Optimal. Leaf size=56 \[ \frac {1}{2} \sqrt {\tan ^4(x)+1}-\frac {\tanh ^{-1}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {\tan ^4(x)+1}}\right )}{\sqrt {2}}-\frac {1}{2} \sinh ^{-1}\left (\tan ^2(x)\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3670, 1248, 735, 844, 215, 725, 206} \[ \frac {1}{2} \sqrt {\tan ^4(x)+1}-\frac {\tanh ^{-1}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {\tan ^4(x)+1}}\right )}{\sqrt {2}}-\frac {1}{2} \sinh ^{-1}\left (\tan ^2(x)\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 725
Rule 735
Rule 844
Rule 1248
Rule 3670
Rubi steps
\begin {align*} \int \tan (x) \sqrt {1+\tan ^4(x)} \, dx &=\operatorname {Subst}\left (\int \frac {x \sqrt {1+x^4}}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{1+x} \, dx,x,\tan ^2(x)\right )\\ &=\frac {1}{2} \sqrt {1+\tan ^4(x)}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1-x}{(1+x) \sqrt {1+x^2}} \, dx,x,\tan ^2(x)\right )\\ &=\frac {1}{2} \sqrt {1+\tan ^4(x)}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\tan ^2(x)\right )+\operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {1+x^2}} \, dx,x,\tan ^2(x)\right )\\ &=-\frac {1}{2} \sinh ^{-1}\left (\tan ^2(x)\right )+\frac {1}{2} \sqrt {1+\tan ^4(x)}-\operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {1-\tan ^2(x)}{\sqrt {1+\tan ^4(x)}}\right )\\ &=-\frac {1}{2} \sinh ^{-1}\left (\tan ^2(x)\right )-\frac {\tanh ^{-1}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {1+\tan ^4(x)}}\right )}{\sqrt {2}}+\frac {1}{2} \sqrt {1+\tan ^4(x)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 74, normalized size = 1.32 \[ \frac {\sqrt {\tan ^4(x)+1} \left (\sqrt {\cos (4 x)+3}-2 \sqrt {2} \cos ^2(x) \sinh ^{-1}(\cos (2 x))-2 \cos ^2(x) \tanh ^{-1}\left (\frac {2 \sin ^2(x)}{\sqrt {\cos (4 x)+3}}\right )\right )}{2 \sqrt {\cos (4 x)+3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 88, normalized size = 1.57 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {3 \, \tan \relax (x)^{4} - 2 \, \tan \relax (x)^{2} + 2 \, \sqrt {\tan \relax (x)^{4} + 1} {\left (\sqrt {2} \tan \relax (x)^{2} - \sqrt {2}\right )} + 3}{\tan \relax (x)^{4} + 2 \, \tan \relax (x)^{2} + 1}\right ) + \frac {1}{2} \, \sqrt {\tan \relax (x)^{4} + 1} + \frac {1}{2} \, \log \left (-\tan \relax (x)^{2} + \sqrt {\tan \relax (x)^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 79, normalized size = 1.41 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\tan \relax (x)^{2} + \sqrt {2} - \sqrt {\tan \relax (x)^{4} + 1} + 1}{\tan \relax (x)^{2} - \sqrt {2} - \sqrt {\tan \relax (x)^{4} + 1} + 1}\right ) + \frac {1}{2} \, \sqrt {\tan \relax (x)^{4} + 1} + \frac {1}{2} \, \log \left (-\tan \relax (x)^{2} + \sqrt {\tan \relax (x)^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 64, normalized size = 1.14 \[ -\frac {\arcsinh \left (\tan ^{2}\relax (x )\right )}{2}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (-2 \left (\tan ^{2}\relax (x )\right )+2\right ) \sqrt {2}}{4 \sqrt {-2 \left (\tan ^{2}\relax (x )\right )+\left (\tan ^{2}\relax (x )+1\right )^{2}}}\right )}{2}+\frac {\sqrt {-2 \left (\tan ^{2}\relax (x )\right )+\left (\tan ^{2}\relax (x )+1\right )^{2}}}{2} \]
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 \[ \int \sqrt {\tan \relax (x)^{4} + 1} \tan \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {tan}\relax (x)\,\sqrt {{\mathrm {tan}\relax (x)}^4+1} \,d x \]
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 \[ \int \sqrt {\tan ^{4}{\relax (x )} + 1} \tan {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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