3.37 \(\int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx\)

Optimal. Leaf size=210 \[ \frac{a^3 \sec ^{11}(c+d x)}{11 d}+\frac{3 a^3 \sec ^{10}(c+d x)}{10 d}-\frac{a^3 \sec ^9(c+d x)}{9 d}-\frac{11 a^3 \sec ^8(c+d x)}{8 d}-\frac{6 a^3 \sec ^7(c+d x)}{7 d}+\frac{7 a^3 \sec ^6(c+d x)}{3 d}+\frac{14 a^3 \sec ^5(c+d x)}{5 d}-\frac{3 a^3 \sec ^4(c+d x)}{2 d}-\frac{11 a^3 \sec ^3(c+d x)}{3 d}-\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d} \]

[Out]

-((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d - (a^3*Sec[c + d*x]^2)/(2*d) - (11*a^3*Sec[c + d*x]^3)/(
3*d) - (3*a^3*Sec[c + d*x]^4)/(2*d) + (14*a^3*Sec[c + d*x]^5)/(5*d) + (7*a^3*Sec[c + d*x]^6)/(3*d) - (6*a^3*Se
c[c + d*x]^7)/(7*d) - (11*a^3*Sec[c + d*x]^8)/(8*d) - (a^3*Sec[c + d*x]^9)/(9*d) + (3*a^3*Sec[c + d*x]^10)/(10
*d) + (a^3*Sec[c + d*x]^11)/(11*d)

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Rubi [A]  time = 0.10263, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac{a^3 \sec ^{11}(c+d x)}{11 d}+\frac{3 a^3 \sec ^{10}(c+d x)}{10 d}-\frac{a^3 \sec ^9(c+d x)}{9 d}-\frac{11 a^3 \sec ^8(c+d x)}{8 d}-\frac{6 a^3 \sec ^7(c+d x)}{7 d}+\frac{7 a^3 \sec ^6(c+d x)}{3 d}+\frac{14 a^3 \sec ^5(c+d x)}{5 d}-\frac{3 a^3 \sec ^4(c+d x)}{2 d}-\frac{11 a^3 \sec ^3(c+d x)}{3 d}-\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^9,x]

[Out]

-((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d - (a^3*Sec[c + d*x]^2)/(2*d) - (11*a^3*Sec[c + d*x]^3)/(
3*d) - (3*a^3*Sec[c + d*x]^4)/(2*d) + (14*a^3*Sec[c + d*x]^5)/(5*d) + (7*a^3*Sec[c + d*x]^6)/(3*d) - (6*a^3*Se
c[c + d*x]^7)/(7*d) - (11*a^3*Sec[c + d*x]^8)/(8*d) - (a^3*Sec[c + d*x]^9)/(9*d) + (3*a^3*Sec[c + d*x]^10)/(10
*d) + (a^3*Sec[c + d*x]^11)/(11*d)

