Optimal. Leaf size=43 \[ \frac{1}{6} (-\sinh (x)+i)^6-\frac{4}{5} i (-\sinh (x)+i)^5-(-\sinh (x)+i)^4 \]
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Rubi [A] time = 0.0451925, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2667, 43} \[ \frac{1}{6} (-\sinh (x)+i)^6-\frac{4}{5} i (-\sinh (x)+i)^5-(-\sinh (x)+i)^4 \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cosh ^7(x)}{i+\sinh (x)} \, dx &=-\operatorname{Subst}\left (\int (i-x)^3 (i+x)^2 \, dx,x,\sinh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-4 (i-x)^3-4 i (i-x)^4+(i-x)^5\right ) \, dx,x,\sinh (x)\right )\\ &=-(i-\sinh (x))^4-\frac{4}{5} i (i-\sinh (x))^5+\frac{1}{6} (i-\sinh (x))^6\\ \end{align*}
Mathematica [A] time = 0.029949, size = 42, normalized size = 0.98 \[ \frac{1}{30} \sinh (x) \left (5 \sinh ^5(x)-6 i \sinh ^4(x)+15 \sinh ^3(x)-20 i \sinh ^2(x)+15 \sinh (x)-30 i\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 142, normalized size = 3.3 \begin{align*}{{\frac{11}{16}}-{\frac{7\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{{\frac{7}{8}}-{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}}-{{\frac{1}{2}}-{\frac{i}{5}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}-{{\frac{5}{16}}-i \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{{\frac{11}{12}}-{\frac{11\,i}{12}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-6}}+{{\frac{11}{12}}+{\frac{11\,i}{12}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}+{{\frac{11}{16}}+{\frac{7\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{{\frac{7}{8}}+{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-4}}+{{\frac{1}{2}}+{\frac{i}{5}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-5}}+{{\frac{5}{16}}+i \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{1}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.24185, size = 101, normalized size = 2.35 \begin{align*} -\frac{1}{1920} \,{\left (12 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} + 100 i \, e^{\left (-3 \, x\right )} - 75 \, e^{\left (-4 \, x\right )} + 600 i \, e^{\left (-5 \, x\right )} - 5\right )} e^{\left (6 \, x\right )} + \frac{5}{16} i \, e^{\left (-x\right )} + \frac{5}{128} \, e^{\left (-2 \, x\right )} + \frac{5}{96} i \, e^{\left (-3 \, x\right )} + \frac{1}{64} \, e^{\left (-4 \, x\right )} + \frac{1}{160} i \, e^{\left (-5 \, x\right )} + \frac{1}{384} \, e^{\left (-6 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85776, size = 240, normalized size = 5.58 \begin{align*} \frac{1}{1920} \,{\left (5 \, e^{\left (12 \, x\right )} - 12 i \, e^{\left (11 \, x\right )} + 30 \, e^{\left (10 \, x\right )} - 100 i \, e^{\left (9 \, x\right )} + 75 \, e^{\left (8 \, x\right )} - 600 i \, e^{\left (7 \, x\right )} + 600 i \, e^{\left (5 \, x\right )} + 75 \, e^{\left (4 \, x\right )} + 100 i \, e^{\left (3 \, x\right )} + 30 \, e^{\left (2 \, x\right )} + 12 i \, e^{x} + 5\right )} e^{\left (-6 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.59726, size = 100, normalized size = 2.33 \begin{align*} \frac{e^{6 x}}{384} - \frac{i e^{5 x}}{160} + \frac{e^{4 x}}{64} - \frac{5 i e^{3 x}}{96} + \frac{5 e^{2 x}}{128} - \frac{5 i e^{x}}{16} + \frac{5 i e^{- x}}{16} + \frac{5 e^{- 2 x}}{128} + \frac{5 i e^{- 3 x}}{96} + \frac{e^{- 4 x}}{64} + \frac{i e^{- 5 x}}{160} + \frac{e^{- 6 x}}{384} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18407, size = 96, normalized size = 2.23 \begin{align*} -\frac{1}{1920} \,{\left (-600 i \, e^{\left (5 \, x\right )} - 75 \, e^{\left (4 \, x\right )} - 100 i \, e^{\left (3 \, x\right )} - 30 \, e^{\left (2 \, x\right )} - 12 i \, e^{x} - 5\right )} e^{\left (-6 \, x\right )} + \frac{1}{384} \, e^{\left (6 \, x\right )} - \frac{1}{160} i \, e^{\left (5 \, x\right )} + \frac{1}{64} \, e^{\left (4 \, x\right )} - \frac{5}{96} i \, e^{\left (3 \, x\right )} + \frac{5}{128} \, e^{\left (2 \, x\right )} - \frac{5}{16} i \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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