3.222 \(\int \frac{\sin ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=239 \[ \frac{2 \left (a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^5 d}-\frac{b \left (3 a^2-5 b^2\right ) \cos ^2(c+d x)}{a^6 d}-\frac{\left (-12 a^2 b^2+a^4+15 b^4\right ) \cos (c+d x)}{a^7 d}+\frac{b^2 \left (-10 a^2 b^2+3 a^4+7 b^4\right )}{a^8 d (a \cos (c+d x)+b)}-\frac{b^3 \left (a^2-b^2\right )^2}{2 a^8 d (a \cos (c+d x)+b)^2}+\frac{b \left (-20 a^2 b^2+3 a^4+21 b^4\right ) \log (a \cos (c+d x)+b)}{a^8 d}+\frac{3 b \cos ^4(c+d x)}{4 a^4 d}-\frac{\cos ^5(c+d x)}{5 a^3 d} \]

[Out]

-(((a^4 - 12*a^2*b^2 + 15*b^4)*Cos[c + d*x])/(a^7*d)) - (b*(3*a^2 - 5*b^2)*Cos[c + d*x]^2)/(a^6*d) + (2*(a^2 -
3*b^2)*Cos[c + d*x]^3)/(3*a^5*d) + (3*b*Cos[c + d*x]^4)/(4*a^4*d) - Cos[c + d*x]^5/(5*a^3*d) - (b^3*(a^2 - b^
2)^2)/(2*a^8*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^4 - 10*a^2*b^2 + 7*b^4))/(a^8*d*(b + a*Cos[c + d*x])) + (b*
(3*a^4 - 20*a^2*b^2 + 21*b^4)*Log[b + a*Cos[c + d*x]])/(a^8*d)

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Rubi [A]  time = 0.364761, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2837, 12, 948} \[ \frac{2 \left (a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^5 d}-\frac{b \left (3 a^2-5 b^2\right ) \cos ^2(c+d x)}{a^6 d}-\frac{\left (-12 a^2 b^2+a^4+15 b^4\right ) \cos (c+d x)}{a^7 d}+\frac{b^2 \left (-10 a^2 b^2+3 a^4+7 b^4\right )}{a^8 d (a \cos (c+d x)+b)}-\frac{b^3 \left (a^2-b^2\right )^2}{2 a^8 d (a \cos (c+d x)+b)^2}+\frac{b \left (-20 a^2 b^2+3 a^4+21 b^4\right ) \log (a \cos (c+d x)+b)}{a^8 d}+\frac{3 b \cos ^4(c+d x)}{4 a^4 d}-\frac{\cos ^5(c+d x)}{5 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]

[Out]

-(((a^4 - 12*a^2*b^2 + 15*b^4)*Cos[c + d*x])/(a^7*d)) - (b*(3*a^2 - 5*b^2)*Cos[c + d*x]^2)/(a^6*d) + (2*(a^2 -
3*b^2)*Cos[c + d*x]^3)/(3*a^5*d) + (3*b*Cos[c + d*x]^4)/(4*a^4*d) - Cos[c + d*x]^5/(5*a^3*d) - (b^3*(a^2 - b^
2)^2)/(2*a^8*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^4 - 10*a^2*b^2 + 7*b^4))/(a^8*d*(b + a*Cos[c + d*x])) + (b*
(3*a^4 - 20*a^2*b^2 + 21*b^4)*Log[b + a*Cos[c + d*x]])/(a^8*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2837

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 948

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))

Rubi steps

\begin{align*} \int \frac{\sin ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=-\int \frac{\cos ^3(c+d x) \sin ^5(c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \left (a^2-x^2\right )^2}{a^3 (-b+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \left (a^2-x^2\right )^2}{(-b+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a^8 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^4 \left (1+\frac{3 b^2 \left (-4 a^2+5 b^2\right )}{a^4}\right )-\frac{b^3 \left (-a^2+b^2\right )^2}{(b-x)^3}+\frac{3 a^4 b^2-10 a^2 b^4+7 b^6}{(b-x)^2}+\frac{-3 a^4 b+20 a^2 b^3-21 b^5}{b-x}+2 b \left (-3 a^2+5 b^2\right ) x-2 \left (a^2-3 b^2\right ) x^2+3 b x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^8 d}\\ &=-\frac{\left (a^4-12 a^2 b^2+15 b^4\right ) \cos (c+d x)}{a^7 d}-\frac{b \left (3 a^2-5 b^2\right ) \cos ^2(c+d x)}{a^6 d}+\frac{2 \left (a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^5 d}+\frac{3 b \cos ^4(c+d x)}{4 a^4 d}-\frac{\cos ^5(c+d x)}{5 a^3 d}-\frac{b^3 \left (a^2-b^2\right )^2}{2 a^8 d (b+a \cos (c+d x))^2}+\frac{b^2 \left (3 a^4-10 a^2 b^2+7 b^4\right )}{a^8 d (b+a \cos (c+d x))}+\frac{b \left (3 a^4-20 a^2 b^2+21 b^4\right ) \log (b+a \cos (c+d x))}{a^8 d}\\ \end{align*}

