3.17 \(\int \cot ^8(c+d x) (a+a \sec (c+d x)) \, dx\)

Optimal. Leaf size=111 \[ -\frac {\cot ^7(c+d x) (a \sec (c+d x)+a)}{7 d}+\frac {\cot ^5(c+d x) (6 a \sec (c+d x)+7 a)}{35 d}-\frac {\cot ^3(c+d x) (24 a \sec (c+d x)+35 a)}{105 d}+\frac {\cot (c+d x) (16 a \sec (c+d x)+35 a)}{35 d}+a x \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac {\cot ^7(c+d x) (a \sec (c+d x)+a)}{7 d}+\frac {\cot ^5(c+d x) (6 a \sec (c+d x)+7 a)}{35 d}-\frac {\cot ^3(c+d x) (24 a \sec (c+d x)+35 a)}{105 d}+\frac {\cot (c+d x) (16 a \sec (c+d x)+35 a)}{35 d}+a x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^8*(a + a*Sec[c + d*x]),x]

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.06, size = 92, normalized size = 0.83 \[ -\frac {a \cot ^7(c+d x) \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};-\tan ^2(c+d x)\right )}{7 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{d}+\frac {a \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^8*(a + a*Sec[c + d*x]),x]

[Out]

(a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*Cot[c +
d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, -Tan[c + d*x]^2])/(7*d)

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fricas [B]  time = 0.74, size = 210, normalized size = 1.89 \[ \frac {176 \, a \cos \left (d x + c\right )^{6} - 71 \, a \cos \left (d x + c\right )^{5} - 335 \, a \cos \left (d x + c\right )^{4} + 125 \, a \cos \left (d x + c\right )^{3} + 225 \, a \cos \left (d x + c\right )^{2} - 57 \, a \cos \left (d x + c\right ) + 105 \, {\left (a d x \cos \left (d x + c\right )^{5} - a d x \cos \left (d x + c\right )^{4} - 2 \, a d x \cos \left (d x + c\right )^{3} + 2 \, a d x \cos \left (d x + c\right )^{2} + a d x \cos \left (d x + c\right ) - a d x\right )} \sin \left (d x + c\right ) - 48 \, a}{105 \, {\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^8*(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.35, size = 113, normalized size = 1.02 \[ -\frac {21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6720 \, {\left (d x + c\right )} a + 3045 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {6720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1015 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 168 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{6720 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^8*(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

-1/6720*(21*a*tan(1/2*d*x + 1/2*c)^5 - 280*a*tan(1/2*d*x + 1/2*c)^3 - 6720*(d*x + c)*a + 3045*a*tan(1/2*d*x +
1/2*c) - (6720*a*tan(1/2*d*x + 1/2*c)^6 - 1015*a*tan(1/2*d*x + 1/2*c)^4 + 168*a*tan(1/2*d*x + 1/2*c)^2 - 15*a)
/tan(1/2*d*x + 1/2*c)^7)/d

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maple [A]  time = 0.95, size = 162, normalized size = 1.46 \[ \frac {a \left (-\frac {\left (\cot ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{8}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{8}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{8}\left (d x +c \right )}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^8*(a+a*sec(d*x+c)),x)

[Out]

1/d*(a*(-1/7*cot(d*x+c)^7+1/5*cot(d*x+c)^5-1/3*cot(d*x+c)^3+cot(d*x+c)+d*x+c)+a*(-1/7/sin(d*x+c)^7*cos(d*x+c)^
8+1/35/sin(d*x+c)^5*cos(d*x+c)^8-1/35/sin(d*x+c)^3*cos(d*x+c)^8+1/7/sin(d*x+c)*cos(d*x+c)^8+1/7*(16/5+cos(d*x+
c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c)))

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maxima [A]  time = 1.03, size = 100, normalized size = 0.90 \[ \frac {{\left (105 \, d x + 105 \, c + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a + \frac {3 \, {\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} a}{\sin \left (d x + c\right )^{7}}}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^8*(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

1/105*((105*d*x + 105*c + (105*tan(d*x + c)^6 - 35*tan(d*x + c)^4 + 21*tan(d*x + c)^2 - 15)/tan(d*x + c)^7)*a
+ 3*(35*sin(d*x + c)^6 - 35*sin(d*x + c)^4 + 21*sin(d*x + c)^2 - 5)*a/sin(d*x + c)^7)/d

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mupad [B]  time = 1.98, size = 204, normalized size = 1.84 \[ -\frac {a\,\left (15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+3045\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1015\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-168\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (c+d\,x\right )\right )}{6720\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^8*(a + a/cos(c + d*x)),x)

[Out]

-(a*(15*cos(c/2 + (d*x)/2)^12 + 21*sin(c/2 + (d*x)/2)^12 - 280*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 30
45*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 + 1015*cos(c/2 +
(d*x)/2)^8*sin(c/2 + (d*x)/2)^4 - 168*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 - 6720*cos(c/2 + (d*x)/2)^5*
sin(c/2 + (d*x)/2)^7*(c + d*x)))/(6720*d*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**8*(a+a*sec(d*x+c)),x)

[Out]

Timed out

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