3.84 \(\int \sec ^4(x) \, dx\)

Optimal. Leaf size=11 \[ \frac {\tan ^3(x)}{3}+\tan (x) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 11, 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} \[ \frac {\tan ^3(x)}{3}+\tan (x) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^4,x]

[Out]

Tan[x] + Tan[x]^3/3

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.55 \[ \frac {2 \tan (x)}{3}+\frac {1}{3} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^4,x]

[Out]

(2*Tan[x])/3 + (Sec[x]^2*Tan[x])/3

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fricas [A]  time = 0.40, size = 16, normalized size = 1.45 \[ \frac {{\left (2 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x)}{3 \, \cos \relax (x)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^4,x, algorithm="fricas")

[Out]

1/3*(2*cos(x)^2 + 1)*sin(x)/cos(x)^3

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giac [A]  time = 0.95, size = 9, normalized size = 0.82 \[ \frac {1}{3} \, \tan \relax (x)^{3} + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^4,x, algorithm="giac")

[Out]

1/3*tan(x)^3 + tan(x)

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maple [A]  time = 0.10, size = 13, normalized size = 1.18 \[ -\left (-\frac {\left (\sec ^{2}\relax (x )\right )}{3}-\frac {2}{3}\right ) \tan \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^4,x)

[Out]

-(-2/3-1/3*sec(x)^2)*tan(x)

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maxima [A]  time = 0.44, size = 9, normalized size = 0.82 \[ \frac {1}{3} \, \tan \relax (x)^{3} + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^4,x, algorithm="maxima")

[Out]

1/3*tan(x)^3 + tan(x)

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mupad [B]  time = 0.03, size = 17, normalized size = 1.55 \[ \frac {2\,\sin \relax (x)\,{\cos \relax (x)}^2+\sin \relax (x)}{3\,{\cos \relax (x)}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x)^4,x)

[Out]

(sin(x) + 2*cos(x)^2*sin(x))/(3*cos(x)^3)

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sympy [B]  time = 0.07, size = 19, normalized size = 1.73 \[ \frac {2 \sin {\relax (x )}}{3 \cos {\relax (x )}} + \frac {\sin {\relax (x )}}{3 \cos ^{3}{\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**4,x)

[Out]

2*sin(x)/(3*cos(x)) + sin(x)/(3*cos(x)**3)

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