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+1/105*cot(d*x+c)^3*(35-24*sec(d*x+c))/a/d-1/35*cot(d*x+c)*(35-16*sec(d*x+c))/a/d-1/35*cot(d*x+c)^5*(7-6*s
ec(d*x+c))/a/d+1/7*cot(d*x+c)^7*(1-sec(d*x+c))/a/d

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Rubi [A]  time = 0.16, antiderivative size = 117, normalized size of antiderivative = 1.00, 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 8

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

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 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]

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*}

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Mathematica [B]  time = 1.10, 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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fricas [A]  time = 0.47, size = 198, normalized size = 1.69 \[ -\frac {176 \, \cos \left (d x + c\right )^{6} + 71 \, \cos \left (d x + c\right )^{5} - 335 \, \cos \left (d x + c\right )^{4} - 125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 105 \, {\left (d x \cos \left (d x + c\right )^{5} + d x \cos \left (d x + c\right )^{4} - 2 \, d x \cos \left (d x + c\right )^{3} - 2 \, d x \cos \left (d x + c\right )^{2} + d x \cos \left (d x + c\right ) + d x\right )} \sin \left (d x + c\right ) + 57 \, \cos \left (d x + c\right ) - 48}{105 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )} \]

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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giac [A]  time = 0.39, size = 127, normalized size = 1.09 \[ -\frac {\frac {6720 \, {\left (d x + c\right )}}{a} + \frac {7 \, {\left (435 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {15 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 168 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1015 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6720 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{7}}}{6720 \, d} \]

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

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maple [A]  time = 0.77, size = 150, normalized size = 1.28 \[ -\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{448 a d}+\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{40 a d}-\frac {29 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {1}{320 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{24 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {29}{64 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d} \]

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)-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)-2/a/d*arcta
n(tan(1/2*d*x+1/2*c))

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maxima [A]  time = 0.59, size = 177, normalized size = 1.51 \[ \frac {\frac {\frac {6720 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1015 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {168 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a} - \frac {13440 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {7 \, {\left (\frac {40 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {435 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a \sin \left (d x + c\right )^{5}}}{6720 \, d} \]

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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mupad [B]  time = 2.04, size = 206, normalized size = 1.76 \[ -\frac {21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-168\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+1015\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3045\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )}{6720\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(21*cos(c/2 + (d*x)/2)^12 + 15*sin(c/2 + (d*x)/2)^12 - 168*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 1015*
cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 - 6720*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6 + 3045*cos(c/2 + (d
*x)/2)^8*sin(c/2 + (d*x)/2)^4 - 280*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 6720*cos(c/2 + (d*x)/2)^7*sin
(c/2 + (d*x)/2)^5*(c + d*x))/(6720*a*d*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{6}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]

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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