Optimal. Leaf size=34 \[ -\frac {\tanh ^{-1}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {\tan ^4(x)+1}}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3670, 1248, 725, 206} \[ -\frac {\tanh ^{-1}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {\tan ^4(x)+1}}\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 1248
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {1+x^2}} \, dx,x,\tan ^2(x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {1-\tan ^2(x)}{\sqrt {1+\tan ^4(x)}}\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {1+\tan ^4(x)}}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 55, normalized size = 1.62 \[ -\frac {\sqrt {\cos (4 x)+3} \sec ^2(x) \log \left (\sqrt {2} \cos (2 x)+\sqrt {\cos (4 x)+3}\right )}{4 \sqrt {2} \sqrt {\tan ^4(x)+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 186, normalized size = 5.47 \[ \frac {1}{32} \, \sqrt {2} \log \left (\frac {577 \, \tan \relax (x)^{16} - 1912 \, \tan \relax (x)^{14} + 4124 \, \tan \relax (x)^{12} - 6216 \, \tan \relax (x)^{10} + 7110 \, \tan \relax (x)^{8} - 6216 \, \tan \relax (x)^{6} + 4124 \, \tan \relax (x)^{4} - 1912 \, \tan \relax (x)^{2} + 8 \, {\left (51 \, \sqrt {2} \tan \relax (x)^{14} - 169 \, \sqrt {2} \tan \relax (x)^{12} + 339 \, \sqrt {2} \tan \relax (x)^{10} - 465 \, \sqrt {2} \tan \relax (x)^{8} + 465 \, \sqrt {2} \tan \relax (x)^{6} - 339 \, \sqrt {2} \tan \relax (x)^{4} + 169 \, \sqrt {2} \tan \relax (x)^{2} - 51 \, \sqrt {2}\right )} \sqrt {\tan \relax (x)^{4} + 1} + 577}{\tan \relax (x)^{16} + 8 \, \tan \relax (x)^{14} + 28 \, \tan \relax (x)^{12} + 56 \, \tan \relax (x)^{10} + 70 \, \tan \relax (x)^{8} + 56 \, \tan \relax (x)^{6} + 28 \, \tan \relax (x)^{4} + 8 \, \tan \relax (x)^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 50, normalized size = 1.47 \[ \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\tan \relax (x)^{2} + \sqrt {2} - \sqrt {\tan \relax (x)^{4} + 1} + 1}{\tan \relax (x)^{2} - \sqrt {2} - \sqrt {\tan \relax (x)^{4} + 1} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 37, normalized size = 1.09 \[ -\frac {\sqrt {2}\, \arctanh \left (\frac {\left (-2 \left (\tan ^{2}\relax (x )\right )+2\right ) \sqrt {2}}{4 \sqrt {-2 \left (\tan ^{2}\relax (x )\right )+\left (\tan ^{2}\relax (x )+1\right )^{2}}}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.35, size = 565, normalized size = 16.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {tan}\relax (x)}{\sqrt {{\mathrm {tan}\relax (x)}^4+1}} \,d x \]
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 \[ \int \frac {\tan {\relax (x )}}{\sqrt {\tan ^{4}{\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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