Optimal. Leaf size=371 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}}-2 e^x}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}}-2 e^x}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}}+2 e^x}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}}+2 e^x}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}} \]
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Rubi [A]
time = 0.23, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 9, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {2320, 12, 305,
1141, 1175, 632, 210, 1178, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}}-2 e^x}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}}-2 e^x}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\text {ArcTan}\left (\frac {2 e^x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\text {ArcTan}\left (\frac {2 e^x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (-\sqrt {2-\sqrt {2}} e^x+e^{2 x}+1\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (\sqrt {2-\sqrt {2}} e^x+e^{2 x}+1\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (-\sqrt {2+\sqrt {2}} e^x+e^{2 x}+1\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\log \left (\sqrt {2+\sqrt {2}} e^x+e^{2 x}+1\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 305
Rule 632
Rule 642
Rule 1141
Rule 1175
Rule 1178
Rule 2320
Rubi steps
\begin {align*} \int e^x \text {sech}(4 x) \, dx &=\text {Subst}\left (\int \frac {2 x^4}{1+x^8} \, dx,x,e^x\right )\\ &=2 \text {Subst}\left (\int \frac {x^4}{1+x^8} \, dx,x,e^x\right )\\ &=\frac {\text {Subst}\left (\int \frac {x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,e^x\right )}{\sqrt {2}}-\frac {\text {Subst}\left (\int \frac {x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,e^x\right )}{\sqrt {2}}\\ &=-\frac {\text {Subst}\left (\int \frac {1-x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,e^x\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1+x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,e^x\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1-x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,e^x\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1+x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,e^x\right )}{2 \sqrt {2}}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,e^x\right )}{4 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,e^x\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,e^x\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,e^x\right )}{4 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+2 x}{-1-\sqrt {2-\sqrt {2}} x-x^2} \, dx,x,e^x\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}-2 x}{-1+\sqrt {2-\sqrt {2}} x-x^2} \, dx,x,e^x\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+2 x}{-1-\sqrt {2+\sqrt {2}} x-x^2} \, dx,x,e^x\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}-2 x}{-1+\sqrt {2+\sqrt {2}} x-x^2} \, dx,x,e^x\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}\\ &=-\frac {\log \left (1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+2 e^x\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+2 e^x\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+2 e^x\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+2 e^x\right )}{2 \sqrt {2}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 e^x}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 e^x}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+2 e^x}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+2 e^x}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 24, normalized size = 0.06 \begin {gather*} \frac {2}{5} e^{5 x} \, _2F_1\left (\frac {5}{8},1;\frac {13}{8};-e^{8 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 = 0.88, size = 25, normalized size = 0.07
method | result | size |
risch | \(2 \left (\munderset {\textit {\_R} =\RootOf \left (16777216 \textit {\_Z}^{8}+1\right )}{\sum }\textit {\_R} \ln \left (-32768 \textit {\_R}^{5}+{\mathrm e}^{x}\right )\right )\) | \(25\) |
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. 1087 vs.
\(2 (255) = 510\).
time = 0.52, size = 1087, normalized size = 2.93 \begin {gather*} \frac {1}{8} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} - \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (-\frac {2 \, \sqrt {2} e^{x} - \sqrt {2} \sqrt {2 \, \sqrt {2} \sqrt {\sqrt {2} + 2} e^{x} - 2 \, \sqrt {2} \sqrt {-\sqrt {2} + 2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} + \sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{8} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} - \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (-\frac {2 \, \sqrt {2} e^{x} - \sqrt {2} \sqrt {-2 \, \sqrt {2} \sqrt {\sqrt {2} + 2} e^{x} + 2 \, \sqrt {2} \sqrt {-\sqrt {2} + 2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} - \sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{8} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} + \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (\frac {2 \, \sqrt {2} e^{x} - \sqrt {2} \sqrt {2 \, \sqrt {2} \sqrt {\sqrt {2} + 2} e^{x} + 2 \, \sqrt {2} \sqrt {-\sqrt {2} + 2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} + \sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{8} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} + \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (\frac {2 \, \sqrt {2} e^{x} - \sqrt {2} \sqrt {-2 \, \sqrt {2} \sqrt {\sqrt {2} + 2} e^{x} - 2 \, \sqrt {2} \sqrt {-\sqrt {2} + 2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} - \sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{32} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} - \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \log \left (2 \, \sqrt {2} \sqrt {\sqrt {2} + 2} e^{x} + 2 \, \sqrt {2} \sqrt {-\sqrt {2} + 2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + \frac {1}{32} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} + \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \log \left (2 \, \sqrt {2} \sqrt {\sqrt {2} + 2} e^{x} - 2 \, \sqrt {2} \sqrt {-\sqrt {2} + 2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) - \frac {1}{32} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} + \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \log \left (-2 \, \sqrt {2} \sqrt {\sqrt {2} + 2} e^{x} + 2 \, \sqrt {2} \sqrt {-\sqrt {2} + 2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + \frac {1}{32} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} - \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \log \left (-2 \, \sqrt {2} \sqrt {\sqrt {2} + 2} e^{x} - 2 \, \sqrt {2} \sqrt {-\sqrt {2} + 2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, \sqrt {\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1} - \sqrt {\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, \sqrt {-\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1} + \sqrt {\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, \sqrt {\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1} - \sqrt {-\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, \sqrt {-\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1} + \sqrt {-\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} e^{x} + 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \operatorname {sech}{\left (4 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 249, normalized size = 0.67 \begin {gather*} \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} e^{x} + 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 = 4.56, size = 479, normalized size = 1.29 \begin {gather*} -\ln \left (32768\,{\mathrm {e}}^x\,{\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )}^3-512\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )+\ln \left (32768\,{\mathrm {e}}^x\,{\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )}^3+512\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )-\ln \left (32768\,{\mathrm {e}}^x\,{\left (-\frac {\sqrt {2-\sqrt {2}}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )}^3-512\right )\,\left (-\frac {\sqrt {2-\sqrt {2}}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )+\ln \left (32768\,{\mathrm {e}}^x\,{\left (-\frac {\sqrt {2-\sqrt {2}}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )}^3+512\right )\,\left (-\frac {\sqrt {2-\sqrt {2}}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )+\sqrt {2}\,\ln \left (-512+\sqrt {2}\,{\mathrm {e}}^x\,{\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )}^3\,\left (16384-16384{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (512+\sqrt {2}\,{\mathrm {e}}^x\,{\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )}^3\,\left (16384-16384{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (-512+\sqrt {2}\,{\mathrm {e}}^x\,{\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )}^3\,\left (16384+16384{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (512+\sqrt {2}\,{\mathrm {e}}^x\,{\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )}^3\,\left (16384+16384{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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