Mathematica [A]  time = 2.91796, size = 388, normalized size = 1.62 \[ \frac{2780 a^5 b^2 \cos (3 (c+d x))-84 a^5 b^2 \cos (5 (c+d x))+420 a^4 b^3 \cos (4 (c+d x))-3360 a^3 b^4 \cos (3 (c+d x))-13440 a^4 b^3 \log (a \cos (c+d x)+b)-18240 a^2 b^5 \log (a \cos (c+d x)+b)+5 a^2 b \cos (2 (c+d x)) \left (192 \left (-20 a^2 b^2+3 a^4+21 b^4\right ) \log (a \cos (c+d x)+b)+3888 a^2 b^2-407 a^4-4800 b^4\right )-10 a \cos (c+d x) \left (-384 b^2 \left (-20 a^2 b^2+3 a^4+21 b^4\right ) \log (a \cos (c+d x)+b)-1728 a^4 b^2+1584 a^2 b^4+85 a^6+1536 b^6\right )+26160 a^4 b^3-46080 a^2 b^5-274 a^6 b \cos (4 (c+d x))+21 a^6 b \cos (6 (c+d x))+2880 a^6 b \log (a \cos (c+d x)+b)-1740 a^6 b-206 a^7 \cos (3 (c+d x))+38 a^7 \cos (5 (c+d x))-6 a^7 \cos (7 (c+d x))+40320 b^7 \log (a \cos (c+d x)+b)+12480 b^7}{1920 a^8 d (a \cos (c+d x)+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]

[Out]

(-1740*a^6*b + 26160*a^4*b^3 - 46080*a^2*b^5 + 12480*b^7 - 206*a^7*Cos[3*(c + d*x)] + 2780*a^5*b^2*Cos[3*(c +
d*x)] - 3360*a^3*b^4*Cos[3*(c + d*x)] - 274*a^6*b*Cos[4*(c + d*x)] + 420*a^4*b^3*Cos[4*(c + d*x)] + 38*a^7*Cos
[5*(c + d*x)] - 84*a^5*b^2*Cos[5*(c + d*x)] + 21*a^6*b*Cos[6*(c + d*x)] - 6*a^7*Cos[7*(c + d*x)] + 2880*a^6*b*
Log[b + a*Cos[c + d*x]] - 13440*a^4*b^3*Log[b + a*Cos[c + d*x]] - 18240*a^2*b^5*Log[b + a*Cos[c + d*x]] + 4032
0*b^7*Log[b + a*Cos[c + d*x]] + 5*a^2*b*Cos[2*(c + d*x)]*(-407*a^4 + 3888*a^2*b^2 - 4800*b^4 + 192*(3*a^4 - 20
*a^2*b^2 + 21*b^4)*Log[b + a*Cos[c + d*x]]) - 10*a*Cos[c + d*x]*(85*a^6 - 1728*a^4*b^2 + 1584*a^2*b^4 + 1536*b
^6 - 384*b^2*(3*a^4 - 20*a^2*b^2 + 21*b^4)*Log[b + a*Cos[c + d*x]]))/(1920*a^8*d*(b + a*Cos[c + d*x])^2)

