Optimal. Leaf size=69 \[ \frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1-i}}\right )}{4 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1+i}}\right )}{4 \sqrt {1+i}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}+\frac {\coth (x)}{4} \]
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Rubi [A]
time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3290, 3260,
212, 3254, 3852, 8} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1-i}}\right )}{4 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1+i}}\right )}{4 \sqrt {1+i}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}+\frac {\coth (x)}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 212
Rule 3254
Rule 3260
Rule 3290
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{1-\cosh ^8(x)} \, dx &=\frac {1}{4} \int \frac {1}{1-\cosh ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1-i \cosh ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1+i \cosh ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1+\cosh ^2(x)} \, dx\\ &=-\left (\frac {1}{4} \int \text {csch}^2(x) \, dx\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\coth (x)\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-(1+i) x^2} \, dx,x,\coth (x)\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-(1-i) x^2} \, dx,x,\coth (x)\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1-i}}\right )}{4 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1+i}}\right )}{4 \sqrt {1+i}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}+\frac {1}{4} i \text {Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1-i}}\right )}{4 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1+i}}\right )}{4 \sqrt {1+i}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}+\frac {\coth (x)}{4}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 64, normalized size = 0.93 \begin {gather*} \frac {1}{8} \left (\frac {2 \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1-i}}\right )}{\sqrt {1-i}}+\frac {2 \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1+i}}\right )}{\sqrt {1+i}}+\sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )+2 \coth (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.74, size = 190, normalized size = 2.75
method | result | size |
risch | \(\frac {1}{2 \,{\mathrm e}^{2 x}-2}+\left (\munderset {\textit {\_R} =\RootOf \left (8192 \textit {\_Z}^{4}-128 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-2048 \textit {\_R}^{3}+256 \textit {\_R}^{2}+{\mathrm e}^{2 x}-1\right )\right )+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3-2 \sqrt {2}\right )}{16}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3+2 \sqrt {2}\right )}{16}\) | \(81\) |
default | \(\frac {\tanh \left (\frac {x}{2}\right )}{8}+\frac {1}{8 \tanh \left (\frac {x}{2}\right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (2 \textit {\_Z}^{4}-2 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (2 \tanh \left (\frac {x}{2}\right ) \textit {\_R} +\tanh ^{2}\left (\frac {x}{2}\right )+1\right )\right )}{8}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh ^{2}\left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{32}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh ^{2}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{32}\) | \(190\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 713 vs.
\(2 (42) = 84\).
time = 0.43, size = 713, normalized size = 10.33 \begin {gather*} \frac {4 \, {\left (2^{\frac {1}{4}} e^{\left (2 \, x\right )} - 2^{\frac {1}{4}}\right )} \sqrt {-2 \, \sqrt {2} + 4} \arctan \left (\frac {1}{14} \, {\left (\sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + 8 \, \sqrt {2} + 4\right )} e^{\left (2 \, x\right )} - \frac {1}{28} \, {\left (2 \, \sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} - {\left (2^{\frac {3}{4}} {\left (8 \, \sqrt {2} + 11\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (5 \, \sqrt {2} + 6\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 16 \, \sqrt {2} + 8\right )} \sqrt {{\left (2^{\frac {3}{4}} e^{\left (2 \, x\right )} + 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} + 4\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 4 \, \sqrt {2} + e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 5} + \frac {1}{14} \, \sqrt {2} {\left (3 \, \sqrt {2} - 2\right )} - \frac {1}{28} \, {\left ({\left (2^{\frac {3}{4}} {\left (8 \, \sqrt {2} + 11\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (5 \, \sqrt {2} + 6\right )}\right )} e^{\left (2 \, x\right )} + 2^{\frac {3}{4}} {\left (2 \, \sqrt {2} + 1\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} - 2\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + \frac {1}{7} \, \sqrt {2} - \frac {3}{7}\right ) + 4 \, {\left (2^{\frac {1}{4}} e^{\left (2 \, x\right )} - 