\(\int \tan ^6(x) \, dx\) [341]

Optimal result

Integrand size = 4, antiderivative size = 22 \[ \int \tan ^6(x) \, dx=-x+\tan (x)-\frac {\tan ^3(x)}{3}+\frac {\tan ^5(x)}{5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3554, 8} \[ \int \tan ^6(x) \, dx=-x+\frac {\tan ^5(x)}{5}-\frac {\tan ^3(x)}{3}+\tan (x) \]

[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 3554

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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \tan ^6(x) \, dx=-\arctan (\tan (x))+\tan (x)-\frac {\tan ^3(x)}{3}+\frac {\tan ^5(x)}{5} \]

[In]

Integrate[Tan[x]^6,x]

[Out]

-ArcTan[Tan[x]] + Tan[x] - Tan[x]^3/3 + Tan[x]^5/5

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \tan ^6(x) \, dx=\frac {1}{5} \, \tan \left (x\right )^{5} - \frac {1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \tan ^6(x) \, dx=- x + \frac {\sin ^{5}{\left (x \right )}}{5 \cos ^{5}{\left (x \right )}} - \frac {\sin ^{3}{\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} + \frac {\sin {\left (x \right )}}{\cos {\left (x \right )}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \tan ^6(x) \, dx=\frac {1}{5} \, \tan \left (x\right )^{5} - \frac {1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \tan ^6(x) \, dx=\frac {1}{5} \, \tan \left (x\right )^{5} - \frac {1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \tan ^6(x) \, dx=\frac {{\mathrm {tan}\left (x\right )}^5}{5}-\frac {{\mathrm {tan}\left (x\right )}^3}{3}+\mathrm {tan}\left (x\right )-x \]

[In]

int(tan(x)^6,x)

[Out]

tan(x) - x - tan(x)^3/3 + tan(x)^5/5