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Maple [A]  time = 0.071, size = 355, normalized size = 1.5 \begin{align*} -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,{a}^{3}d}}+{\frac{3\,b \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,{a}^{4}d}}+{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,{a}^{3}d}}-2\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}{b}^{2}}{d{a}^{5}}}-3\,{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{{a}^{4}d}}+5\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}{b}^{3}}{d{a}^{6}}}-{\frac{\cos \left ( dx+c \right ) }{{a}^{3}d}}+12\,{\frac{\cos \left ( dx+c \right ){b}^{2}}{d{a}^{5}}}-15\,{\frac{{b}^{4}\cos \left ( dx+c \right ) }{d{a}^{7}}}+3\,{\frac{b\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{{a}^{4}d}}-20\,{\frac{{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{6}}}+21\,{\frac{{b}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{8}}}-{\frac{{b}^{3}}{2\,{a}^{4}d \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{5}}{d{a}^{6} \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{7}}{2\,d{a}^{8} \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{{b}^{2}}{{a}^{4}d \left ( b+a\cos \left ( dx+c \right ) \right ) }}-10\,{\frac{{b}^{4}}{d{a}^{6} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+7\,{\frac{{b}^{6}}{d{a}^{8} \left ( b+a\cos \left ( dx+c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^5/(a+b*sec(d*x+c))^3,x)

[Out]

-1/5*cos(d*x+c)^5/a^3/d+3/4*b*cos(d*x+c)^4/a^4/d+2/3*cos(d*x+c)^3/a^3/d-2/d/a^5*cos(d*x+c)^3*b^2-3*b*cos(d*x+c
)^2/a^4/d+5/d/a^6*cos(d*x+c)^2*b^3-cos(d*x+c)/a^3/d+12/d/a^5*cos(d*x+c)*b^2-15/d/a^7*b^4*cos(d*x+c)+3*b*ln(b+a
*cos(d*x+c))/a^4/d-20/d/a^6*b^3*ln(b+a*cos(d*x+c))+21/d/a^8*b^5*ln(b+a*cos(d*x+c))-1/2*b^3/a^4/d/(b+a*cos(d*x+
c))^2+1/d*b^5/a^6/(b+a*cos(d*x+c))^2-1/2/d*b^7/a^8/(b+a*cos(d*x+c))^2+3*b^2/a^4/d/(b+a*cos(d*x+c))-10/d/a^6*b^
4/(b+a*cos(d*x+c))+7/d/a^8*b^6/(b+a*cos(d*x+c))

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Maxima [A]  time = 0.973615, size = 316, normalized size = 1.32 \begin{align*} \frac{\frac{30 \,{\left (5 \, a^{4} b^{3} - 18 \, a^{2} b^{5} + 13 \, b^{7} + 2 \,{\left (3 \, a^{5} b^{2} - 10 \, a^{3} b^{4} + 7 \, a b^{6}\right )} \cos \left (d x + c\right )\right )}}{a^{10} \cos \left (d x + c\right )^{2} + 2 \, a^{9} b \cos \left (d x + c\right ) + a^{8} b^{2}} - \frac{12 \, a^{4} \cos \left (d x + c\right )^{5} - 45 \, a^{3} b \cos \left (d x + c\right )^{4} - 40 \,{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 60 \,{\left (3 \, a^{3} b - 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{2} + 60 \,{\left (a^{4} - 12 \, a^{2} b^{2} + 15 \, b^{4}\right )} \cos \left (d x + c\right )}{a^{7}} + \frac{60 \,{\left (3 \, a^{4} b - 20 \, a^{2} b^{3} + 21 \, b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^5/(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/60*(30*(5*a^4*b^3 - 18*a^2*b^5 + 13*b^7 + 2*(3*a^5*b^2 - 10*a^3*b^4 + 7*a*b^6)*cos(d*x + c))/(a^10*cos(d*x +
c)^2 + 2*a^9*b*cos(d*x + c) + a^8*b^2) - (12*a^4*cos(d*x + c)^5 - 45*a^3*b*cos(d*x + c)^4 - 40*(a^4 - 3*a^2*b
^2)*cos(d*x + c)^3 + 60*(3*a^3*b - 5*a*b^3)*cos(d*x + c)^2 + 60*(a^4 - 12*a^2*b^2 + 15*b^4)*cos(d*x + c))/a^7
+ 60*(3*a^4*b - 20*a^2*b^3 + 21*b^5)*log(a*cos(d*x + c) + b)/a^8)/d

