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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.111427, antiderivative size = 111, normalized size of antiderivative = 1., 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} \[ -\frac{\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac{\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}-\frac{\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac{\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}+a x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.0460286, size = 92, normalized size = 0.83 \[ -\frac{a \cot ^7(c+d x) \text{Hypergeometric2F1}\left (-\frac{7}{2},1,-\frac{5}{2},-\tan ^2(c+d x)\right )}{7 d}-\frac{b \csc ^7(c+d x)}{7 d}+\frac{3 b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{d}+\frac{b \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.051, size = 162, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5}}-{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3}}+\cot \left ( dx+c \right ) +dx+c \right ) +b \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{7\,\sin \left ( dx+c \right ) }}+{\frac{\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^8*(a+b*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)+b*(-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)))

________________________________________________________________________________________

Maxima [A]  time = 1.48094, size = 135, normalized size = 1.22 \begin{align*} \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 )} b}{\sin \left (d x + c\right )^{7}}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^8*(a+b*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)*b/sin(d*x + c)^7)/d

________________________________________________________________________________________

Fricas [A]  time = 0.805925, size = 477, normalized size = 4.3 \begin{align*} \frac{176 \, a \cos \left (d x + c\right )^{7} + 105 \, b \cos \left (d x + c\right )^{6} - 406 \, a \cos \left (d x + c\right )^{5} - 210 \, b \cos \left (d x + c\right )^{4} + 350 \, a \cos \left (d x + c\right )^{3} + 168 \, b \cos \left (d x + c\right )^{2} - 105 \, a \cos \left (d x + c\right ) + 105 \,{\left (a d x \cos \left (d x + c\right )^{6} - 3 \, a d x \cos \left (d x + c\right )^{4} + 3 \, a d x \cos \left (d x + c\right )^{2} - a d x\right )} \sin \left (d x + c\right ) - 48 \, b}{105 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - 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)^8*(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/105*(176*a*cos(d*x + c)^7 + 105*b*cos(d*x + c)^6 - 406*a*cos(d*x + c)^5 - 210*b*cos(d*x + c)^4 + 350*a*cos(d
*x + c)^3 + 168*b*cos(d*x + c)^2 - 105*a*cos(d*x + c) + 105*(a*d*x*cos(d*x + c)^6 - 3*a*d*x*cos(d*x + c)^4 + 3
*a*d*x*cos(d*x + c)^2 - a*d*x)*sin(d*x + c) - 48*b)/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)
^2 - d)*sin(d*x + c))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.47965, size = 304, normalized size = 2.74 \begin{align*} \frac{15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 189 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 147 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1295 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 735 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 13440 \,{\left (d x + c\right )} a - 9765 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3675 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{9765 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3675 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1295 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 735 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 189 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 147 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a - 15 \, b}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13440*(15*a*tan(1/2*d*x + 1/2*c)^7 - 15*b*tan(1/2*d*x + 1/2*c)^7 - 189*a*tan(1/2*d*x + 1/2*c)^5 + 147*b*tan(
1/2*d*x + 1/2*c)^5 + 1295*a*tan(1/2*d*x + 1/2*c)^3 - 735*b*tan(1/2*d*x + 1/2*c)^3 + 13440*(d*x + c)*a - 9765*a
*tan(1/2*d*x + 1/2*c) + 3675*b*tan(1/2*d*x + 1/2*c) + (9765*a*tan(1/2*d*x + 1/2*c)^6 + 3675*b*tan(1/2*d*x + 1/
2*c)^6 - 1295*a*tan(1/2*d*x + 1/2*c)^4 - 735*b*tan(1/2*d*x + 1/2*c)^4 + 189*a*tan(1/2*d*x + 1/2*c)^2 + 147*b*t
an(1/2*d*x + 1/2*c)^2 - 15*a - 15*b)/tan(1/2*d*x + 1/2*c)^7)/d