3.84 \(\int \sec ^4(x) \, dx\)

Optimal. Leaf size=11 \[ \frac{\tan ^3(x)}{3}+\tan (x) \]

[Out]

Tan[x] + Tan[x]^3/3

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Rubi [A]  time = 0.012423, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\tan ^3(x)}{3}+\tan (x) \]

Antiderivative was successfully verified.

[In]  Int[Sec[x]^4,x]

[Out]

Tan[x] + Tan[x]^3/3

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Rubi in Sympy [A]  time = 0.52396, size = 19, normalized size = 1.73 \[ \frac{2 \sin{\left (x \right )}}{3 \cos{\left (x \right )}} + \frac{\sin{\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sec(x)**4,x)

[Out]

2*sin(x)/(3*cos(x)) + sin(x)/(3*cos(x)**3)

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Mathematica [A]  time = 0.00391115, size = 17, normalized size = 1.55 \[ \frac{2 \tan (x)}{3}+\frac{1}{3} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[x]^4,x]

[Out]

(2*Tan[x])/3 + (Sec[x]^2*Tan[x])/3

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Maple [A]  time = 0.043, size = 13, normalized size = 1.2 \[ - \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( x \right ) \right ) ^{2}}{3}} \right ) \tan \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sec(x)^4,x)

[Out]

-(-2/3-1/3*sec(x)^2)*tan(x)

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Maxima [A]  time = 1.34197, size = 12, normalized size = 1.09 \[ \frac{1}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^4,x, algorithm="maxima")

[Out]

1/3*tan(x)^3 + tan(x)

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Fricas [A]  time = 0.206108, size = 22, normalized size = 2. \[ \frac{{\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{3 \, \cos \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^4,x, algorithm="fricas")

[Out]

1/3*(2*cos(x)^2 + 1)*sin(x)/cos(x)^3

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Sympy [A]  time = 0.044559, size = 19, normalized size = 1.73 \[ \frac{2 \sin{\left (x \right )}}{3 \cos{\left (x \right )}} + \frac{\sin{\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)**4,x)

[Out]

2*sin(x)/(3*cos(x)) + sin(x)/(3*cos(x)**3)

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GIAC/XCAS [A]  time = 0.208536, size = 12, normalized size = 1.09 \[ \frac{1}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^4,x, algorithm="giac")

[Out]

1/3*tan(x)^3 + tan(x)