3.355 \(\int \sec ^5(x) \, dx\)

Optimal. Leaf size=26 \[ \frac{3}{8} \tanh ^{-1}(\sin (x))+\frac{1}{4} \tan (x) \sec ^3(x)+\frac{3}{8} \tan (x) \sec (x) \]

[Out]

(3*ArcTanh[Sin[x]])/8 + (3*Sec[x]*Tan[x])/8 + (Sec[x]^3*Tan[x])/4

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Rubi [A]  time = 0.0250671, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{3}{8} \tanh ^{-1}(\sin (x))+\frac{1}{4} \tan (x) \sec ^3(x)+\frac{3}{8} \tan (x) \sec (x) \]

Antiderivative was successfully verified.

[In]  Int[Sec[x]^5,x]

[Out]

(3*ArcTanh[Sin[x]])/8 + (3*Sec[x]*Tan[x])/8 + (Sec[x]^3*Tan[x])/4

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Rubi in Sympy [A]  time = 0.607089, size = 29, normalized size = 1.12 \[ \frac{3 \sin{\left (x \right )}}{8 \cos ^{2}{\left (x \right )}} + \frac{\sin{\left (x \right )}}{4 \cos ^{4}{\left (x \right )}} + \frac{3 \operatorname{atanh}{\left (\sin{\left (x \right )} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sec(x)**5,x)

[Out]

3*sin(x)/(8*cos(x)**2) + sin(x)/(4*cos(x)**4) + 3*atanh(sin(x))/8

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Mathematica [B]  time = 0.187033, size = 58, normalized size = 2.23 \[ \frac{1}{16} \left (\frac{1}{2} (11 \sin (x)+3 \sin (3 x)) \sec ^4(x)-6 \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+6 \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[x]^5,x]

[Out]

(-6*Log[Cos[x/2] - Sin[x/2]] + 6*Log[Cos[x/2] + Sin[x/2]] + (Sec[x]^4*(11*Sin[x]
+ 3*Sin[3*x]))/2)/16

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Maple [A]  time = 0.052, size = 25, normalized size = 1. \[ - \left ( -{\frac{ \left ( \sec \left ( x \right ) \right ) ^{3}}{4}}-{\frac{3\,\sec \left ( x \right ) }{8}} \right ) \tan \left ( x \right ) +{\frac{3\,\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sec(x)^5,x)

[Out]

-(-1/4*sec(x)^3-3/8*sec(x))*tan(x)+3/8*ln(sec(x)+tan(x))

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Maxima [A]  time = 1.32185, size = 57, normalized size = 2.19 \[ -\frac{3 \, \sin \left (x\right )^{3} - 5 \, \sin \left (x\right )}{8 \,{\left (\sin \left (x\right )^{4} - 2 \, \sin \left (x\right )^{2} + 1\right )}} + \frac{3}{16} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{3}{16} \, \log \left (\sin \left (x\right ) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^5,x, algorithm="maxima")

[Out]

-1/8*(3*sin(x)^3 - 5*sin(x))/(sin(x)^4 - 2*sin(x)^2 + 1) + 3/16*log(sin(x) + 1)
- 3/16*log(sin(x) - 1)

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Fricas [A]  time = 0.223194, size = 58, normalized size = 2.23 \[ \frac{3 \, \cos \left (x\right )^{4} \log \left (\sin \left (x\right ) + 1\right ) - 3 \, \cos \left (x\right )^{4} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \,{\left (3 \, \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right )}{16 \, \cos \left (x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^5,x, algorithm="fricas")

[Out]

1/16*(3*cos(x)^4*log(sin(x) + 1) - 3*cos(x)^4*log(-sin(x) + 1) + 2*(3*cos(x)^2 +
2)*sin(x))/cos(x)^4

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Sympy [A]  time = 0.168673, size = 46, normalized size = 1.77 \[ - \frac{3 \sin ^{3}{\left (x \right )} - 5 \sin{\left (x \right )}}{8 \sin ^{4}{\left (x \right )} - 16 \sin ^{2}{\left (x \right )} + 8} - \frac{3 \log{\left (\sin{\left (x \right )} - 1 \right )}}{16} + \frac{3 \log{\left (\sin{\left (x \right )} + 1 \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)**5,x)

[Out]

-(3*sin(x)**3 - 5*sin(x))/(8*sin(x)**4 - 16*sin(x)**2 + 8) - 3*log(sin(x) - 1)/1
6 + 3*log(sin(x) + 1)/16

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GIAC/XCAS [A]  time = 0.20665, size = 51, normalized size = 1.96 \[ -\frac{3 \, \sin \left (x\right )^{3} - 5 \, \sin \left (x\right )}{8 \,{\left (\sin \left (x\right )^{2} - 1\right )}^{2}} + \frac{3}{16} \,{\rm ln}\left (\sin \left (x\right ) + 1\right ) - \frac{3}{16} \,{\rm ln}\left (-\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^5,x, algorithm="giac")

[Out]

-1/8*(3*sin(x)^3 - 5*sin(x))/(sin(x)^2 - 1)^2 + 3/16*ln(sin(x) + 1) - 3/16*ln(-s
in(x) + 1)