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^4 (a+a x)^7}{x^{12}} \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a^{11}}{x^{12}}+\frac{3 a^{11}}{x^{11}}-\frac{a^{11}}{x^{10}}-\frac{11 a^{11}}{x^9}-\frac{6 a^{11}}{x^8}+\frac{14 a^{11}}{x^7}+\frac{14 a^{11}}{x^6}-\frac{6 a^{11}}{x^5}-\frac{11 a^{11}}{x^4}-\frac{a^{11}}{x^3}+\frac{3 a^{11}}{x^2}+\frac{a^{11}}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac{a^3 \log (\cos (c+d x))}{d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \sec ^2(c+d x)}{2 d}-\frac{11 a^3 \sec ^3(c+d x)}{3 d}-\frac{3 a^3 \sec ^4(c+d x)}{2 d}+\frac{14 a^3 \sec ^5(c+d x)}{5 d}+\frac{7 a^3 \sec ^6(c+d x)}{3 d}-\frac{6 a^3 \sec ^7(c+d x)}{7 d}-\frac{11 a^3 \sec ^8(c+d x)}{8 d}-\frac{a^3 \sec ^9(c+d x)}{9 d}+\frac{3 a^3 \sec ^{10}(c+d x)}{10 d}+\frac{a^3 \sec ^{11}(c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 0.729802, size = 214, normalized size = 1.02 \[ -\frac{a^3 \sec ^{11}(c+d x) (-1613260 \cos (2 (c+d x))+960960 \cos (3 (c+d x))-1131504 \cos (4 (c+d x))+314160 \cos (5 (c+d x))-432894 \cos (6 (c+d x))+145530 \cos (7 (c+d x))-106260 \cos (8 (c+d x))+6930 \cos (9 (c+d x))-20790 \cos (10 (c+d x))+1143450 \cos (3 (c+d x)) \log (\cos (c+d x))+571725 \cos (5 (c+d x)) \log (\cos (c+d x))+190575 \cos (7 (c+d x)) \log (\cos (c+d x))+38115 \cos (9 (c+d x)) \log (\cos (c+d x))+3465 \cos (11 (c+d x)) \log (\cos (c+d x))+462 \cos (c+d x) (3465 \log (\cos (c+d x))[phone])}{3548160 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^9,x]

[Out]

-(a^3*(-1151740 - 1613260*Cos[2*(c + d*x)] + 960960*Cos[3*(c + d*x)] - 1131504*Cos[4*(c + d*x)] + 314160*Cos[5
*(c + d*x)] - 432894*Cos[6*(c + d*x)] + 145530*Cos[7*(c + d*x)] - 106260*Cos[8*(c + d*x)] + 6930*Cos[9*(c + d*
x)] - 20790*Cos[10*(c + d*x)] + 1143450*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + 571725*Cos[5*(c + d*x)]*Log[Cos[c
+ d*x]] + 190575*Cos[7*(c + d*x)]*Log[Cos[c + d*x]] + 38115*Cos[9*(c + d*x)]*Log[Cos[c + d*x]] + 3465*Cos[11*
(c + d*x)]*Log[Cos[c + d*x]] + 462*Cos[c + d*x]*(2606 + 3465*Log[Cos[c + d*x]]))*Sec[c + d*x]^11)/(3548160*d)

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Maple [A]  time = 0.063, size = 351, normalized size = 1.7 \begin{align*}{\frac{4352\,{a}^{3}\cos \left ( dx+c \right ) }{3465\,d}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{11\,d \left ( \cos \left ( dx+c \right ) \right ) ^{11}}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{10\,d \left ( \cos \left ( dx+c \right ) \right ) ^{10}}}+{\frac{34\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{99\,d \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}-{\frac{34\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{693\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{34\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{1155\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{34\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{693\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{34\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{99\,d\cos \left ( dx+c \right ) }}+{\frac{34\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{99\,d}}+{\frac{272\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{693\,d}}+{\frac{544\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{1155\,d}}+{\frac{2176\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{3465\,d}}+{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{8}}{8\,d}}-{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x)

[Out]

4352/3465*a^3*cos(d*x+c)/d+1/11/d*a^3*sin(d*x+c)^10/cos(d*x+c)^11+3/10/d*a^3*sin(d*x+c)^10/cos(d*x+c)^10+34/99
/d*a^3*sin(d*x+c)^10/cos(d*x+c)^9-34/693/d*a^3*sin(d*x+c)^10/cos(d*x+c)^7+34/1155/d*a^3*sin(d*x+c)^10/cos(d*x+
c)^5-34/693/d*a^3*sin(d*x+c)^10/cos(d*x+c)^3+34/99/d*a^3*sin(d*x+c)^10/cos(d*x+c)+34/99/d*a^3*cos(d*x+c)*sin(d
*x+c)^8+272/693/d*a^3*cos(d*x+c)*sin(d*x+c)^6+544/1155/d*a^3*cos(d*x+c)*sin(d*x+c)^4+2176/3465/d*a^3*cos(d*x+c
)*sin(d*x+c)^2+1/8/d*a^3*tan(d*x+c)^8-1/6/d*a^3*tan(d*x+c)^6+1/4/d*a^3*tan(d*x+c)^4-1/2/d*a^3*tan(d*x+c)^2-a^3
*ln(cos(d*x+c))/d

