3.196 \(\int \frac{\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx\)

Optimal. Leaf size=223 \[ -\frac{\left (3 a^2-b^2\right ) \cos ^5(c+d x)}{5 a^3 d}+\frac{b \left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 a^4 d}+\frac{\left (-3 a^2 b^2+3 a^4+b^4\right ) \cos ^3(c+d x)}{3 a^5 d}-\frac{b \left (-3 a^2 b^2+3 a^4+b^4\right ) \cos ^2(c+d x)}{2 a^6 d}-\frac{\left (a^2-b^2\right )^3 \cos (c+d x)}{a^7 d}+\frac{b \left (a^2-b^2\right )^3 \log (a \cos (c+d x)+b)}{a^8 d}-\frac{b \cos ^6(c+d x)}{6 a^2 d}+\frac{\cos ^7(c+d x)}{7 a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.250938, antiderivative size = 223, 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, 772} \[ -\frac{\left (3 a^2-b^2\right ) \cos ^5(c+d x)}{5 a^3 d}+\frac{b \left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 a^4 d}+\frac{\left (-3 a^2 b^2+3 a^4+b^4\right ) \cos ^3(c+d x)}{3 a^5 d}-\frac{b \left (-3 a^2 b^2+3 a^4+b^4\right ) \cos ^2(c+d x)}{2 a^6 d}-\frac{\left (a^2-b^2\right )^3 \cos (c+d x)}{a^7 d}+\frac{b \left (a^2-b^2\right )^3 \log (a \cos (c+d x)+b)}{a^8 d}-\frac{b \cos ^6(c+d x)}{6 a^2 d}+\frac{\cos ^7(c+d x)}{7 a d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^7/(a + b*Sec[c + d*x]),x]

[Out]

-(((a^2 - b^2)^3*Cos[c + d*x])/(a^7*d)) - (b*(3*a^4 - 3*a^2*b^2 + b^4)*Cos[c + d*x]^2)/(2*a^6*d) + ((3*a^4 - 3
*a^2*b^2 + b^4)*Cos[c + d*x]^3)/(3*a^5*d) + (b*(3*a^2 - b^2)*Cos[c + d*x]^4)/(4*a^4*d) - ((3*a^2 - b^2)*Cos[c
+ d*x]^5)/(5*a^3*d) - (b*Cos[c + d*x]^6)/(6*a^2*d) + Cos[c + d*x]^7/(7*a*d) + (b*(a^2 - b^2)^3*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 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 1.34391, size = 282, normalized size = 1.26 \[ \frac{-1260 a^5 b^2 \cos (3 (c+d x))+84 a^5 b^2 \cos (5 (c+d x))-210 a^4 b^3 \cos (4 (c+d x))+560 a^3 b^4 \cos (3 (c+d x))-105 a \left (-152 a^4 b^2+176 a^2 b^4+35 a^6-64 b^6\right ) \cos (c+d x)-105 \left (-40 a^4 b^3+16 a^2 b^5+29 a^6 b\right ) \cos (2 (c+d x))-20160 a^4 b^3 \log (a \cos (c+d x)+b)+20160 a^2 b^5 \log (a \cos (c+d x)+b)+420 a^6 b \cos (4 (c+d x))-35 a^6 b \cos (6 (c+d x))+6720 a^6 b \log (a \cos (c+d x)+b)+735 a^7 \cos (3 (c+d x))-147 a^7 \cos (5 (c+d x))+15 a^7 \cos (7 (c+d x))-6720 b^7 \log (a \cos (c+d x)+b)}{6720 a^8 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^7/(a + b*Sec[c + d*x]),x]

[Out]

(-105*a*(35*a^6 - 152*a^4*b^2 + 176*a^2*b^4 - 64*b^6)*Cos[c + d*x] - 105*(29*a^6*b - 40*a^4*b^3 + 16*a^2*b^5)*
Cos[2*(c + d*x)] + 735*a^7*Cos[3*(c + d*x)] - 1260*a^5*b^2*Cos[3*(c + d*x)] + 560*a^3*b^4*Cos[3*(c + d*x)] + 4
20*a^6*b*Cos[4*(c + d*x)] - 210*a^4*b^3*Cos[4*(c + d*x)] - 147*a^7*Cos[5*(c + d*x)] + 84*a^5*b^2*Cos[5*(c + d*
x)] - 35*a^6*b*Cos[6*(c + d*x)] + 15*a^7*Cos[7*(c + d*x)] + 6720*a^6*b*Log[b + a*Cos[c + d*x]] - 20160*a^4*b^3
*Log[b + a*Cos[c + d*x]] + 20160*a^2*b^5*Log[b + a*Cos[c + d*x]] - 6720*b^7*Log[b + a*Cos[c + d*x]])/(6720*a^8
*d)

