Optimal. Leaf size=55 \[ \frac{3 e^{3 x}}{4 \left (1-e^{4 x}\right )}-\frac{e^{3 x}}{\left (1-e^{4 x}\right )^2}+\frac{5}{8} \tan ^{-1}\left (e^x\right )-\frac{5}{8} \tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.0480443, antiderivative size = 55, 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, 463, 457, 298, 203, 206} \[ \frac{3 e^{3 x}}{4 \left (1-e^{4 x}\right )}-\frac{e^{3 x}}{\left (1-e^{4 x}\right )^2}+\frac{5}{8} \tan ^{-1}\left (e^x\right )-\frac{5}{8} \tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 463
Rule 457
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int e^x \coth ^2(2 x) \text{csch}(2 x) \, dx &=\operatorname{Subst}\left (\int \frac{2 x^2 \left (1+x^4\right )^2}{\left (-1+x^4\right )^3} \, dx,x,e^x\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x^2 \left (1+x^4\right )^2}{\left (-1+x^4\right )^3} \, dx,x,e^x\right )\\ &=-\frac{e^{3 x}}{\left (1-e^{4 x}\right )^2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2 \left (4+8 x^4\right )}{\left (-1+x^4\right )^2} \, dx,x,e^x\right )\\ &=-\frac{e^{3 x}}{\left (1-e^{4 x}\right )^2}+\frac{3 e^{3 x}}{4 \left (1-e^{4 x}\right )}+\frac{5}{4} \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,e^x\right )\\ &=-\frac{e^{3 x}}{\left (1-e^{4 x}\right )^2}+\frac{3 e^{3 x}}{4 \left (1-e^{4 x}\right )}-\frac{5}{8} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^x\right )+\frac{5}{8} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^x\right )\\ &=-\frac{e^{3 x}}{\left (1-e^{4 x}\right )^2}+\frac{3 e^{3 x}}{4 \left (1-e^{4 x}\right )}+\frac{5}{8} \tan ^{-1}\left (e^x\right )-\frac{5}{8} \tanh ^{-1}\left (e^x\right )\\ \end{align*}
Mathematica [C] time = 3.15484, size = 161, normalized size = 2.93 \[ -\frac{16 e^{7 x} \left (e^{4 x}+1\right )^2 \, _5F_4\left (\frac{7}{4},2,2,2,2;1,1,1,\frac{19}{4};e^{4 x}\right )}{1155}-\frac{8 e^{7 x} \left (26 e^{4 x}+11 e^{8 x}+15\right ) \, _4F_3\left (\frac{7}{4},2,2,2;1,1,\frac{19}{4};e^{4 x}\right )}{1155}+\frac{e^{-5 x} \left (-7 \left (24152 e^{4 x}-10058 e^{8 x}-9048 e^{12 x}+513 e^{16 x}+25289\right ) \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};e^{4 x}\right )+244931 e^{4 x}+43161 e^{8 x}-26091 e^{12 x}+177023\right )}{10752} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.097, size = 56, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{3\,x}} \left ( 3\,{{\rm e}^{4\,x}}+1 \right ) }{4\, \left ({{\rm e}^{4\,x}}-1 \right ) ^{2}}}-{\frac{5\,\ln \left ({{\rm e}^{x}}+1 \right ) }{16}}+{\frac{5\,i}{16}}\ln \left ({{\rm e}^{x}}+i \right ) -{\frac{5\,i}{16}}\ln \left ({{\rm e}^{x}}-i \right ) +{\frac{5\,\ln \left ({{\rm e}^{x}}-1 \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53406, size = 63, normalized size = 1.15 \begin{align*} -\frac{3 \, e^{\left (7 \, x\right )} + e^{\left (3 \, x\right )}}{4 \,{\left (e^{\left (8 \, x\right )} - 2 \, e^{\left (4 \, x\right )} + 1\right )}} + \frac{5}{8} \, \arctan \left (e^{x}\right ) - \frac{5}{16} \, \log \left (e^{x} + 1\right ) + \frac{5}{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 = 1.89445, size = 1868, normalized size = 33.96 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15557, size = 57, normalized size = 1.04 \begin{align*} -\frac{3 \, e^{\left (7 \, x\right )} + e^{\left (3 \, x\right )}}{4 \,{\left (e^{\left (4 \, x\right )} - 1\right )}^{2}} + \frac{5}{8} \, \arctan \left (e^{x}\right ) - \frac{5}{16} \, \log \left (e^{x} + 1\right ) + \frac{5}{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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