Optimal. Leaf size=38 \[ \frac {i x}{8}+\frac {\cosh ^3(x)}{3}-\frac {1}{4} i \sinh (x) \cosh ^3(x)+\frac {1}{8} i \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.13, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3872, 2839, 2565, 30, 2568, 2635, 8} \[ \frac {i x}{8}+\frac {\cosh ^3(x)}{3}-\frac {1}{4} i \sinh (x) \cosh ^3(x)+\frac {1}{8} i \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2565
Rule 2568
Rule 2635
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {\cosh ^4(x)}{i+\text {csch}(x)} \, dx &=i \int \frac {\cosh ^4(x) \sinh (x)}{i-\sinh (x)} \, dx\\ &=-\left (i \int \cosh ^2(x) \sinh ^2(x) \, dx\right )+\int \cosh ^2(x) \sinh (x) \, dx\\ &=-\frac {1}{4} i \cosh ^3(x) \sinh (x)+\frac {1}{4} i \int \cosh ^2(x) \, dx+\operatorname {Subst}\left (\int x^2 \, dx,x,\cosh (x)\right )\\ &=\frac {\cosh ^3(x)}{3}+\frac {1}{8} i \cosh (x) \sinh (x)-\frac {1}{4} i \cosh ^3(x) \sinh (x)+\frac {1}{8} i \int 1 \, dx\\ &=\frac {i x}{8}+\frac {\cosh ^3(x)}{3}+\frac {1}{8} i \cosh (x) \sinh (x)-\frac {1}{4} i \cosh ^3(x) \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 0.84 \[ \frac {i x}{8}-\frac {1}{32} i \sinh (4 x)+\frac {\cosh (x)}{4}+\frac {1}{12} \cosh (3 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 43, normalized size = 1.13 \[ \frac {1}{192} \, {\left (24 i \, x e^{\left (4 \, x\right )} - 3 i \, e^{\left (8 \, x\right )} + 8 \, e^{\left (7 \, x\right )} + 24 \, e^{\left (5 \, x\right )} + 24 \, e^{\left (3 \, x\right )} + 8 \, e^{x} + 3 i\right )} e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 38, normalized size = 1.00 \[ \frac {1}{192} \, {\left (24 \, e^{\left (3 \, x\right )} + 8 \, e^{x} + 3 i\right )} e^{\left (-4 \, x\right )} + \frac {1}{8} i \, x - \frac {1}{64} i \, e^{\left (4 \, x\right )} + \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 170, normalized size = 4.47 \[ -\frac {i}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {i}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8}+\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 42, normalized size = 1.11 \[ \frac {1}{192} \, {\left (8 \, e^{\left (-x\right )} + 24 \, e^{\left (-3 \, x\right )} - 3 i\right )} e^{\left (4 \, x\right )} + \frac {1}{8} i \, x + \frac {1}{8} \, e^{\left (-x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {1}{64} i \, e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.55, size = 41, normalized size = 1.08 \[ \frac {{\mathrm {e}}^{-x}}{8}+\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{24}+\frac {{\mathrm {e}}^x}{8}+\frac {x\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{-4\,x}\,1{}\mathrm {i}}{64}-\frac {{\mathrm {e}}^{4\,x}\,1{}\mathrm {i}}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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