Optimal. Leaf size=20 \[ \frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2282, 12, 264} \[ \frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 2282
Rubi steps
\begin {align*} \int e^{2 x} \text {csch}^4(x) \, dx &=\operatorname {Subst}\left (\int \frac {16 x^5}{\left (1-x^2\right )^4} \, dx,x,e^x\right )\\ &=16 \operatorname {Subst}\left (\int \frac {x^5}{\left (1-x^2\right )^4} \, dx,x,e^x\right )\\ &=\frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ \frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{2 x} \text {csch}^4(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.31, size = 75, normalized size = 3.75 \[ -\frac {8 \, {\left (4 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 4 \, \sinh \relax (x)^{2} - 3\right )}}{3 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 2\right )} \sinh \relax (x)^{2} - 4 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 24, normalized size = 1.20 \[ -\frac {8 \, {\left (3 \, e^{\left (4 \, x\right )} - 3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 25, normalized size = 1.25
method | result | size |
risch | \(-\frac {8 \left (3 \,{\mathrm e}^{4 x}-3 \,{\mathrm e}^{2 x}+1\right )}{3 \left (-1+{\mathrm e}^{2 x}\right )^{3}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 22, normalized size = 1.10 \[ \frac {8}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 24, normalized size = 1.20 \[ -\frac {8\,\left (3\,{\mathrm {e}}^{4\,x}-3\,{\mathrm {e}}^{2\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{2 x}}{\sinh ^{4}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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