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

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

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Rubi [A]  time = 0.02, antiderivative size = 41, 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} \[ \frac {\tan ^{11}(x)}{11}+\frac {5 \tan ^9(x)}{9}+\frac {10 \tan ^7(x)}{7}+2 \tan ^5(x)+\frac {5 \tan ^3(x)}{3}+\tan (x) \]

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*}

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Mathematica [A]  time = 0.01, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec ^{12}(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Sec[x]^12,x]

[Out]

Could not integrate

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fricas [A]  time = 0.81, size = 40, normalized size = 0.98 \[ \frac {{\left (256 \, \cos \relax (x)^{10} + 128 \, \cos \relax (x)^{8} + 96 \, \cos \relax (x)^{6} + 80 \, \cos \relax (x)^{4} + 70 \, \cos \relax (x)^{2} + 63\right )} \sin \relax (x)}{693 \, \cos \relax (x)^{11}} \]

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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giac [A]  time = 0.61, size = 33, normalized size = 0.80 \[ \frac {1}{11} \, \tan \relax (x)^{11} + \frac {5}{9} \, \tan \relax (x)^{9} + \frac {10}{7} \, \tan \relax (x)^{7} + 2 \, \tan \relax (x)^{5} + \frac {5}{3} \, \tan \relax (x)^{3} + \tan \relax (x) \]

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)

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maple [A]  time = 0.33, size = 37, normalized size = 0.90

method	result	size
default	\(-\left (-\frac {256}{693}-\frac {\left (\sec ^{10}\relax (x )\right )}{11}-\frac {10 \left (\sec ^{8}\relax (x )\right )}{99}-\frac {80 \left (\sec ^{6}\relax (x )\right )}{693}-\frac {32 \left (\sec ^{4}\relax (x )\right )}{231}-\frac {128 \left (\sec ^{2}\relax (x )\right )}{693}\right ) \tan \relax (x )\)	\(37\)
risch	\(\frac {512 i \left (462 \,{\mathrm e}^{10 i x}+330 \,{\mathrm e}^{8 i x}+165 \,{\mathrm e}^{6 i x}+55 \,{\mathrm e}^{4 i x}+11 \,{\mathrm e}^{2 i x}+1\right )}{693 \left (1+{\mathrm e}^{2 i x}\right )^{11}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x)^12,x,method=_RETURNVERBOSE)

[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 = 0.49, size = 33, normalized size = 0.80 \[ \frac {1}{11} \, \tan \relax (x)^{11} + \frac {5}{9} \, \tan \relax (x)^{9} + \frac {10}{7} \, \tan \relax (x)^{7} + 2 \, \tan \relax (x)^{5} + \frac {5}{3} \, \tan \relax (x)^{3} + \tan \relax (x) \]

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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mupad [B]  time = 0.21, size = 51, normalized size = 1.24 \[ \frac {256\,\sin \relax (x)\,{\cos \relax (x)}^{10}+128\,\sin \relax (x)\,{\cos \relax (x)}^8+96\,\sin \relax (x)\,{\cos \relax (x)}^6+80\,\sin \relax (x)\,{\cos \relax (x)}^4+70\,\sin \relax (x)\,{\cos \relax (x)}^2+63\,\sin \relax (x)}{693\,{\cos \relax (x)}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.07, size = 66, normalized size = 1.61 \[ \frac {256 \sin {\relax (x )}}{693 \cos {\relax (x )}} + \frac {128 \sin {\relax (x )}}{693 \cos ^{3}{\relax (x )}} + \frac {32 \sin {\relax (x )}}{231 \cos ^{5}{\relax (x )}} + \frac {80 \sin {\relax (x )}}{693 \cos ^{7}{\relax (x )}} + \frac {10 \sin {\relax (x )}}{99 \cos ^{9}{\relax (x )}} + \frac {\sin {\relax (x )}}{11 \cos ^{11}{\relax (x )}} \]

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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