3.341 \(\int \tan ^6(x) \, dx\)

Optimal. Leaf size=22 \[ -x+\frac {\tan ^5(x)}{5}-\frac {\tan ^3(x)}{3}+\tan (x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 8} \[ -x+\frac {\tan ^5(x)}{5}-\frac {\tan ^3(x)}{3}+\tan (x) \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]^6,x]

[Out]

-x + Tan[x] - Tan[x]^3/3 + Tan[x]^5/5

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 ^6(x) \, dx &=\frac {\tan ^5(x)}{5}-\int \tan ^4(x) \, dx\\ &=-\frac {1}{3} \tan ^3(x)+\frac {\tan ^5(x)}{5}+\int \tan ^2(x) \, dx\\ &=\tan (x)-\frac {\tan ^3(x)}{3}+\frac {\tan ^5(x)}{5}-\int 1 \, dx\\ &=-x+\tan (x)-\frac {\tan ^3(x)}{3}+\frac {\tan ^5(x)}{5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 30, normalized size = 1.36 \[ -x+\frac {23 \tan (x)}{15}+\frac {1}{5} \tan (x) \sec ^4(x)-\frac {11}{15} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]^6,x]

[Out]

-x + (23*Tan[x])/15 - (11*Sec[x]^2*Tan[x])/15 + (Sec[x]^4*Tan[x])/5

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tan ^6(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Tan[x]^6,x]

[Out]

Could not integrate

________________________________________________________________________________________

fricas [A]  time = 0.79, size = 18, normalized size = 0.82 \[ \frac {1}{5} \, \tan \relax (x)^{5} - \frac {1}{3} \, \tan \relax (x)^{3} - x + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^6,x, algorithm="fricas")

[Out]

1/5*tan(x)^5 - 1/3*tan(x)^3 - x + tan(x)

________________________________________________________________________________________

giac [A]  time = 0.59, size = 18, normalized size = 0.82 \[ \frac {1}{5} \, \tan \relax (x)^{5} - \frac {1}{3} \, \tan \relax (x)^{3} - x + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^6,x, algorithm="giac")

[Out]

1/5*tan(x)^5 - 1/3*tan(x)^3 - x + tan(x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 19, normalized size = 0.86

method	result	size
norman	\(-x +\tan \relax (x )-\frac {\left (\tan ^{3}\relax (x )\right )}{3}+\frac {\left (\tan ^{5}\relax (x )\right )}{5}\)	\(19\)
derivativedivides	\(\frac {\left (\tan ^{5}\relax (x )\right )}{5}-\frac {\left (\tan ^{3}\relax (x )\right )}{3}+\tan \relax (x )-\arctan \left (\tan \relax (x )\right )\)	\(21\)
default	\(\frac {\left (\tan ^{5}\relax (x )\right )}{5}-\frac {\left (\tan ^{3}\relax (x )\right )}{3}+\tan \relax (x )-\arctan \left (\tan \relax (x )\right )\)	\(21\)
risch	\(-x +\frac {2 i \left (45 \,{\mathrm e}^{8 i x}+90 \,{\mathrm e}^{6 i x}+140 \,{\mathrm e}^{4 i x}+70 \,{\mathrm e}^{2 i x}+23\right )}{15 \left (1+{\mathrm e}^{2 i x}\right )^{5}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^6,x,method=_RETURNVERBOSE)

[Out]

-x+tan(x)-1/3*tan(x)^3+1/5*tan(x)^5

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 18, normalized size = 0.82 \[ \frac {1}{5} \, \tan \relax (x)^{5} - \frac {1}{3} \, \tan \relax (x)^{3} - x + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^6,x, algorithm="maxima")

[Out]

1/5*tan(x)^5 - 1/3*tan(x)^3 - x + tan(x)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 18, normalized size = 0.82 \[ \frac {{\mathrm {tan}\relax (x)}^5}{5}-\frac {{\mathrm {tan}\relax (x)}^3}{3}+\mathrm {tan}\relax (x)-x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^6,x)

[Out]

tan(x) - x - tan(x)^3/3 + tan(x)^5/5

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 31, normalized size = 1.41 \[ - x + \frac {\sin ^{5}{\relax (x )}}{5 \cos ^{5}{\relax (x )}} - \frac {\sin ^{3}{\relax (x )}}{3 \cos ^{3}{\relax (x )}} + \frac {\sin {\relax (x )}}{\cos {\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)**6,x)

[Out]

-x + sin(x)**5/(5*cos(x)**5) - sin(x)**3/(3*cos(x)**3) + sin(x)/cos(x)

________________________________________________________________________________________