3.85 \(\int \sec ^6(x) \, dx\)

Optimal. Leaf size=19 \[ \frac {\tan ^5(x)}{5}+\frac {2 \tan ^3(x)}{3}+\tan (x) \]

[Out]

tan(x)+2/3*tan(x)^3+1/5*tan(x)^5

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3767} \[ \frac {\tan ^5(x)}{5}+\frac {2 \tan ^3(x)}{3}+\tan (x) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^6,x]

[Out]

Tan[x] + (2*Tan[x]^3)/3 + Tan[x]^5/5

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \sec ^6(x) \, dx &=-\operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (x)\right )\\ &=\tan (x)+\frac {2 \tan ^3(x)}{3}+\frac {\tan ^5(x)}{5}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 27, normalized size = 1.42 \[ \frac {8 \tan (x)}{15}+\frac {1}{5} \tan (x) \sec ^4(x)+\frac {4}{15} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^6,x]

[Out]

(8*Tan[x])/15 + (4*Sec[x]^2*Tan[x])/15 + (Sec[x]^4*Tan[x])/5

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fricas [A]  time = 0.40, size = 22, normalized size = 1.16 \[ \frac {{\left (8 \, \cos \relax (x)^{4} + 4 \, \cos \relax (x)^{2} + 3\right )} \sin \relax (x)}{15 \, \cos \relax (x)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6,x, algorithm="fricas")

[Out]

1/15*(8*cos(x)^4 + 4*cos(x)^2 + 3)*sin(x)/cos(x)^5

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giac [A]  time = 0.97, size = 15, normalized size = 0.79 \[ \frac {1}{5} \, \tan \relax (x)^{5} + \frac {2}{3} \, \tan \relax (x)^{3} + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6,x, algorithm="giac")

[Out]

1/5*tan(x)^5 + 2/3*tan(x)^3 + tan(x)

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maple [A]  time = 0.11, size = 19, normalized size = 1.00 \[ -\left (-\frac {\left (\sec ^{4}\relax (x )\right )}{5}-\frac {4 \left (\sec ^{2}\relax (x )\right )}{15}-\frac {8}{15}\right ) \tan \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^6,x)

[Out]

-(-8/15-1/5*sec(x)^4-4/15*sec(x)^2)*tan(x)

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maxima [A]  time = 0.44, size = 15, normalized size = 0.79 \[ \frac {1}{5} \, \tan \relax (x)^{5} + \frac {2}{3} \, \tan \relax (x)^{3} + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6,x, algorithm="maxima")

[Out]

1/5*tan(x)^5 + 2/3*tan(x)^3 + tan(x)

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mupad [B]  time = 0.04, size = 27, normalized size = 1.42 \[ \frac {8\,\sin \relax (x)\,{\cos \relax (x)}^4+4\,\sin \relax (x)\,{\cos \relax (x)}^2+3\,\sin \relax (x)}{15\,{\cos \relax (x)}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x)^6,x)

[Out]

(3*sin(x) + 4*cos(x)^2*sin(x) + 8*cos(x)^4*sin(x))/(15*cos(x)^5)

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sympy [A]  time = 0.07, size = 31, normalized size = 1.63 \[ \frac {8 \sin {\relax (x )}}{15 \cos {\relax (x )}} + \frac {4 \sin {\relax (x )}}{15 \cos ^{3}{\relax (x )}} + \frac {\sin {\relax (x )}}{5 \cos ^{5}{\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**6,x)

[Out]

8*sin(x)/(15*cos(x)) + 4*sin(x)/(15*cos(x)**3) + sin(x)/(5*cos(x)**5)

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