3.341 \(\int \tan ^6(x) \, dx\)

Optimal. Leaf size=22 \[ -x+\frac{\tan ^5(x)}{5}-\frac{\tan ^3(x)}{3}+\tan (x) \]

[Out]

-x + Tan[x] - Tan[x]^3/3 + Tan[x]^5/5

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Rubi [A]  time = 0.0230807, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -x+\frac{\tan ^5(x)}{5}-\frac{\tan ^3(x)}{3}+\tan (x) \]

Antiderivative was successfully verified.

[In]  Int[Tan[x]^6,x]

[Out]

-x + Tan[x] - Tan[x]^3/3 + Tan[x]^5/5

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Rubi in Sympy [A]  time = 0.53301, size = 17, normalized size = 0.77 \[ - x + \frac{\tan ^{5}{\left (x \right )}}{5} - \frac{\tan ^{3}{\left (x \right )}}{3} + \tan{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(tan(x)**6,x)

[Out]

-x + tan(x)**5/5 - tan(x)**3/3 + tan(x)

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Mathematica [A]  time = 0.00637214, size = 30, normalized size = 1.36 \[ -x+\frac{23 \tan (x)}{15}+\frac{1}{5} \tan (x) \sec ^4(x)-\frac{11}{15} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Tan[x]^6,x]

[Out]

-x + (23*Tan[x])/15 - (11*Sec[x]^2*Tan[x])/15 + (Sec[x]^4*Tan[x])/5

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Maple [A]  time = 0.004, size = 19, normalized size = 0.9 \[ -x+\tan \left ( x \right ) -{\frac{ \left ( \tan \left ( x \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \tan \left ( x \right ) \right ) ^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(tan(x)^6,x)

[Out]

-x+tan(x)-1/3*tan(x)^3+1/5*tan(x)^5

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Maxima [A]  time = 1.61045, size = 24, normalized size = 1.09 \[ \frac{1}{5} \, \tan \left (x\right )^{5} - \frac{1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)^6,x, algorithm="maxima")

[Out]

1/5*tan(x)^5 - 1/3*tan(x)^3 - x + tan(x)

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Fricas [A]  time = 0.218354, size = 24, normalized size = 1.09 \[ \frac{1}{5} \, \tan \left (x\right )^{5} - \frac{1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)^6,x, algorithm="fricas")

[Out]

1/5*tan(x)^5 - 1/3*tan(x)^3 - x + tan(x)

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Sympy [A]  time = 0.060743, size = 31, normalized size = 1.41 \[ - x + \frac{\sin ^{5}{\left (x \right )}}{5 \cos ^{5}{\left (x \right )}} - \frac{\sin ^{3}{\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)**6,x)

[Out]

-x + sin(x)**5/(5*cos(x)**5) - sin(x)**3/(3*cos(x)**3) + sin(x)/cos(x)

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GIAC/XCAS [A]  time = 0.20299, size = 24, normalized size = 1.09 \[ \frac{1}{5} \, \tan \left (x\right )^{5} - \frac{1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)^6,x, algorithm="giac")

[Out]

1/5*tan(x)^5 - 1/3*tan(x)^3 - x + tan(x)