2^{\frac {1}{4}}\right )} \sqrt {-2 \, \sqrt {2} + 4} \arctan \left (-\frac {1}{14} \, {\left (\sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + 8 \, \sqrt {2} + 4\right )} e^{\left (2 \, x\right )} + \frac {1}{28} \, {\left (2 \, \sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + {\left (2^{\frac {3}{4}} {\left (8 \, \sqrt {2} + 11\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (5 \, \sqrt {2} + 6\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 16 \, \sqrt {2} + 8\right )} \sqrt {-{\left (2^{\frac {3}{4}} e^{\left (2 \, x\right )} + 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} + 4\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 4 \, \sqrt {2} + e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 5} - \frac {1}{14} \, \sqrt {2} {\left (3 \, \sqrt {2} - 2\right )} - \frac {1}{28} \, {\left ({\left (2^{\frac {3}{4}} {\left (8 \, \sqrt {2} + 11\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (5 \, \sqrt {2} + 6\right )}\right )} e^{\left (2 \, x\right )} + 2^{\frac {3}{4}} {\left (2 \, \sqrt {2} + 1\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} - 2\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} - \frac {1}{7} \, \sqrt {2} + \frac {3}{7}\right ) - {\left (2^{\frac {1}{4}} {\left (\sqrt {2} + 1\right )} e^{\left (2 \, x\right )} - 2^{\frac {1}{4}} {\left (\sqrt {2} + 1\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} \log \left ({\left (2^{\frac {3}{4}} e^{\left (2 \, x\right )} + 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} + 4\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 4 \, \sqrt {2} + e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 5\right ) + {\left (2^{\frac {1}{4}} {\left (\sqrt {2} + 1\right )} e^{\left (2 \, x\right )} - 2^{\frac {1}{4}} {\left (\sqrt {2} + 1\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} \log \left (-{\left (2^{\frac {3}{4}} e^{\left (2 \, x\right )} + 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} + 4\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 4 \, \sqrt {2} + e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 5\right ) + 2 \, {\left (\sqrt {2} e^{\left (2 \, x\right )} - \sqrt {2}\right )} \log \left (-\frac {2 \, {\left (2 \, \sqrt {2} - 3\right )} e^{\left (2 \, x\right )} + 12 \, \sqrt {2} - e^{\left (4 \, x\right )} - 17}{e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1}\right ) + 16}{32 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 45, normalized size = 0.65 \begin {gather*} \frac {1}{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 ) + \frac {1}{2 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.63, size = 271, normalized size = 3.93 \begin {gather*} \frac {\sqrt {2}\,\ln \left (582732658686033920\,{\mathrm {e}}^{2\,x}+70697326355677184\,\sqrt {2}+412054214575915008\,\sqrt {2}\,{\mathrm {e}}^{2\,x}+99981117754441728\right )}{16}-\frac {\sqrt {2}\,\ln \left (70697326355677184\,\sqrt {2}-582732658686033920\,{\mathrm {e}}^{2\,x}+412054214575915008\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-99981117754441728\right )}{16}+\frac {1}{2\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left (70836483296067584+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (-54684829282729984+21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-12296353929494528+271474128182050816{}\mathrm {i}\right )+{\mathrm {e}}^{2\,x}\,\left (-155613434002538496-429723297714798592{}\mathrm {i}\right )-69311013991743488{}\mathrm {i}\right )}{16}+\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left (70836483296067584+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (54684829282729984-21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (12296353929494528-271474128182050816{}\mathrm {i}\right )+{\mathrm {e}}^{2\,x}\,\left (-155613434002538496-429723297714798592{}\mathrm {i}\right )-69311013991743488{}\mathrm {i}\right )}{16}-\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left (70836483296067584+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (-54684829282729984-21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-12296353929494528-271474128182050816{}\mathrm {i}\right )+{\mathrm {e}}^{2\,x}\,\left (-155613434002538496+429723297714798592{}\mathrm {i}\right )+69311013991743488{}\mathrm {i}\right )}{16}+\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left (70836483296067584+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (54684829282729984+21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (12296353929494528+271474128182050816{}\mathrm {i}\right )+{\mathrm {e}}^{2\,x}\,\left (-155613434002538496+429723297714798592{}\mathrm {i}\right )+69311013991743488{}\mathrm {i}\right )}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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