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 = 0.0423016, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 2282
Rule 12
Rule 457
Rule 288
Rule 212
Rule 206
Rule 203
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*}
Mathematica [A] time = 0.0910944, 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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Maple [C] time = 0.065, size = 54, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{x}} \left ( 5\,{{\rm e}^{4\,x}}-1 \right ) }{4\, \left ({{\rm e}^{4\,x}}-1 \right ) ^{2}}}-{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{16}}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{16}}+{\frac{i}{16}}\ln \left ({{\rm e}^{x}}-i \right ) -{\frac{i}{16}}\ln \left ({{\rm e}^{x}}+i \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50556, size = 63, normalized size = 1.19 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89066, size = 1750, normalized size = 33.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x} \coth{\left (2 x \right )} \operatorname{csch}^{2}{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19141, size = 57, normalized size = 1.08 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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