3.43 \(\int \tan ^4(x) \, dx\)

Optimal. Leaf size=14 \[ x+\frac {\tan ^3(x)}{3}-\tan (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 8} \[ x+\frac {\tan ^3(x)}{3}-\tan (x) \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]^4,x]

[Out]

x - Tan[x] + Tan[x]^3/3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 18, normalized size = 1.29 \[ x-\frac {4 \tan (x)}{3}+\frac {1}{3} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]^4,x]

[Out]

x - (4*Tan[x])/3 + (Sec[x]^2*Tan[x])/3

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 12, normalized size = 0.86 \[ \frac {1}{3} \, \tan \relax (x)^{3} + x - \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tan(x)^3 + x - tan(x)

________________________________________________________________________________________

giac [A]  time = 0.01, size = 12, normalized size = 0.86 \[ \frac {1}{3} \, \tan \relax (x)^{3} + x - \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tan(x)^3 + x - tan(x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 13, normalized size = 0.93 \[ \frac {\left (\tan ^{3}\relax (x )\right )}{3}+x -\tan \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^4,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.27, size = 12, normalized size = 0.86 \[ \frac {1}{3} \, \tan \relax (x)^{3} + x - \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tan(x)^3 + x - tan(x)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 12, normalized size = 0.86 \[ \frac {{\mathrm {tan}\relax (x)}^3}{3}-\mathrm {tan}\relax (x)+x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^4,x)

[Out]

x - tan(x) + tan(x)^3/3

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 19, normalized size = 1.36 \[ x + \frac {\sin ^{3}{\relax (x )}}{3 \cos ^{3}{\relax (x )}} - \frac {\sin {\relax (x )}}{\cos {\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)**4,x)

[Out]

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

________________________________________________________________________________________