3.70 \(\int \frac{\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=117 \[ \frac{\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d}-\frac{\cot ^5(c+d x) (7-6 \sec (c+d x))}{35 a d}+\frac{\cot ^3(c+d x) (35-24 \sec (c+d x))}{105 a d}-\frac{\cot (c+d x) (35-16 \sec (c+d x))}{35 a d}-\frac{x}{a} \]

[Out]

-(x/a) + (Cot[c + d*x]^3*(35 - 24*Sec[c + d*x]))/(105*a*d) - (Cot[c + d*x]*(35 - 16*Sec[c + d*x]))/(35*a*d) -
(Cot[c + d*x]^5*(7 - 6*Sec[c + d*x]))/(35*a*d) + (Cot[c + d*x]^7*(1 - Sec[c + d*x]))/(7*a*d)

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Rubi [A]  time = 0.161961, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3888, 3882, 8} \[ \frac{\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d}-\frac{\cot ^5(c+d x) (7-6 \sec (c+d x))}{35 a d}+\frac{\cot ^3(c+d x) (35-24 \sec (c+d x))}{105 a d}-\frac{\cot (c+d x) (35-16 \sec (c+d x))}{35 a d}-\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x]),x]

[Out]

-(x/a) + (Cot[c + d*x]^3*(35 - 24*Sec[c + d*x]))/(105*a*d) - (Cot[c + d*x]*(35 - 16*Sec[c + d*x]))/(35*a*d) -
(Cot[c + d*x]^5*(7 - 6*Sec[c + d*x]))/(35*a*d) + (Cot[c + d*x]^7*(1 - Sec[c + d*x]))/(7*a*d)

Rule 3888

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rule 3882

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[((e*Cot[c
+ d*x])^(m + 1)*(a + b*Csc[c + d*x]))/(d*e*(m + 1)), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)*(
a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx &=\frac{\int \cot ^8(c+d x) (-a+a \sec (c+d x)) \, dx}{a^2}\\ &=\frac{\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d}+\frac{\int \cot ^6(c+d x) (7 a-6 a \sec (c+d x)) \, dx}{7 a^2}\\ &=-\frac{\cot ^5(c+d x) (7-6 \sec (c+d x))}{35 a d}+\frac{\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d}+\frac{\int \cot ^4(c+d x) (-35 a+24 a \sec (c+d x)) \, dx}{35 a^2}\\ &=\frac{\cot ^3(c+d x) (35-24 \sec (c+d x))}{105 a d}-\frac{\cot ^5(c+d x) (7-6 \sec (c+d x))}{35 a d}+\frac{\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d}+\frac{\int \cot ^2(c+d x) (105 a-48 a \sec (c+d x)) \, dx}{105 a^2}\\ &=\frac{\cot ^3(c+d x) (35-24 \sec (c+d x))}{105 a d}-\frac{\cot (c+d x) (35-16 \sec (c+d x))}{35 a d}-\frac{\cot ^5(c+d x) (7-6 \sec (c+d x))}{35 a d}+\frac{\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d}+\frac{\int -105 a \, dx}{105 a^2}\\ &=-\frac{x}{a}+\frac{\cot ^3(c+d x) (35-24 \sec (c+d x))}{105 a d}-\frac{\cot (c+d x) (35-16 \sec (c+d x))}{35 a d}-\frac{\cot ^5(c+d x) (7-6 \sec (c+d x))}{35 a d}+\frac{\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d}\\ \end{align*}

Mathematica [B]  time = 1.07246, size = 359, normalized size = 3.07 \[ \frac{\csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \csc ^5(c+d x) \sec (c+d x) (-22860 \sin (c+d x)-5715 \sin (2 (c+d x))+11430 \sin (3 (c+d x))+4572 \sin (4 (c+d x))-2286 \sin (5 (c+d x))-1143 \sin (6 (c+d x))+26208 \sin (2 c+d x)+14080 \sin (c+2 d x)-16400 \sin (2 c+3 d x)-11760 \sin (4 c+3 d x)-7904 \sin (3 c+4 d x)-3360 \sin (5 c+4 d x)+3952 \sin (4 c+5 d x)+1680 \sin (6 c+5 d x)+2816 \sin (5 c+6 d x)+16800 d x \cos (2 c+d x)-4200 d x \cos (c+2 d x)+4200 d x \cos (3 c+2 d x)+8400 d x \cos (2 c+3 d x)-8400 d x \cos (4 c+3 d x)+3360 d x \cos (3 c+4 d x)-3360 d x \cos (5 c+4 d x)-1680 d x \cos (4 c+5 d x)+1680 d x \cos (6 c+5 d x)-840 d x \cos (5 c+6 d x)+840 d x \cos (7 c+6 d x)+3136 \sin (c)+30112 \sin (d x)-16800 d x \cos (d x))}{107520 a d (\sec (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x]),x]

[Out]

