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 = 0.0298864, 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 \[ \frac{\tan ^{11}(x)}{11}+\frac{5 \tan ^9(x)}{9}+\frac{10 \tan ^7(x)}{7}+2 \tan ^5(x)+\frac{5 \tan ^3(x)}{3}+\tan (x) \]

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

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Rubi in Sympy [A]  time = 1.01575, size = 66, normalized size = 1.61 \[ \frac{256 \sin{\left (x \right )}}{693 \cos{\left (x \right )}} + \frac{128 \sin{\left (x \right )}}{693 \cos ^{3}{\left (x \right )}} + \frac{32 \sin{\left (x \right )}}{231 \cos ^{5}{\left (x \right )}} + \frac{80 \sin{\left (x \right )}}{693 \cos ^{7}{\left (x \right )}} + \frac{10 \sin{\left (x \right )}}{99 \cos ^{9}{\left (x \right )}} + \frac{\sin{\left (x \right )}}{11 \cos ^{11}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.00694843, 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*Se
c[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 = 0.052, size = 37, normalized size = 0.9 \[ - \left ( -{\frac{256}{693}}-{\frac{ \left ( \sec \left ( x \right ) \right ) ^{10}}{11}}-{\frac{10\, \left ( \sec \left ( x \right ) \right ) ^{8}}{99}}-{\frac{80\, \left ( \sec \left ( x \right ) \right ) ^{6}}{693}}-{\frac{32\, \left ( \sec \left ( x \right ) \right ) ^{4}}{231}}-{\frac{128\, \left ( \sec \left ( x \right ) \right ) ^{2}}{693}} \right ) \tan \left ( x \right ) \]

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 = 1.57325, size = 45, normalized size = 1.1 \[ \frac{1}{11} \, \tan \left (x\right )^{11} + \frac{5}{9} \, \tan \left (x\right )^{9} + \frac{10}{7} \, \tan \left (x\right )^{7} + 2 \, \tan \left (x\right )^{5} + \frac{5}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(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 = 0.217848, size = 54, normalized size = 1.32 \[ \frac{{\left (256 \, \cos \left (x\right )^{10} + 128 \, \cos \left (x\right )^{8} + 96 \, \cos \left (x\right )^{6} + 80 \, \cos \left (x\right )^{4} + 70 \, \cos \left (x\right )^{2} + 63\right )} \sin \left (x\right )}{693 \, \cos \left (x\right )^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(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 = 0.043031, size = 66, normalized size = 1.61 \[ \frac{256 \sin{\left (x \right )}}{693 \cos{\left (x \right )}} + \frac{128 \sin{\left (x \right )}}{693 \cos ^{3}{\left (x \right )}} + \frac{32 \sin{\left (x \right )}}{231 \cos ^{5}{\left (x \right )}} + \frac{80 \sin{\left (x \right )}}{693 \cos ^{7}{\left (x \right )}} + \frac{10 \sin{\left (x \right )}}{99 \cos ^{9}{\left (x \right )}} + \frac{\sin{\left (x \right )}}{11 \cos ^{11}{\left (x \right )}} \]

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/XCAS [A]  time = 0.199023, size = 45, normalized size = 1.1 \[ \frac{1}{11} \, \tan \left (x\right )^{11} + \frac{5}{9} \, \tan \left (x\right )^{9} + \frac{10}{7} \, \tan \left (x\right )^{7} + 2 \, \tan \left (x\right )^{5} + \frac{5}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(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)