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 - (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)

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Rubi [A]  time = 0.113559, 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} \[ -\frac{\cot ^7(c+d x) (a \sec (c+d x)+a)}{7 d}+\frac{\cot ^5(c+d x) (6 a \sec (c+d x)+7 a)}{35 d}-\frac{\cot ^3(c+d x) (24 a \sec (c+d x)+35 a)}{105 d}+\frac{\cot (c+d x) (16 a \sec (c+d x)+35 a)}{35 d}+a x \]

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 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+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*}

Mathematica [C]  time = 0.0496859, size = 92, normalized size = 0.83 \[ -\frac{a \cot ^7(c+d x) \text{Hypergeometric2F1}\left (-\frac{7}{2},1,-\frac{5}{2},-\tan ^2(c+d x)\right )}{7 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d} \]

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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Maple [A]  time = 0.077, 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 ) +a \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+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 = 1.78986, 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 )} a}{\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+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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Fricas [B]  time = 0.896661, size = 551, normalized size = 4.96 \begin{align*} \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 )} \end{align*}

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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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+a*sec(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.38131, size = 153, normalized size = 1.38 \begin{align*} -\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} \end{align*}

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