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Maxima [A]  time = 1.64076, size = 219, normalized size = 1.04 \begin{align*} -\frac{27720 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac{83160 \, a^{3} \cos \left (d x + c\right )^{10} - 13860 \, a^{3} \cos \left (d x + c\right )^{9} - 101640 \, a^{3} \cos \left (d x + c\right )^{8} - 41580 \, a^{3} \cos \left (d x + c\right )^{7} + 77616 \, a^{3} \cos \left (d x + c\right )^{6} + 64680 \, a^{3} \cos \left (d x + c\right )^{5} - 23760 \, a^{3} \cos \left (d x + c\right )^{4} - 38115 \, a^{3} \cos \left (d x + c\right )^{3} - 3080 \, a^{3} \cos \left (d x + c\right )^{2} + 8316 \, a^{3} \cos \left (d x + c\right ) + 2520 \, a^{3}}{\cos \left (d x + c\right )^{11}}}{27720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x, algorithm="maxima")

[Out]

-1/27720*(27720*a^3*log(cos(d*x + c)) - (83160*a^3*cos(d*x + c)^10 - 13860*a^3*cos(d*x + c)^9 - 101640*a^3*cos
(d*x + c)^8 - 41580*a^3*cos(d*x + c)^7 + 77616*a^3*cos(d*x + c)^6 + 64680*a^3*cos(d*x + c)^5 - 23760*a^3*cos(d
*x + c)^4 - 38115*a^3*cos(d*x + c)^3 - 3080*a^3*cos(d*x + c)^2 + 8316*a^3*cos(d*x + c) + 2520*a^3)/cos(d*x + c
)^11)/d

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Fricas [A]  time = 1.05857, size = 481, normalized size = 2.29 \begin{align*} -\frac{27720 \, a^{3} \cos \left (d x + c\right )^{11} \log \left (-\cos \left (d x + c\right )\right ) - 83160 \, a^{3} \cos \left (d x + c\right )^{10} + 13860 \, a^{3} \cos \left (d x + c\right )^{9} + 101640 \, a^{3} \cos \left (d x + c\right )^{8} + 41580 \, a^{3} \cos \left (d x + c\right )^{7} - 77616 \, a^{3} \cos \left (d x + c\right )^{6} - 64680 \, a^{3} \cos \left (d x + c\right )^{5} + 23760 \, a^{3} \cos \left (d x + c\right )^{4} + 38115 \, a^{3} \cos \left (d x + c\right )^{3} + 3080 \, a^{3} \cos \left (d x + c\right )^{2} - 8316 \, a^{3} \cos \left (d x + c\right ) - 2520 \, a^{3}}{27720 \, d \cos \left (d x + c\right )^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x, algorithm="fricas")

[Out]

-1/27720*(27720*a^3*cos(d*x + c)^11*log(-cos(d*x + c)) - 83160*a^3*cos(d*x + c)^10 + 13860*a^3*cos(d*x + c)^9
+ 101640*a^3*cos(d*x + c)^8 + 41580*a^3*cos(d*x + c)^7 - 77616*a^3*cos(d*x + c)^6 - 64680*a^3*cos(d*x + c)^5 +
23760*a^3*cos(d*x + c)^4 + 38115*a^3*cos(d*x + c)^3 + 3080*a^3*cos(d*x + c)^2 - 8316*a^3*cos(d*x + c) - 2520*
a^3)/(d*cos(d*x + c)^11)

