Optimal. Leaf size=34 \[ -\frac{\tanh ^{-1}\left (\frac{1-\tan ^2(x)}{\sqrt{2} \sqrt{\tan ^4(x)+1}}\right )}{2 \sqrt{2}} \]
[Out]
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Rubi [A] time = 0.104084, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{\tanh ^{-1}\left (\frac{1-\tan ^2(x)}{\sqrt{2} \sqrt{\tan ^4(x)+1}}\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[Tan[x]/Sqrt[1 + Tan[x]^4],x]
[Out]
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Rubi in Sympy [A] time = 5.99853, size = 32, normalized size = 0.94 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (- \tan ^{2}{\left (x \right )} + 1\right )}{2 \sqrt{\tan ^{4}{\left (x \right )} + 1}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(tan(x)/(1+tan(x)**4)**(1/2),x)
[Out]
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Mathematica [A] time = 0.107521, 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.
[In] Integrate[Tan[x]/Sqrt[1 + Tan[x]^4],x]
[Out]
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Maple [A] time = 0.036, size = 37, normalized size = 1.1 \[ -{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{ \left ( -2\, \left ( \tan \left ( x \right ) \right ) ^{2}+2 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{ \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{2}-2\, \left ( \tan \left ( x \right ) \right ) ^{2}}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(tan(x)/(1+tan(x)^4)^(1/2),x)
[Out]
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Maxima [A] time = 1.98263, size = 19, normalized size = 0.56 \[ -\frac{1}{4} \, \sqrt{2} \operatorname{arsinh}\left (2 \, \sin \left (x\right )^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)/sqrt(tan(x)^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232494, size = 255, normalized size = 7.5 \[ \frac{1}{32} \, \sqrt{2} \log \left (\frac{577 \, \sqrt{2} \tan \left (x\right )^{16} - 1912 \, \sqrt{2} \tan \left (x\right )^{14} + 4124 \, \sqrt{2} \tan \left (x\right )^{12} - 6216 \, \sqrt{2} \tan \left (x\right )^{10} + 7110 \, \sqrt{2} \tan \left (x\right )^{8} - 6216 \, \sqrt{2} \tan \left (x\right )^{6} + 4124 \, \sqrt{2} \tan \left (x\right )^{4} - 1912 \, \sqrt{2} \tan \left (x\right )^{2} + 16 \,{\left (51 \, \tan \left (x\right )^{14} - 169 \, \tan \left (x\right )^{12} + 339 \, \tan \left (x\right )^{10} - 465 \, \tan \left (x\right )^{8} + 465 \, \tan \left (x\right )^{6} - 339 \, \tan \left (x\right )^{4} + 169 \, \tan \left (x\right )^{2} - 51\right )} \sqrt{\tan \left (x\right )^{4} + 1} + 577 \, \sqrt{2}}{\tan \left (x\right )^{16} + 8 \, \tan \left (x\right )^{14} + 28 \, \tan \left (x\right )^{12} + 56 \, \tan \left (x\right )^{10} + 70 \, \tan \left (x\right )^{8} + 56 \, \tan \left (x\right )^{6} + 28 \, \tan \left (x\right )^{4} + 8 \, \tan \left (x\right )^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)/sqrt(tan(x)^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\tan{\left (x \right )}}{\sqrt{\tan ^{4}{\left (x \right )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)/(1+tan(x)**4)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.208158, size = 68, normalized size = 2. \[ \frac{1}{4} \, \sqrt{2}{\rm ln}\left (-\frac{\tan \left (x\right )^{2} + \sqrt{2} - \sqrt{\tan \left (x\right )^{4} + 1} + 1}{\tan \left (x\right )^{2} - \sqrt{2} - \sqrt{\tan \left (x\right )^{4} + 1} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)/sqrt(tan(x)^4 + 1),x, algorithm="giac")
[Out]