3.1.85 \(\int \sec ^6(x) \, dx\) [85]

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

[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 = {3852} \begin {gather*} \frac {\tan ^5(x)}{5}+\frac {2 \tan ^3(x)}{3}+\tan (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[x]^6,x]

[Out]

Tan[x] + (2*Tan[x]^3)/3 + Tan[x]^5/5

Rule 3852

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 &=-\text {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 \begin {gather*} \frac {8 \tan (x)}{15}+\frac {4}{15} \sec ^2(x) \tan (x)+\frac {1}{5} \sec ^4(x) \tan (x) \end {gather*}

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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Mathics [A]
time = 2.12, size = 22, normalized size = 1.16 \begin {gather*} \frac {\left (15-20 \text {Sin}\left [x\right ]^2+8 \text {Sin}\left [x\right ]^4\right ) \text {Sin}\left [x\right ]}{15 \text {Cos}\left [x\right ]^5} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sec[x]^6,x]')

[Out]

(15 - 20 Sin[x] ^ 2 + 8 Sin[x] ^ 4) Sin[x] / (15 Cos[x] ^ 5)

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Maple [A]
time = 0.07, size = 19, normalized size = 1.00

method	result	size
default	\(-\left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (x \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (x \right )\right )}{15}\right ) \tan \left (x \right )\)	\(19\)
risch	\(\frac {16 i \left (10 \,{\mathrm e}^{4 i x}+5 \,{\mathrm e}^{2 i x}+1\right )}{15 \left ({\mathrm e}^{2 i x}+1\right )^{5}}\)	\(29\)
norman	\(\frac {\frac {8 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}-\frac {116 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{15}+\frac {8 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3}-2 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right )}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{5}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^6,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{5} \, \tan \left (x\right )^{5} + \frac {2}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \end {gather*}

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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Fricas [A]
time = 0.32, size = 22, normalized size = 1.16 \begin {gather*} \frac {{\left (8 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}{15 \, \cos \left (x\right )^{5}} \end {gather*}

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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Sympy [A]
time = 0.03, size = 31, normalized size = 1.63 \begin {gather*} \frac {8 \sin {\left (x \right )}}{15 \cos {\left (x \right )}} + \frac {4 \sin {\left (x \right )}}{15 \cos ^{3}{\left (x \right )}} + \frac {\sin {\left (x \right )}}{5 \cos ^{5}{\left (x \right )}} \end {gather*}

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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Giac [A]
time = 0.00, size = 23, normalized size = 1.21 \begin {gather*} \frac {2}{2} \left (\frac {1}{5} \tan ^{5}x+\frac {2}{3} \tan ^{3}x+\tan x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6,x)

[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 \begin {gather*} \frac {8\,\sin \left (x\right )\,{\cos \left (x\right )}^4+4\,\sin \left (x\right )\,{\cos \left (x\right )}^2+3\,\sin \left (x\right )}{15\,{\cos \left (x\right )}^5} \end {gather*}

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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