3.6 \(\int \tan ^6(c+d x) \, dx\)

Optimal. Leaf size=44 \[ \frac{\tan ^5(c+d x)}{5 d}-\frac{\tan ^3(c+d x)}{3 d}+\frac{\tan (c+d x)}{d}-x \]

[Out]

-x + Tan[c + d*x]/d - Tan[c + d*x]^3/(3*d) + Tan[c + d*x]^5/(5*d)

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Rubi [A]  time = 0.0246057, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ \frac{\tan ^5(c+d x)}{5 d}-\frac{\tan ^3(c+d x)}{3 d}+\frac{\tan (c+d x)}{d}-x \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^6,x]

[Out]

-x + Tan[c + d*x]/d - Tan[c + d*x]^3/(3*d) + Tan[c + d*x]^5/(5*d)

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

Mathematica [A]  time = 0.0150459, size = 53, normalized size = 1.2 \[ \frac{\tan ^5(c+d x)}{5 d}-\frac{\tan ^3(c+d x)}{3 d}-\frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^6,x]

[Out]

-(ArcTan[Tan[c + d*x]]/d) + Tan[c + d*x]/d - Tan[c + d*x]^3/(3*d) + Tan[c + d*x]^5/(5*d)

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Maple [A]  time = 0.003, size = 50, normalized size = 1.1 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{\tan \left ( dx+c \right ) }{d}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^6,x)

[Out]

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

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Maxima [A]  time = 1.86456, size = 55, normalized size = 1.25 \begin{align*} \frac{3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6,x, algorithm="maxima")

[Out]

1/15*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x - 15*c + 15*tan(d*x + c))/d

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Fricas [A]  time = 1.53716, size = 99, normalized size = 2.25 \begin{align*} \frac{3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x + 15 \, \tan \left (d x + c\right )}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6,x, algorithm="fricas")

[Out]

1/15*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x + 15*tan(d*x + c))/d

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Sympy [A]  time = 0.636812, size = 39, normalized size = 0.89 \begin{align*} \begin{cases} - x + \frac{\tan ^{5}{\left (c + d x \right )}}{5 d} - \frac{\tan ^{3}{\left (c + d x \right )}}{3 d} + \frac{\tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \tan ^{6}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**6,x)

[Out]

Piecewise((-x + tan(c + d*x)**5/(5*d) - tan(c + d*x)**3/(3*d) + tan(c + d*x)/d, Ne(d, 0)), (x*tan(c)**6, True)
)

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Giac [B]  time = 4.1907, size = 1335, normalized size = 30.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6,x, algorithm="giac")

[Out]

1/60*(15*pi - 60*d*x*tan(d*x)^5*tan(c)^5 - 15*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) -
2*tan(c))*tan(d*x)^5*tan(c)^5 - 15*pi*tan(d*x)^5*tan(c)^5 + 30*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c)
))*tan(d*x)^5*tan(c)^5 + 30*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^5*tan(c)^5 + 300*d*x*ta
n(d*x)^4*tan(c)^4 + 75*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*ta
n(c)^4 + 75*pi*tan(d*x)^4*tan(c)^4 - 150*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c)))*tan(d*x)^4*tan(c)^4
- 150*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4*tan(c)^4 - 60*tan(d*x)^5*tan(c)^4 - 60*tan
(d*x)^4*tan(c)^5 - 600*d*x*tan(d*x)^3*tan(c)^3 - 150*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(
d*x) - 2*tan(c))*tan(d*x)^3*tan(c)^3 + 20*tan(d*x)^5*tan(c)^2 - 150*pi*tan(d*x)^3*tan(c)^3 + 300*arctan((tan(d
*x)*tan(c) - 1)/(tan(d*x) + tan(c)))*tan(d*x)^3*tan(c)^3 + 300*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1
))*tan(d*x)^3*tan(c)^3 + 300*tan(d*x)^4*tan(c)^3 + 300*tan(d*x)^3*tan(c)^4 + 20*tan(d*x)^2*tan(c)^5 + 600*d*x*
tan(d*x)^2*tan(c)^2 + 150*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*tan(c))*tan(d*x)^2
*tan(c)^2 - 12*tan(d*x)^5 - 100*tan(d*x)^4*tan(c) + 150*pi*tan(d*x)^2*tan(c)^2 - 300*arctan((tan(d*x)*tan(c) -
1)/(tan(d*x) + tan(c)))*tan(d*x)^2*tan(c)^2 - 300*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^
2*tan(c)^2 - 600*tan(d*x)^3*tan(c)^2 - 600*tan(d*x)^2*tan(c)^3 - 100*tan(d*x)*tan(c)^4 - 12*tan(c)^5 - 300*d*x
*tan(d*x)*tan(c) - 75*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*tan(c))*tan(d*x)*tan(c
) + 20*tan(d*x)^3 - 75*pi*tan(d*x)*tan(c) + 150*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c)))*tan(d*x)*tan
(c) + 150*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)*tan(c) + 300*tan(d*x)^2*tan(c) + 300*tan(
d*x)*tan(c)^2 + 20*tan(c)^3 + 60*d*x + 15*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*ta
n(c)) - 30*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c))) - 30*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c)
- 1)) - 60*tan(d*x) - 60*tan(c))/(d*tan(d*x)^5*tan(c)^5 - 5*d*tan(d*x)^4*tan(c)^4 + 10*d*tan(d*x)^3*tan(c)^3 -
10*d*tan(d*x)^2*tan(c)^2 + 5*d*tan(d*x)*tan(c) - d)