3.53 \(\int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=141 \[ -\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x \]

[Out]

a^3*x + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x]^5)/(5*d) - (4*a^3*Cot[c + d*x]^7
)/(7*d) + (3*a^3*Csc[c + d*x])/d - (10*a^3*Csc[c + d*x]^3)/(3*d) + (11*a^3*Csc[c + d*x]^5)/(5*d) - (4*a^3*Csc[
c + d*x]^7)/(7*d)

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Rubi [A]  time = 0.178499, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30, 270} \[ -\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^8*(a + a*Sec[c + d*x])^3,x]

[Out]

a^3*x + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x]^5)/(5*d) - (4*a^3*Cot[c + d*x]^7
)/(7*d) + (3*a^3*Csc[c + d*x])/d - (10*a^3*Csc[c + d*x]^3)/(3*d) + (11*a^3*Csc[c + d*x]^5)/(5*d) - (4*a^3*Csc[
c + d*x]^7)/(7*d)

Rule 3886

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

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

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

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cot ^8(c+d x)+3 a^3 \cot ^7(c+d x) \csc (c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+a^3 \cot ^5(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^8(c+d x) \, dx+a^3 \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^7(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot ^7(c+d x)}{7 d}-a^3 \int \cot ^6(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+a^3 \int \cot ^4(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}-a^3 \int \cot ^2(c+d x) \, dx\\ &=\frac{a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}+a^3 \int 1 \, dx\\ &=a^3 x+\frac{a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.986898, size = 252, normalized size = 1.79 \[ \frac{a^3 \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \csc ^7\left (\frac{1}{2} (c+d x)\right ) \sec \left (\frac{1}{2} (c+d x)\right ) (-23282 \sin (c+d x)+23282 \sin (2 (c+d x))-9978 \sin (3 (c+d x))+1663 \sin (4 (c+d x))-13720 \sin (2 c+d x)+15512 \sin (c+2 d x)+9240 \sin (3 c+2 d x)-8088 \sin (2 c+3 d x)-2520 \sin (4 c+3 d x)+1768 \sin (3 c+4 d x)-5880 d x \cos (2 c+d x)-5880 d x \cos (c+2 d x)+5880 d x \cos (3 c+2 d x)+2520 d x \cos (2 c+3 d x)-2520 d x \cos (4 c+3 d x)-420 d x \cos (3 c+4 d x)+420 d x \cos (5 c+4 d x)+4200 \sin (c)-11032 \sin (d x)+5880 d x \cos (d x))}{215040 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^8*(a + a*Sec[c + d*x])^3,x]

[Out]

(a^3*Csc[c/2]*Csc[(c + d*x)/2]^7*Sec[c/2]*Sec[(c + d*x)/2]*(5880*d*x*Cos[d*x] - 5880*d*x*Cos[2*c + d*x] - 5880
*d*x*Cos[c + 2*d*x] + 5880*d*x*Cos[3*c + 2*d*x] + 2520*d*x*Cos[2*c + 3*d*x] - 2520*d*x*Cos[4*c + 3*d*x] - 420*
d*x*Cos[3*c + 4*d*x] + 420*d*x*Cos[5*c + 4*d*x] + 4200*Sin[c] - 11032*Sin[d*x] - 23282*Sin[c + d*x] + 23282*Si
n[2*(c + d*x)] - 9978*Sin[3*(c + d*x)] + 1663*Sin[4*(c + d*x)] - 13720*Sin[2*c + d*x] + 15512*Sin[c + 2*d*x] +
9240*Sin[3*c + 2*d*x] - 8088*Sin[2*c + 3*d*x] - 2520*Sin[4*c + 3*d*x] + 1768*Sin[3*c + 4*d*x]))/(215040*d)

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Maple [B]  time = 0.08, size = 293, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ({a}^{3} \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 ) +3\,{a}^{3} \left ( -1/7\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+1/35\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-1/35\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+1/7\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{\sin \left ( dx+c \right ) }}+1/7\, \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+6/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+8/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{a}^{3} \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{105\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{35\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{35} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \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))^3,x)

[Out]

1/d*(a^3*(-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)+3*a^3*(-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+co
s(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))-3/7*a^3/sin(d*x+c)^7*cos(d*x+c)^7+a^3*(-1/7/sin(d*x+
c)^7*cos(d*x+c)^6-1/35/sin(d*x+c)^5*cos(d*x+c)^6+1/105/sin(d*x+c)^3*cos(d*x+c)^6-1/35/sin(d*x+c)*cos(d*x+c)^6-
1/35*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)))

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Maxima [A]  time = 1.67757, size = 205, normalized size = 1.45 \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^{3} + \frac{9 \,{\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^{3}}{\sin \left (d x + c\right )^{7}} - \frac{{\left (35 \, \sin \left (d x + c\right )^{4} - 42 \, \sin \left (d x + c\right )^{2} + 15\right )} a^{3}}{\sin \left (d x + c\right )^{7}} - \frac{45 \, a^{3}}{\tan \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))^3,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 + 9*(35*sin(d*x + c)^6 - 35*sin(d*x + c)^4 + 21*sin(d*x + c)^2 - 5)*a^3/sin(d*x + c)^7 - (35*sin(d*x + c)^4
- 42*sin(d*x + c)^2 + 15)*a^3/sin(d*x + c)^7 - 45*a^3/tan(d*x + c)^7)/d

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Fricas [A]  time = 1.13793, size = 402, normalized size = 2.85 \begin{align*} \frac{221 \, a^{3} \cos \left (d x + c\right )^{4} - 348 \, a^{3} \cos \left (d x + c\right )^{3} - 25 \, a^{3} \cos \left (d x + c\right )^{2} + 303 \, a^{3} \cos \left (d x + c\right ) - 136 \, a^{3} + 105 \,{\left (a^{3} d x \cos \left (d x + c\right )^{3} - 3 \, a^{3} d x \cos \left (d x + c\right )^{2} + 3 \, a^{3} d x \cos \left (d x + c\right ) - a^{3} d x\right )} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + 3 \, 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))^3,x, algorithm="fricas")

[Out]

1/105*(221*a^3*cos(d*x + c)^4 - 348*a^3*cos(d*x + c)^3 - 25*a^3*cos(d*x + c)^2 + 303*a^3*cos(d*x + c) - 136*a^
3 + 105*(a^3*d*x*cos(d*x + c)^3 - 3*a^3*d*x*cos(d*x + c)^2 + 3*a^3*d*x*cos(d*x + c) - a^3*d*x)*sin(d*x + c))/(
(d*cos(d*x + c)^3 - 3*d*cos(d*x + c)^2 + 3*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))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.58607, size = 130, normalized size = 0.92 \begin{align*} \frac{1680 \,{\left (d x + c\right )} a^{3} - 105 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{2730 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 560 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 126 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{1680 \, 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))^3,x, algorithm="giac")

[Out]

1/1680*(1680*(d*x + c)*a^3 - 105*a^3*tan(1/2*d*x + 1/2*c) + (2730*a^3*tan(1/2*d*x + 1/2*c)^6 - 560*a^3*tan(1/2
*d*x + 1/2*c)^4 + 126*a^3*tan(1/2*d*x + 1/2*c)^2 - 15*a^3)/tan(1/2*d*x + 1/2*c)^7)/d