(Csc[c/2]*Csc[c + d*x]^5*Sec[c/2]*Sec[c + d*x]*(-16800*d*x*Cos[d*x] + 16800*d*x*Cos[2*c + d*x] - 4200*d*x*Cos[
c + 2*d*x] + 4200*d*x*Cos[3*c + 2*d*x] + 8400*d*x*Cos[2*c + 3*d*x] - 8400*d*x*Cos[4*c + 3*d*x] + 3360*d*x*Cos[
3*c + 4*d*x] - 3360*d*x*Cos[5*c + 4*d*x] - 1680*d*x*Cos[4*c + 5*d*x] + 1680*d*x*Cos[6*c + 5*d*x] - 840*d*x*Cos
[5*c + 6*d*x] + 840*d*x*Cos[7*c + 6*d*x] + 3136*Sin[c] + 30112*Sin[d*x] - 22860*Sin[c + d*x] - 5715*Sin[2*(c +
d*x)] + 11430*Sin[3*(c + d*x)] + 4572*Sin[4*(c + d*x)] - 2286*Sin[5*(c + d*x)] - 1143*Sin[6*(c + d*x)] + 2620
8*Sin[2*c + d*x] + 14080*Sin[c + 2*d*x] - 16400*Sin[2*c + 3*d*x] - 11760*Sin[4*c + 3*d*x] - 7904*Sin[3*c + 4*d
*x] - 3360*Sin[5*c + 4*d*x] + 3952*Sin[4*c + 5*d*x] + 1680*Sin[6*c + 5*d*x] + 2816*Sin[5*c + 6*d*x]))/(107520*
a*d*(1 + Sec[c + d*x]))

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Maple [A]  time = 0.067, size = 150, normalized size = 1.3 \begin{align*} -{\frac{1}{448\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{1}{40\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{29}{192\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}-{\frac{1}{320\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{1}{24\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{29}{64\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6/(a+a*sec(d*x+c)),x)

[Out]

-1/448/a/d*tan(1/2*d*x+1/2*c)^7+1/40/a/d*tan(1/2*d*x+1/2*c)^5-29/192/a/d*tan(1/2*d*x+1/2*c)^3+1/a/d*tan(1/2*d*
x+1/2*c)-2/d/a*arctan(tan(1/2*d*x+1/2*c))-1/320/a/d/tan(1/2*d*x+1/2*c)^5+1/24/a/d/tan(1/2*d*x+1/2*c)^3-29/64/a
/d/tan(1/2*d*x+1/2*c)

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Maxima [A]  time = 1.69761, size = 239, normalized size = 2.04 \begin{align*} \frac{\frac{\frac{6720 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1015 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{168 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a} - \frac{13440 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{7 \,{\left (\frac{40 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{435 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 3\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a \sin \left (d x + c\right )^{5}}}{6720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

1/6720*((6720*sin(d*x + c)/(cos(d*x + c) + 1) - 1015*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 168*sin(d*x + c)^5/
(cos(d*x + c) + 1)^5 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a - 13440*arctan(sin(d*x + c)/(cos(d*x + c) + 1
))/a + 7*(40*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 435*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3)*(cos(d*x + c)
+ 1)^5/(a*sin(d*x + c)^5))/d

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Fricas [A]  time = 1.16881, size = 533, normalized size = 4.56 \begin{align*} -\frac{176 \, \cos \left (d x + c\right )^{6} + 71 \, \cos \left (d x + c\right )^{5} - 335 \, \cos \left (d x + c\right )^{4} - 125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 105 \,{\left (d x \cos \left (d x + c\right )^{5} + d x \cos \left (d x + c\right )^{4} - 2 \, d x \cos \left (d x + c\right )^{3} - 2 \, d x \cos \left (d x + c\right )^{2} + d x \cos \left (d x + c\right ) + d x\right )} \sin \left (d x + c\right ) + 57 \, \cos \left (d x + c\right ) - 48}{105 \,{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

-1/105*(176*cos(d*x + c)^6 + 71*cos(d*x + c)^5 - 335*cos(d*x + c)^4 - 125*cos(d*x + c)^3 + 225*cos(d*x + c)^2
+ 105*(d*x*cos(d*x + c)^5 + d*x*cos(d*x + c)^4 - 2*d*x*cos(d*x + c)^3 - 2*d*x*cos(d*x + c)^2 + d*x*cos(d*x + c
) + d*x)*sin(d*x + c) + 57*cos(d*x + c) - 48)/((a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3
- 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d)*sin(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot ^{6}{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6/(a+a*sec(d*x+c)),x)

[Out]

Integral(cot(c + d*x)**6/(sec(c + d*x) + 1), x)/a

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Giac [A]  time = 1.32248, size = 171, normalized size = 1.46 \begin{align*} -\frac{\frac{6720 \,{\left (d x + c\right )}}{a} + \frac{7 \,{\left (435 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 40 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3\right )}}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}} + \frac{15 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 168 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1015 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6720 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{7}}}{6720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

-1/6720*(6720*(d*x + c)/a + 7*(435*tan(1/2*d*x + 1/2*c)^4 - 40*tan(1/2*d*x + 1/2*c)^2 + 3)/(a*tan(1/2*d*x + 1/
2*c)^5) + (15*a^6*tan(1/2*d*x + 1/2*c)^7 - 168*a^6*tan(1/2*d*x + 1/2*c)^5 + 1015*a^6*tan(1/2*d*x + 1/2*c)^3 -
6720*a^6*tan(1/2*d*x + 1/2*c))/a^7)/d