Optimal. Leaf size=53 \[ \frac {e^x}{4 \left (1-e^{4 x}\right )}-\frac {e^{5 x}}{\left (1-e^{4 x}\right )^2}-\frac {1}{8} \tan ^{-1}\left (e^x\right )-\frac {1}{8} \tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2282, 12, 457, 288, 212, 206, 203} \[ \frac {e^x}{4 \left (1-e^{4 x}\right )}-\frac {e^{5 x}}{\left (1-e^{4 x}\right )^2}-\frac {1}{8} \tan ^{-1}\left (e^x\right )-\frac {1}{8} \tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 206
Rule 212
Rule 288
Rule 457
Rule 2282
Rubi steps
\begin {align*} \int e^x \coth (2 x) \text {csch}^2(2 x) \, dx &=\operatorname {Subst}\left (\int \frac {4 x^4 \left (-1-x^4\right )}{\left (1-x^4\right )^3} \, dx,x,e^x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x^4 \left (-1-x^4\right )}{\left (1-x^4\right )^3} \, dx,x,e^x\right )\\ &=-\frac {e^{5 x}}{\left (1-e^{4 x}\right )^2}+\operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^4\right )^2} \, dx,x,e^x\right )\\ &=-\frac {e^{5 x}}{\left (1-e^{4 x}\right )^2}+\frac {e^x}{4 \left (1-e^{4 x}\right )}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,e^x\right )\\ &=-\frac {e^{5 x}}{\left (1-e^{4 x}\right )^2}+\frac {e^x}{4 \left (1-e^{4 x}\right )}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^x\right )-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=-\frac {e^{5 x}}{\left (1-e^{4 x}\right )^2}+\frac {e^x}{4 \left (1-e^{4 x}\right )}-\frac {1}{8} \tan ^{-1}\left (e^x\right )-\frac {1}{8} \tanh ^{-1}\left (e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 54, normalized size = 1.02 \[ -\frac {-2 e^x+10 e^{5 x}+\left (e^{4 x}-1\right )^2 \tan ^{-1}\left (e^x\right )+\left (e^{4 x}-1\right )^2 \tanh ^{-1}\left (e^x\right )}{8 \left (e^{4 x}-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 522, normalized size = 9.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 42, normalized size = 0.79 \[ -\frac {5 \, e^{\left (5 \, x\right )} - e^{x}}{4 \, {\left (e^{\left (4 \, x\right )} - 1\right )}^{2}} - \frac {1}{8} \, \arctan \left (e^{x}\right ) - \frac {1}{16} \, \log \left (e^{x} + 1\right ) + \frac {1}{16} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.38, size = 54, normalized size = 1.02 \[ -\frac {{\mathrm e}^{x} \left (5 \,{\mathrm e}^{4 x}-1\right )}{4 \left ({\mathrm e}^{4 x}-1\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{16}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{16}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{16}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 47, normalized size = 0.89 \[ -\frac {5 \, e^{\left (5 \, x\right )} - e^{x}}{4 \, {\left (e^{\left (8 \, x\right )} - 2 \, e^{\left (4 \, x\right )} + 1\right )}} - \frac {1}{8} \, \arctan \left (e^{x}\right ) - \frac {1}{16} \, \log \left (e^{x} + 1\right ) + \frac {1}{16} \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.00, size = 80, normalized size = 1.51 \[ \frac {\ln \left (\frac {1}{4}-\frac {{\mathrm {e}}^x}{4}\right )}{16}-\frac {\ln \left (\frac {{\mathrm {e}}^x}{4}+\frac {1}{4}\right )}{16}-\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{8}-\frac {{\mathrm {e}}^{5\,x}}{2\,\left ({\mathrm {e}}^{8\,x}-2\,{\mathrm {e}}^{4\,x}+1\right )}-\frac {3\,{\mathrm {e}}^x}{4\,\left ({\mathrm {e}}^{4\,x}-1\right )}-\frac {{\mathrm {e}}^x}{2\,\left ({\mathrm {e}}^{8\,x}-2\,{\mathrm {e}}^{4\,x}+1\right )} \]
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 \[ \int e^{x} \coth {\left (2 x \right )} \operatorname {csch}^{2}{\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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