### 3.91 $$\int \tan ^6(x) \, dx$$

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

[Out]

-x + Tan[x] - Tan[x]^3/3 + Tan[x]^5/5

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Rubi [A]  time = 0.0131138, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, 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 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]

Rule 8

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

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 = 0.003632, 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

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Maple [A]  time = 0.002, size = 19, normalized size = 0.9 \begin{align*} -x+\tan \left ( x \right ) -{\frac{ \left ( \tan \left ( x \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \tan \left ( x \right ) \right ) ^{5}}{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^6,x)

[Out]

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

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Maxima [A]  time = 1.41082, size = 24, normalized size = 1.09 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} - \frac{1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \end{align*}

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)

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Fricas [A]  time = 2.03569, size = 57, normalized size = 2.59 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} - \frac{1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \end{align*}

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)

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Sympy [A]  time = 0.067084, size = 31, normalized size = 1.41 \begin{align*} - 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 )}} \end{align*}

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)

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Giac [A]  time = 1.05793, size = 24, normalized size = 1.09 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} - \frac{1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \end{align*}

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)