Optimal. Leaf size=20 \[ \frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]
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Rubi [A]
time = 0.01, 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 = {2320, 12, 270}
\begin {gather*} \frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 2320
Rubi steps
\begin {align*} \int e^{2 x} \text {csch}^4(x) \, dx &=\text {Subst}\left (\int \frac {16 x^5}{\left (1-x^2\right )^4} \, dx,x,e^x\right )\\ &=16 \text {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 \begin {gather*} \frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 20, normalized size = 1.00
method | result | size |
default | \(-\frac {1}{\tanh \left (x \right )}-\frac {1}{3 \tanh \left (x \right )^{3}}-\frac {1}{\tanh \left (x \right )^{2}}\) | \(20\) |
risch | \(-\frac {8 \left (3 \,{\mathrm e}^{4 x}-3 \,{\mathrm e}^{2 x}+1\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{3}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.93, size = 22, normalized size = 1.10 \begin {gather*} \frac {8}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (14) = 28\).
time = 0.94, size = 75, normalized size = 3.75 \begin {gather*} -\frac {8 \, {\left (4 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sinh \left (x\right )^{2} - 3\right )}}{3 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 2\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\right )}} \end {gather*}
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 \begin {gather*} \int \frac {e^{2 x}}{\sinh ^{4}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 24, normalized size = 1.20 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} -\frac {8\,\left (3\,{\mathrm {e}}^{4\,x}-3\,{\mathrm {e}}^{2\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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