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

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Rubi [A]  time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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(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*}

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Mathematica [A]  time = 0.02, size = 53, normalized size = 1.20 \[ -\frac {\tan ^{-1}(\tan (c+d x))}{d}+\frac {\tan ^5(c+d x)}{5 d}-\frac {\tan ^3(c+d x)}{3 d}+\frac {\tan (c+d x)}{d} \]

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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fricas [A]  time = 0.56, size = 38, normalized size = 0.86 \[ \frac {3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x + 15 \, \tan \left (d x + c\right )}{15 \, d} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check si
gn: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check si
gn: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)(-60*d*x*tan(c)^5*tan(d*x)^5+300*d*x*tan(c)^4*tan(d*x)^4-600*d
*x*tan(c)^3*tan(d*x)^3+600*d*x*tan(c)^2*tan(d*x)^2-300*d*x*tan(c)*tan(d*x)+60*d*x-15*pi*sign(2*tan(c)^2*tan(d*
x)+2*tan(c)*tan(d*x)^2-2*tan(c)-2*tan(d*x))*tan(c)^5*tan(d*x)^5+75*pi*sign(2*tan(c)^2*tan(d*x)+2*tan(c)*tan(d*
x)^2-2*tan(c)-2*tan(d*x))*tan(c)^4*tan(d*x)^4-150*pi*sign(2*tan(c)^2*tan(d*x)+2*tan(c)*tan(d*x)^2-2*tan(c)-2*t
an(d*x))*tan(c)^3*tan(d*x)^3+150*pi*sign(2*tan(c)^2*tan(d*x)+2*tan(c)*tan(d*x)^2-2*tan(c)-2*tan(d*x))*tan(c)^2
*tan(d*x)^2-75*pi*sign(2*tan(c)^2*tan(d*x)+2*tan(c)*tan(d*x)^2-2*tan(c)-2*tan(d*x))*tan(c)*tan(d*x)+15*pi*sign
(2*tan(c)^2*tan(d*x)+2*tan(c)*tan(d*x)^2-2*tan(c)-2*tan(d*x))-15*pi*tan(c)^5*tan(d*x)^5+75*pi*tan(c)^4*tan(d*x
)^4-150*pi*tan(c)^3*tan(d*x)^3+150*pi*tan(c)^2*tan(d*x)^2-75*pi*tan(c)*tan(d*x)+15*pi+30*atan((tan(c)*tan(d*x)
-1)/(tan(c)+tan(d*x)))*tan(c)^5*tan(d*x)^5-150*atan((tan(c)*tan(d*x)-1)/(tan(c)+tan(d*x)))*tan(c)^4*tan(d*x)^4
+300*atan((tan(c)*tan(d*x)-1)/(tan(c)+tan(d*x)))*tan(c)^3*tan(d*x)^3-300*atan((tan(c)*tan(d*x)-1)/(tan(c)+tan(
d*x)))*tan(c)^2*tan(d*x)^2+150*atan((tan(c)*tan(d*x)-1)/(tan(c)+tan(d*x)))*tan(c)*tan(d*x)-30*atan((tan(c)*tan
(d*x)-1)/(tan(c)+tan(d*x)))+30*atan((tan(c)+tan(d*x))/(tan(c)*tan(d*x)-1))*tan(c)^5*tan(d*x)^5-150*atan((tan(c
)+tan(d*x))/(tan(c)*tan(d*x)-1))*tan(c)^4*tan(d*x)^4+300*atan((tan(c)+tan(d*x))/(tan(c)*tan(d*x)-1))*tan(c)^3*
tan(d*x)^3-300*atan((tan(c)+tan(d*x))/(tan(c)*tan(d*x)-1))*tan(c)^2*tan(d*x)^2+150*atan((tan(c)+tan(d*x))/(tan
(c)*tan(d*x)-1))*tan(c)*tan(d*x)-30*atan((tan(c)+tan(d*x))/(tan(c)*tan(d*x)-1))-60*tan(c)^5*tan(d*x)^4+20*tan(
c)^5*tan(d*x)^2-12*tan(c)^5-60*tan(c)^4*tan(d*x)^5+300*tan(c)^4*tan(d*x)^3-100*tan(c)^4*tan(d*x)+300*tan(c)^3*
tan(d*x)^4-600*tan(c)^3*tan(d*x)^2+20*tan(c)^3+20*tan(c)^2*tan(d*x)^5-600*tan(c)^2*tan(d*x)^3+300*tan(c)^2*tan
(d*x)-100*tan(c)*tan(d*x)^4+300*tan(c)*tan(d*x)^2-60*tan(c)-12*tan(d*x)^5+20*tan(d*x)^3-60*tan(d*x))/(60*d*tan
(c)^5*tan(d*x)^5-300*d*tan(c)^4*tan(d*x)^4+600*d*tan(c)^3*tan(d*x)^3-600*d*tan(c)^2*tan(d*x)^2+300*d*tan(c)*ta
n(d*x)-60*d)

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maple [A]  time = 0.01, size = 50, normalized size = 1.14 \[ \frac {\tan ^{5}\left (d x +c \right )}{5 d}-\frac {\tan ^{3}\left (d x +c \right )}{3 d}+\frac {\tan \left (d x +c \right )}{d}-\frac {\arctan \left (\tan \left (d x +c \right )\right )}{d} \]

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 = 0.55, size = 41, normalized size = 0.93 \[ \frac {3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )}{15 \, d} \]

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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mupad [B]  time = 2.53, size = 35, normalized size = 0.80 \[ \frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^5}{5}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+\mathrm {tan}\left (c+d\,x\right )}{d}-x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^6,x)

[Out]

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

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sympy [A]  time = 0.46, size = 39, normalized size = 0.89 \[ \begin {cases} - x + \frac {\tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {\tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {\tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \tan ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]

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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