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Sympy [A]  time = 115.078, size = 439, normalized size = 2.09 \begin{align*} \begin{cases} \frac{a^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{3} \tan ^{8}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{11 d} + \frac{3 a^{3} \tan ^{8}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac{a^{3} \tan ^{8}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{3 d} + \frac{a^{3} \tan ^{8}{\left (c + d x \right )}}{8 d} - \frac{8 a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{99 d} - \frac{3 a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac{8 a^{3} \tan ^{6}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{21 d} - \frac{a^{3} \tan ^{6}{\left (c + d x \right )}}{6 d} + \frac{16 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{231 d} + \frac{3 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac{16 a^{3} \tan ^{4}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{35 d} + \frac{a^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{64 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{1155 d} - \frac{3 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac{64 a^{3} \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{105 d} - \frac{a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac{128 a^{3} \sec ^{3}{\left (c + d x \right )}}{3465 d} + \frac{3 a^{3} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac{128 a^{3} \sec{\left (c + d x \right )}}{105 d} & \text{for}\: d \neq 0 \\x \left (a \sec{\left (c \right )} + a\right )^{3} \tan ^{9}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**3*tan(d*x+c)**9,x)

[Out]

Piecewise((a**3*log(tan(c + d*x)**2 + 1)/(2*d) + a**3*tan(c + d*x)**8*sec(c + d*x)**3/(11*d) + 3*a**3*tan(c +
d*x)**8*sec(c + d*x)**2/(10*d) + a**3*tan(c + d*x)**8*sec(c + d*x)/(3*d) + a**3*tan(c + d*x)**8/(8*d) - 8*a**3
*tan(c + d*x)**6*sec(c + d*x)**3/(99*d) - 3*a**3*tan(c + d*x)**6*sec(c + d*x)**2/(10*d) - 8*a**3*tan(c + d*x)*
*6*sec(c + d*x)/(21*d) - a**3*tan(c + d*x)**6/(6*d) + 16*a**3*tan(c + d*x)**4*sec(c + d*x)**3/(231*d) + 3*a**3
*tan(c + d*x)**4*sec(c + d*x)**2/(10*d) + 16*a**3*tan(c + d*x)**4*sec(c + d*x)/(35*d) + a**3*tan(c + d*x)**4/(
4*d) - 64*a**3*tan(c + d*x)**2*sec(c + d*x)**3/(1155*d) - 3*a**3*tan(c + d*x)**2*sec(c + d*x)**2/(10*d) - 64*a
**3*tan(c + d*x)**2*sec(c + d*x)/(105*d) - a**3*tan(c + d*x)**2/(2*d) + 128*a**3*sec(c + d*x)**3/(3465*d) + 3*
a**3*sec(c + d*x)**2/(10*d) + 128*a**3*sec(c + d*x)/(105*d), Ne(d, 0)), (x*(a*sec(c) + a)**3*tan(c)**9, True))

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Giac [A]  time = 14.2469, size = 495, normalized size = 2.36 \begin{align*} \frac{27720 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 27720 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{153343 \, a^{3} + \frac{1742213 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{9043705 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{28369275 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{59954070 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{67458930 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{57997170 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{36975510 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{16879995 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{5213945 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac{976261 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{83711 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{11}}}{27720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x, algorithm="giac")

[Out]

1/27720*(27720*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 27720*a^3*log(abs(-(cos(d*x + c) - 1
)/(cos(d*x + c) + 1) - 1)) + (153343*a^3 + 1742213*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9043705*a^3*(co
s(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 28369275*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 59954070*a^3
*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 67458930*a^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 57997170
*a^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 36975510*a^3*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 1687
9995*a^3*(cos(d*x + c) - 1)^8/(cos(d*x + c) + 1)^8 + 5213945*a^3*(cos(d*x + c) - 1)^9/(cos(d*x + c) + 1)^9 + 9
76261*a^3*(cos(d*x + c) - 1)^10/(cos(d*x + c) + 1)^10 + 83711*a^3*(cos(d*x + c) - 1)^11/(cos(d*x + c) + 1)^11)
/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^11)/d