3.339 \(\int \sec ^{12}(x) \, dx\)

Optimal. Leaf size=41 \[ \frac{\tan ^{11}(x)}{11}+\frac{5 \tan ^9(x)}{9}+\frac{10 \tan ^7(x)}{7}+2 \tan ^5(x)+\frac{5 \tan ^3(x)}{3}+\tan (x) \]

[Out]

Tan[x] + (5*Tan[x]^3)/3 + 2*Tan[x]^5 + (10*Tan[x]^7)/7 + (5*Tan[x]^9)/9 + Tan[x]^11/11

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Rubi [A]  time = 0.0165278, antiderivative size = 41, 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, Rules used = {3767} \[ \frac{\tan ^{11}(x)}{11}+\frac{5 \tan ^9(x)}{9}+\frac{10 \tan ^7(x)}{7}+2 \tan ^5(x)+\frac{5 \tan ^3(x)}{3}+\tan (x) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^12,x]

[Out]

Tan[x] + (5*Tan[x]^3)/3 + 2*Tan[x]^5 + (10*Tan[x]^7)/7 + (5*Tan[x]^9)/9 + Tan[x]^11/11

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 ^{12}(x) \, dx &=-\operatorname{Subst}\left (\int \left (1+5 x^2+10 x^4+10 x^6+5 x^8+x^{10}\right ) \, dx,x,-\tan (x)\right )\\ &=\tan (x)+\frac{5 \tan ^3(x)}{3}+2 \tan ^5(x)+\frac{10 \tan ^7(x)}{7}+\frac{5 \tan ^9(x)}{9}+\frac{\tan ^{11}(x)}{11}\\ \end{align*}

Mathematica [A]  time = 0.0038647, size = 57, normalized size = 1.39 \[ \frac{256 \tan (x)}{693}+\frac{1}{11} \tan (x) \sec ^{10}(x)+\frac{10}{99} \tan (x) \sec ^8(x)+\frac{80}{693} \tan (x) \sec ^6(x)+\frac{32}{231} \tan (x) \sec ^4(x)+\frac{128}{693} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^12,x]

[Out]

(256*Tan[x])/693 + (128*Sec[x]^2*Tan[x])/693 + (32*Sec[x]^4*Tan[x])/231 + (80*Sec[x]^6*Tan[x])/693 + (10*Sec[x
]^8*Tan[x])/99 + (Sec[x]^10*Tan[x])/11

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Maple [A]  time = 0.033, size = 37, normalized size = 0.9 \begin{align*} - \left ( -{\frac{256}{693}}-{\frac{ \left ( \sec \left ( x \right ) \right ) ^{10}}{11}}-{\frac{10\, \left ( \sec \left ( x \right ) \right ) ^{8}}{99}}-{\frac{80\, \left ( \sec \left ( x \right ) \right ) ^{6}}{693}}-{\frac{32\, \left ( \sec \left ( x \right ) \right ) ^{4}}{231}}-{\frac{128\, \left ( \sec \left ( x \right ) \right ) ^{2}}{693}} \right ) \tan \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x)^12,x)

[Out]

-(-256/693-1/11*sec(x)^10-10/99*sec(x)^8-80/693*sec(x)^6-32/231*sec(x)^4-128/693*sec(x)^2)*tan(x)

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Maxima [A]  time = 0.932061, size = 45, normalized size = 1.1 \begin{align*} \frac{1}{11} \, \tan \left (x\right )^{11} + \frac{5}{9} \, \tan \left (x\right )^{9} + \frac{10}{7} \, \tan \left (x\right )^{7} + 2 \, \tan \left (x\right )^{5} + \frac{5}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^12,x, algorithm="maxima")

[Out]

1/11*tan(x)^11 + 5/9*tan(x)^9 + 10/7*tan(x)^7 + 2*tan(x)^5 + 5/3*tan(x)^3 + tan(x)

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Fricas [A]  time = 1.609, size = 138, normalized size = 3.37 \begin{align*} \frac{{\left (256 \, \cos \left (x\right )^{10} + 128 \, \cos \left (x\right )^{8} + 96 \, \cos \left (x\right )^{6} + 80 \, \cos \left (x\right )^{4} + 70 \, \cos \left (x\right )^{2} + 63\right )} \sin \left (x\right )}{693 \, \cos \left (x\right )^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^12,x, algorithm="fricas")

[Out]

1/693*(256*cos(x)^10 + 128*cos(x)^8 + 96*cos(x)^6 + 80*cos(x)^4 + 70*cos(x)^2 + 63)*sin(x)/cos(x)^11

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Sympy [A]  time = 0.059934, size = 66, normalized size = 1.61 \begin{align*} \frac{256 \sin{\left (x \right )}}{693 \cos{\left (x \right )}} + \frac{128 \sin{\left (x \right )}}{693 \cos ^{3}{\left (x \right )}} + \frac{32 \sin{\left (x \right )}}{231 \cos ^{5}{\left (x \right )}} + \frac{80 \sin{\left (x \right )}}{693 \cos ^{7}{\left (x \right )}} + \frac{10 \sin{\left (x \right )}}{99 \cos ^{9}{\left (x \right )}} + \frac{\sin{\left (x \right )}}{11 \cos ^{11}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)**12,x)

[Out]

256*sin(x)/(693*cos(x)) + 128*sin(x)/(693*cos(x)**3) + 32*sin(x)/(231*cos(x)**5) + 80*sin(x)/(693*cos(x)**7) +
10*sin(x)/(99*cos(x)**9) + sin(x)/(11*cos(x)**11)

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Giac [A]  time = 1.04604, size = 45, normalized size = 1.1 \begin{align*} \frac{1}{11} \, \tan \left (x\right )^{11} + \frac{5}{9} \, \tan \left (x\right )^{9} + \frac{10}{7} \, \tan \left (x\right )^{7} + 2 \, \tan \left (x\right )^{5} + \frac{5}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^12,x, algorithm="giac")

[Out]

1/11*tan(x)^11 + 5/9*tan(x)^9 + 10/7*tan(x)^7 + 2*tan(x)^5 + 5/3*tan(x)^3 + tan(x)