6 (1-i \sinh (x))^2}-\frac {7 i \cosh (x) \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {1-i \sinh (x)} \sqrt {1+i \sinh (x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 91, normalized size = 1.57 \[ -\frac {21 \, {\left (4 \, x - 3\right )} e^{\left (5 \, x\right )} - {\left (-252 i \, x - 147 i\right )} e^{\left (4 \, x\right )} - 3 \, {\left (84 \, x + 127\right )} e^{\left (3 \, x\right )} - {\left (84 i \, x + 239 i\right )} e^{\left (2 \, x\right )} - 3 \, e^{\left (7 \, x\right )} + 15 i \, e^{\left (6 \, x\right )} + 15 \, e^{x} - 3 i}{24 \, e^{\left (5 \, x\right )} + 72 i \, e^{\left (4 \, x\right )} - 72 \, e^{\left (3 \, x\right )} - 24 i \, e^{\left (2 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 50, normalized size = 0.86 \[ -\frac {7}{2} \, x - \frac {{\left (216 i \, e^{\left (4 \, x\right )} - 405 \, e^{\left (3 \, x\right )} - 239 i \, e^{\left (2 \, x\right )} + 15 \, e^{x} - 3 i\right )} e^{\left (-2 \, x\right )}}{24 \, {\left (e^{x} + i\right )}^{3}} + \frac {1}{8} \, e^{\left (2 \, x\right )} - i \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 116, normalized size = 2.00 \[ \frac {1}{2 \tanh \left (\frac {x}{2}\right )-2}+\frac {2 i}{\tanh \left (\frac {x}{2}\right )-1}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2}-\frac {2 i}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {6}{\tanh \left (\frac {x}{2}\right )+i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 71, normalized size = 1.22 \[ -\frac {7}{2} \, x + \frac {30 \, e^{\left (-x\right )} + 478 i \, e^{\left (-2 \, x\right )} - 810 \, e^{\left (-3 \, x\right )} - 432 i \, e^{\left (-4 \, x\right )} + 6 i}{16 \, {\left (3 i \, e^{\left (-2 \, x\right )} - 9 \, e^{\left (-3 \, x\right )} - 9 i \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}\right )}} - i \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 97, normalized size = 1.67 \[ \frac {{\mathrm {e}}^{2\,x}}{8}-{\mathrm {e}}^{-x}\,1{}\mathrm {i}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {7\,x}{2}-{\mathrm {e}}^x\,1{}\mathrm {i}-\frac {-2+\frac {{\mathrm {e}}^x\,8{}\mathrm {i}}{3}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}+\frac {4\,{\mathrm {e}}^x-\frac {{\mathrm {e}}^{2\,x}\,8{}\mathrm {i}}{3}+\frac {8}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {8{}\mathrm {i}}{3\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 68, normalized size = 1.17 \[ - \frac {7 x}{2} + \frac {24 e^{2 x} + 42 i e^{x} - 22}{3 i e^{3 x} - 9 e^{2 x} - 9 i e^{x} + 3} + \frac {e^{2 x}}{8} - i e^{x} - i e^{- x} - \frac {e^{- 2 x}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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