3.43 \(\int \frac{\tan (x)}{\sqrt{1+\tan ^4(x)}} \, dx\)

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]

-ArcTanh[(1 - Tan[x]^2)/(Sqrt[2]*Sqrt[1 + Tan[x]^4])]/(2*Sqrt[2])

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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]

-ArcTanh[(1 - Tan[x]^2)/(Sqrt[2]*Sqrt[1 + Tan[x]^4])]/(2*Sqrt[2])

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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]

-sqrt(2)*atanh(sqrt(2)*(-tan(x)**2 + 1)/(2*sqrt(tan(x)**4 + 1)))/4

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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]

-(Sqrt[3 + Cos[4*x]]*Log[Sqrt[2]*Cos[2*x] + Sqrt[3 + Cos[4*x]]]*Sec[x]^2)/(4*Sqr
t[2]*Sqrt[1 + Tan[x]^4])

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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]

-1/4*2^(1/2)*arctanh(1/4*(-2*tan(x)^2+2)*2^(1/2)/((1+tan(x)^2)^2-2*tan(x)^2)^(1/
2))

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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]

-1/4*sqrt(2)*arcsinh(2*sin(x)^2 - 1)

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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]

1/32*sqrt(2)*log((577*sqrt(2)*tan(x)^16 - 1912*sqrt(2)*tan(x)^14 + 4124*sqrt(2)*
tan(x)^12 - 6216*sqrt(2)*tan(x)^10 + 7110*sqrt(2)*tan(x)^8 - 6216*sqrt(2)*tan(x)
^6 + 4124*sqrt(2)*tan(x)^4 - 1912*sqrt(2)*tan(x)^2 + 16*(51*tan(x)^14 - 169*tan(
x)^12 + 339*tan(x)^10 - 465*tan(x)^8 + 465*tan(x)^6 - 339*tan(x)^4 + 169*tan(x)^
2 - 51)*sqrt(tan(x)^4 + 1) + 577*sqrt(2))/(tan(x)^16 + 8*tan(x)^14 + 28*tan(x)^1
2 + 56*tan(x)^10 + 70*tan(x)^8 + 56*tan(x)^6 + 28*tan(x)^4 + 8*tan(x)^2 + 1))

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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]

Integral(tan(x)/sqrt(tan(x)**4 + 1), x)

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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]

1/4*sqrt(2)*ln(-(tan(x)^2 + sqrt(2) - sqrt(tan(x)^4 + 1) + 1)/(tan(x)^2 - sqrt(2
) - sqrt(tan(x)^4 + 1) + 1))