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Fricas [A]  time = 2.39359, size = 792, normalized size = 3.31 \begin{align*} -\frac{96 \, a^{7} \cos \left (d x + c\right )^{7} - 168 \, a^{6} b \cos \left (d x + c\right )^{6} - 1785 \, a^{4} b^{3} + 5520 \, a^{2} b^{5} - 3120 \, b^{7} - 16 \,{\left (20 \, a^{7} - 21 \, a^{5} b^{2}\right )} \cos \left (d x + c\right )^{5} + 40 \,{\left (20 \, a^{6} b - 21 \, a^{4} b^{3}\right )} \cos \left (d x + c\right )^{4} + 160 \,{\left (3 \, a^{7} - 20 \, a^{5} b^{2} + 21 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (25 \, a^{6} b - 592 \, a^{4} b^{3} + 800 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - 30 \,{\left (71 \, a^{5} b^{2} - 48 \, a^{3} b^{4} - 128 \, a b^{6}\right )} \cos \left (d x + c\right ) - 480 \,{\left (3 \, a^{4} b^{3} - 20 \, a^{2} b^{5} + 21 \, b^{7} +{\left (3 \, a^{6} b - 20 \, a^{4} b^{3} + 21 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, a^{5} b^{2} - 20 \, a^{3} b^{4} + 21 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{480 \,{\left (a^{10} d \cos \left (d x + c\right )^{2} + 2 \, a^{9} b d \cos \left (d x + c\right ) + a^{8} b^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^5/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/480*(96*a^7*cos(d*x + c)^7 - 168*a^6*b*cos(d*x + c)^6 - 1785*a^4*b^3 + 5520*a^2*b^5 - 3120*b^7 - 16*(20*a^7
- 21*a^5*b^2)*cos(d*x + c)^5 + 40*(20*a^6*b - 21*a^4*b^3)*cos(d*x + c)^4 + 160*(3*a^7 - 20*a^5*b^2 + 21*a^3*b
^4)*cos(d*x + c)^3 + 15*(25*a^6*b - 592*a^4*b^3 + 800*a^2*b^5)*cos(d*x + c)^2 - 30*(71*a^5*b^2 - 48*a^3*b^4 -
128*a*b^6)*cos(d*x + c) - 480*(3*a^4*b^3 - 20*a^2*b^5 + 21*b^7 + (3*a^6*b - 20*a^4*b^3 + 21*a^2*b^5)*cos(d*x +
c)^2 + 2*(3*a^5*b^2 - 20*a^3*b^4 + 21*a*b^6)*cos(d*x + c))*log(a*cos(d*x + c) + b))/(a^10*d*cos(d*x + c)^2 +
2*a^9*b*d*cos(d*x + c) + a^8*b^2*d)

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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(sin(d*x+c)**5/(a+b*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.39382, size = 1805, normalized size = 7.55 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^5/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/60*(60*(3*a^5*b - 3*a^4*b^2 - 20*a^3*b^3 + 20*a^2*b^4 + 21*a*b^5 - 21*b^6)*log(abs(a + b + a*(cos(d*x + c) -
1)/(cos(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/(a^9 - a^8*b) - 60*(3*a^4*b - 20*a^2*b^3 +
21*b^5)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^8 - 30*(9*a^6*b + 6*a^5*b^2 - 75*a^4*b^3 - 108*
a^3*b^4 + 51*a^2*b^5 + 150*a*b^6 + 63*b^7 + 18*a^6*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 12*a^5*b^2*(cos(d
*x + c) - 1)/(cos(d*x + c) + 1) - 142*a^4*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 36*a^3*b^4*(cos(d*x + c)
- 1)/(cos(d*x + c) + 1) + 250*a^2*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 24*a*b^6*(cos(d*x + c) - 1)/(co
s(d*x + c) + 1) - 126*b^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9*a^6*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) +
1)^2 - 18*a^5*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 51*a^4*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) +
1)^2 + 120*a^3*b^4*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 3*a^2*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) +
1)^2 - 126*a*b^6*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 63*b^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2
)/((a + b + a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))^2*a^8) + (64*a^
5 - 411*a^4*b - 1200*a^3*b^2 + 2740*a^2*b^3 + 1800*a*b^4 - 2877*b^5 - 320*a^5*(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) + 2415*a^4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 5280*a^3*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)
- 14900*a^2*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 7200*a*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 143
85*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 640*a^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 5910*a^4*b*
(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 7680*a^3*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 31000*a^2
*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 10800*a*b^4*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 28770
*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 5910*a^4*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 4320*a
^3*b^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 31000*a^2*b^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 7
200*a*b^4*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 28770*b^5*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 24
15*a^4*b*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 720*a^3*b^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 1
4900*a^2*b^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 1800*a*b^4*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4
- 14385*b^5*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 411*a^4*b*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 -
2740*a^2*b^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 2877*b^5*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5)/(
a^8*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^5))/d