Optimal. Leaf size=176 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {1+\sqrt {2}}-2 \tanh (x)}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\text {ArcTan}\left (\frac {\sqrt {1+\sqrt {2}}+2 \tanh (x)}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \tanh (x)+2 \tanh ^2(x)\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \tanh (x)+\sqrt {2} \tanh ^2(x)\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3288, 1183,
648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {1+\sqrt {2}}-2 \tanh (x)}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\text {ArcTan}\left (\frac {2 \tanh (x)+\sqrt {1+\sqrt {2}}}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (2 \tanh ^2(x)-2 \sqrt {1+\sqrt {2}} \tanh (x)+\sqrt {2}\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (\sqrt {2} \tanh ^2(x)+\sqrt {2 \left (1+\sqrt {2}\right )} \tanh (x)+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 3288
Rubi steps
\begin {align*} \int \frac {1}{1+\sinh ^4(x)} \, dx &=\text {Subst}\left (\int \frac {1-x^2}{1-2 x^2+2 x^4} \, dx,x,\tanh (x)\right )\\ &=\frac {\text {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}-\left (1+\frac {1}{\sqrt {2}}\right ) x}{\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\tanh (x)\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\text {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}+\left (1+\frac {1}{\sqrt {2}}\right ) x}{\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\tanh (x)\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\\ &=\frac {1}{8} \sqrt {3-2 \sqrt {2}} \text {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\tanh (x)\right )+\frac {1}{8} \sqrt {3-2 \sqrt {2}} \text {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\tanh (x)\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \text {Subst}\left (\int \frac {-\sqrt {1+\sqrt {2}}+2 x}{\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\tanh (x)\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}+2 x}{\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \tanh (x)+2 \tanh ^2(x)\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \tanh (x)+\sqrt {2} \tanh ^2(x)\right )-\frac {1}{4} \sqrt {3-2 \sqrt {2}} \text {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,-\sqrt {1+\sqrt {2}}+2 \tanh (x)\right )-\frac {1}{4} \sqrt {3-2 \sqrt {2}} \text {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,\sqrt {1+\sqrt {2}}+2 \tanh (x)\right )\\ &=-\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}-2 \tanh (x)}{\sqrt {-1+\sqrt {2}}}\right )+\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}+2 \tanh (x)}{\sqrt {-1+\sqrt {2}}}\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \tanh (x)+2 \tanh ^2(x)\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \tanh (x)+\sqrt {2} \tanh ^2(x)\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.06, size = 45, normalized size = 0.26 \begin {gather*} \frac {\tanh ^{-1}\left (\sqrt {1-i} \tanh (x)\right )}{2 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\sqrt {1+i} \tanh (x)\right )}{2 \sqrt {1+i}} \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 = 0.54, size = 44, normalized size = 0.25
method | result | size |
risch | \(\munderset {\textit {\_R} =\RootOf \left (512 \textit {\_Z}^{4}-32 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (256 \textit {\_R}^{3}-64 \textit {\_R}^{2}+{\mathrm e}^{2 x}+1\right )\) | \(36\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (2 \textit {\_Z}^{4}-2 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+\left (-4 \textit {\_R}^{3}+4 \textit {\_R} \right ) \tanh \left (\frac {x}{2}\right )+1\right )\right )}{4}\) | \(44\) |
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. 596 vs.
\(2 (124) = 248\).
time = 0.38, size = 596, normalized size = 3.39 \begin {gather*} \frac {1}{16} \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 1\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 ) - \frac {1}{16} \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 1\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 ) - \frac {1}{4} \cdot 2^{\frac {1}{4}} \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 ) - \frac {1}{4} \cdot 2^{\frac {1}{4}} \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 ) \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 [C] Result contains complex when optimal does not.
time = 0.45, size = 281, normalized size = 1.60 \begin {gather*} \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} - 2} {\left (-\frac {i}{\sqrt {2} - 1} + 1\right )} \log \left (\left (20 i + 10\right ) \, \sqrt {2} e^{\left (2 \, x\right )} + 10 \, \sqrt {2} \sqrt {10 \, \sqrt {2} + 14} - 50 \, \sqrt {2} - \left (2 i - 14\right ) \, \sqrt {10 \, \sqrt {2} + 14} + \left (28 i + 14\right ) \, e^{\left (2 \, x\right )} - 70\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} - 2} {\left (-\frac {i}{\sqrt {2} - 1} + 1\right )} \log \left (\left (20 i + 10\right ) \, \sqrt {2} e^{\left (2 \, x\right )} - 10 \, \sqrt {2} \sqrt {10 \, \sqrt {2} + 14} - 50 \, \sqrt {2} + \left (2 i - 14\right ) \, \sqrt {10 \, \sqrt {2} + 14} + \left (28 i + 14\right ) \, e^{\left (2 \, x\right )} - 70\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} + 2} {\left (-\frac {i}{\sqrt {2} + 1} + 1\right )} \log \left (2 \, \sqrt {2} e^{\left (2 \, x\right )} + 2 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} - \left (4 i + 2\right ) \, \sqrt {2} + \left (2 i - 2\right ) \, \sqrt {2 \, \sqrt {2} - 2} - 2 \, e^{\left (2 \, x\right )} + 4 i + 2\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} + 2} {\left (-\frac {i}{\sqrt {2} + 1} + 1\right )} \log \left (2 \, \sqrt {2} e^{\left (2 \, x\right )} - 2 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} - \left (4 i + 2\right ) \, \sqrt {2} - \left (2 i - 2\right ) \, \sqrt {2 \, \sqrt {2} - 2} - 2 \, e^{\left (2 \, x\right )} + 4 i + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 205, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152+91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (-9830400+56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-218890240-149422080{}\mathrm {i}\right )-21168128-94306304{}\mathrm {i}\right )}{8}+\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152+91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (9830400-56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (218890240+149422080{}\mathrm {i}\right )-21168128-94306304{}\mathrm {i}\right )}{8}-\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152-91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (-9830400-56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-218890240+149422080{}\mathrm {i}\right )-21168128+94306304{}\mathrm {i}\right )}{8}+\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152-91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (9830400+56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (218890240-149422080{}\mathrm {i}\right )-21168128+94306304{}\mathrm {i}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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