3.7 \(\int \frac{\tan (x)}{\sqrt{1+\sec ^3(x)}} \, dx\)

Optimal. Leaf size=15 \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{\sec ^3(x)+1}\right ) \]

[Out]

(-2*ArcTanh[Sqrt[1 + Sec[x]^3]])/3

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Rubi [A]  time = 0.0525434, antiderivative size = 15, 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 \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{\sec ^3(x)+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Tan[x]/Sqrt[1 + Sec[x]^3],x]

[Out]

(-2*ArcTanh[Sqrt[1 + Sec[x]^3]])/3

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(tan(x)/(1+sec(x)**3)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 11.6268, size = 296, normalized size = 19.73 \[ -\frac{i \cos ^2(x) \sqrt{\frac{\sqrt{3}-3 i \tan ^2\left (\frac{x}{2}\right )}{\sqrt{3}-3 i}} \sqrt{\frac{\sqrt{3}+3 i \tan ^2\left (\frac{x}{2}\right )}{\sqrt{3}+3 i}} \sec ^4\left (\frac{x}{2}\right ) \sqrt{(3 \cos (x)+\cos (3 x)+4) \sec ^3(x)} \left (F\left (i \sinh ^{-1}\left (\sqrt{3} \sqrt{\frac{i \cos (x) \sec ^2\left (\frac{x}{2}\right )}{-3 i+\sqrt{3}}}\right )|\frac{3 i-\sqrt{3}}{3 i+\sqrt{3}}\right )-\Pi \left (\frac{1}{6} \left (3+i \sqrt{3}\right );i \sinh ^{-1}\left (\sqrt{3} \sqrt{\frac{i \cos (x) \sec ^2\left (\frac{x}{2}\right )}{-3 i+\sqrt{3}}}\right )|\frac{3 i-\sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (3 \tan ^4\left (\frac{x}{2}\right )+1\right ) \sqrt{\frac{\cos (x) \sec ^2\left (\frac{x}{2}\right )}{-3-i \sqrt{3}}}} \]

Antiderivative was successfully verified.

[In]  Integrate[Tan[x]/Sqrt[1 + Sec[x]^3],x]

[Out]

((-I)*Cos[x]^2*(EllipticF[I*ArcSinh[Sqrt[3]*Sqrt[(I*Cos[x]*Sec[x/2]^2)/(-3*I + S
qrt[3])]], (3*I - Sqrt[3])/(3*I + Sqrt[3])] - EllipticPi[(3 + I*Sqrt[3])/6, I*Ar
cSinh[Sqrt[3]*Sqrt[(I*Cos[x]*Sec[x/2]^2)/(-3*I + Sqrt[3])]], (3*I - Sqrt[3])/(3*
I + Sqrt[3])])*Sec[x/2]^4*Sqrt[(4 + 3*Cos[x] + Cos[3*x])*Sec[x]^3]*Sqrt[(Sqrt[3]
- (3*I)*Tan[x/2]^2)/(-3*I + Sqrt[3])]*Sqrt[(Sqrt[3] + (3*I)*Tan[x/2]^2)/(3*I +
Sqrt[3])])/(Sqrt[3]*Sqrt[(Cos[x]*Sec[x/2]^2)/(-3 - I*Sqrt[3])]*(1 + 3*Tan[x/2]^4
))

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Maple [A]  time = 0.062, size = 12, normalized size = 0.8 \[ -{\frac{2}{3}{\it Artanh} \left ( \sqrt{1+ \left ( \sec \left ( x \right ) \right ) ^{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(tan(x)/(1+sec(x)^3)^(1/2),x)

[Out]

-2/3*arctanh((1+sec(x)^3)^(1/2))

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Maxima [A]  time = 1.48455, size = 36, normalized size = 2.4 \[ -\frac{1}{3} \, \log \left (\sqrt{\frac{1}{\cos \left (x\right )^{3}} + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{\frac{1}{\cos \left (x\right )^{3}} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)/sqrt(sec(x)^3 + 1),x, algorithm="maxima")

[Out]

-1/3*log(sqrt(1/cos(x)^3 + 1) + 1) + 1/3*log(sqrt(1/cos(x)^3 + 1) - 1)

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Fricas [A]  time = 0.287803, size = 41, normalized size = 2.73 \[ \frac{1}{3} \, \log \left (2 \, \sqrt{\frac{\cos \left (x\right )^{3} + 1}{\cos \left (x\right )^{3}}} \cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)/sqrt(sec(x)^3 + 1),x, algorithm="fricas")

[Out]

1/3*log(2*sqrt((cos(x)^3 + 1)/cos(x)^3)*cos(x)^3 - 2*cos(x)^3 - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\tan{\left (x \right )}}{\sqrt{\left (\sec{\left (x \right )} + 1\right ) \left (\sec ^{2}{\left (x \right )} - \sec{\left (x \right )} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)/(1+sec(x)**3)**(1/2),x)

[Out]

Integral(tan(x)/sqrt((sec(x) + 1)*(sec(x)**2 - sec(x) + 1)), x)

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GIAC/XCAS [A]  time = 0.220186, size = 38, normalized size = 2.53 \[ -\frac{1}{3} \,{\rm ln}\left (\sqrt{\frac{1}{\cos \left (x\right )^{3}} + 1} + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | \sqrt{\frac{1}{\cos \left (x\right )^{3}} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)/sqrt(sec(x)^3 + 1),x, algorithm="giac")

[Out]

-1/3*ln(sqrt(1/cos(x)^3 + 1) + 1) + 1/3*ln(abs(sqrt(1/cos(x)^3 + 1) - 1))