3.94 \(\int \sec ^6(x) \tan (x) \, dx\)

Optimal. Leaf size=8 \[ \frac{\sec ^6(x)}{6} \]

[Out]

Sec[x]^6/6

_______________________________________________________________________________________

Rubi [A]  time = 0.0189001, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\sec ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]  Int[Sec[x]^6*Tan[x],x]

[Out]

Sec[x]^6/6

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\tan{\left (x \right )}}{\cos ^{6}{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sec(x)**6*tan(x),x)

[Out]

Integral(tan(x)/cos(x)**6, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0030136, size = 8, normalized size = 1. \[ \frac{\sec ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[x]^6*Tan[x],x]

[Out]

Sec[x]^6/6

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 7, normalized size = 0.9 \[{\frac{ \left ( \sec \left ( x \right ) \right ) ^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sec(x)^6*tan(x),x)

[Out]

1/6*sec(x)^6

_______________________________________________________________________________________

Maxima [A]  time = 1.36296, size = 14, normalized size = 1.75 \[ -\frac{1}{6 \,{\left (\sin \left (x\right )^{2} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/(sin(x)^2 - 1)^3

_______________________________________________________________________________________

Fricas [A]  time = 0.216379, size = 8, normalized size = 1. \[ \frac{1}{6 \, \cos \left (x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6/cos(x)^6

_______________________________________________________________________________________

Sympy [A]  time = 0.050575, size = 7, normalized size = 0.88 \[ \frac{1}{6 \cos ^{6}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)**6*tan(x),x)

[Out]

1/(6*cos(x)**6)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.198959, size = 8, normalized size = 1. \[ \frac{1}{6 \, \cos \left (x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6/cos(x)^6