Optimal. Leaf size=54 \[ x-\frac {7 \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2} \]
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Rubi [A] time = 0.09, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {470, 578, 522, 206} \[ x-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac {7 \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 470
Rule 522
Rule 578
Rubi steps
\begin {align*} \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx &=\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right ) \left (2-x^2\right )^3} \, dx,x,\tanh (x)\right )\\ &=-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^2 \left (6-2 x^2\right )}{\left (1-x^2\right ) \left (2-x^2\right )^2} \, dx,x,\tanh (x)\right )\\ &=-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {-2-6 x^2}{\left (1-x^2\right ) \left (2-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac {7}{4} \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\tanh (x)\right )+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=x-\frac {7 \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 66, normalized size = 1.22 \[ \frac {76 x-2 \sinh (2 x)-3 \sinh (4 x)+48 x \cosh (2 x)+4 x \cosh (4 x)-7 \sqrt {2} (\cosh (2 x)+3)^2 \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{8 (\cosh (2 x)+3)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 715, normalized size = 13.24 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 72, normalized size = 1.33 \[ -\frac {7}{16} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + x + \frac {17 \, e^{\left (6 \, x\right )} + 57 \, e^{\left (4 \, x\right )} + 19 \, e^{\left (2 \, x\right )} + 3}{2 \, {\left (e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 145, normalized size = 2.69 \[ -\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\frac {-\frac {\left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{4}-\frac {5 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{4}-\frac {5 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{4}-\frac {\tanh \left (\frac {x}{2}\right )}{4}}{\left (\tanh ^{4}\left (\frac {x}{2}\right )+1\right )^{2}}-\frac {7 \sqrt {2}\, \ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )+\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}{\tanh ^{2}\left (\frac {x}{2}\right )-\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}\right )}{32}+\frac {7 \sqrt {2}\, \ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )-\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}{\tanh ^{2}\left (\frac {x}{2}\right )+\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}\right )}{32}+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 84, normalized size = 1.56 \[ \frac {7}{16} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) + x - \frac {19 \, e^{\left (-2 \, x\right )} + 57 \, e^{\left (-4 \, x\right )} + 17 \, e^{\left (-6 \, x\right )} + 3}{2 \, {\left (12 \, e^{\left (-2 \, x\right )} + 38 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.67, size = 112, normalized size = 2.07 \[ x+\frac {136\,{\mathrm {e}}^{2\,x}+24}{12\,{\mathrm {e}}^{2\,x}+38\,{\mathrm {e}}^{4\,x}+12\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {7\,\sqrt {2}\,\ln \left (7\,{\mathrm {e}}^{2\,x}-\frac {7\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{16}\right )}{16}-\frac {7\,\sqrt {2}\,\ln \left (7\,{\mathrm {e}}^{2\,x}+\frac {7\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{16}\right )}{16}+\frac {\frac {17\,{\mathrm {e}}^{2\,x}}{2}-\frac {45}{2}}{6\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1} \]
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 \[ \int \frac {1}{\left (\coth ^{2}{\relax (x )} + \operatorname {csch}^{2}{\relax (x )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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