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 363, normalized size = 1.6 \begin{align*}{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,ad}}-{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6\,{a}^{2}d}}-{\frac{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,ad}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}{b}^{2}}{5\,d{a}^{3}}}+{\frac{3\,b \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,{a}^{2}d}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}{b}^{3}}{4\,d{a}^{4}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{ad}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}{b}^{2}}{d{a}^{3}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}{b}^{4}}{3\,d{a}^{5}}}-{\frac{3\,b \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,{a}^{2}d}}+{\frac{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}{b}^{3}}{2\,d{a}^{4}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}{b}^{5}}{2\,d{a}^{6}}}-{\frac{\cos \left ( dx+c \right ) }{ad}}+3\,{\frac{{b}^{2}\cos \left ( dx+c \right ) }{d{a}^{3}}}-3\,{\frac{{b}^{4}\cos \left ( dx+c \right ) }{d{a}^{5}}}+{\frac{{b}^{6}\cos \left ( dx+c \right ) }{d{a}^{7}}}+{\frac{b\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{{a}^{2}d}}-3\,{\frac{{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{4}}}+3\,{\frac{{b}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{6}}}-{\frac{{b}^{7}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^7/(a+b*sec(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.05525, size = 302, normalized size = 1.35 \begin{align*} \frac{\frac{60 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 84 \,{\left (3 \, a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \,{\left (3 \, a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \,{\left (3 \, a^{6} - 3 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \,{\left (3 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} - 420 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )}{a^{7}} + \frac{420 \,{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+b*sec(d*x+c)),x, algorithm="maxima")

[Out]

1/420*((60*a^6*cos(d*x + c)^7 - 70*a^5*b*cos(d*x + c)^6 - 84*(3*a^6 - a^4*b^2)*cos(d*x + c)^5 + 105*(3*a^5*b -
a^3*b^3)*cos(d*x + c)^4 + 140*(3*a^6 - 3*a^4*b^2 + a^2*b^4)*cos(d*x + c)^3 - 210*(3*a^5*b - 3*a^3*b^3 + a*b^5
)*cos(d*x + c)^2 - 420*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(d*x + c))/a^7 + 420*(a^6*b - 3*a^4*b^3 + 3*a^2*
b^5 - b^7)*log(a*cos(d*x + c) + b)/a^8)/d

________________________________________________________________________________________

Fricas [A]  time = 1.9669, size = 504, normalized size = 2.26 \begin{align*} \frac{60 \, a^{7} \cos \left (d x + c\right )^{7} - 70 \, a^{6} b \cos \left (d x + c\right )^{6} - 84 \,{\left (3 \, a^{7} - a^{5} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \,{\left (3 \, a^{6} b - a^{4} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \,{\left (3 \, a^{7} - 3 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \,{\left (3 \, a^{6} b - 3 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - 420 \,{\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right ) + 420 \,{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{420 \, a^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/420*(60*a^7*cos(d*x + c)^7 - 70*a^6*b*cos(d*x + c)^6 - 84*(3*a^7 - a^5*b^2)*cos(d*x + c)^5 + 105*(3*a^6*b -
a^4*b^3)*cos(d*x + c)^4 + 140*(3*a^7 - 3*a^5*b^2 + a^3*b^4)*cos(d*x + c)^3 - 210*(3*a^6*b - 3*a^4*b^3 + a^2*b^
5)*cos(d*x + c)^2 - 420*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c) + 420*(a^6*b - 3*a^4*b^3 + 3*a^2*b^
5 - b^7)*log(a*cos(d*x + c) + b))/(a^8*d)

________________________________________________________________________________________

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)**7/(a+b*sec(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.28884, size = 2105, normalized size = 9.44 \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)^7/(a+b*sec(d*x+c)),x, algorithm="giac")

[Out]

1/420*(420*(a^7*b - a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 - 3*a^2*b^6 - a*b^7 + b^8)*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) - 420*(a^6*b -
3*a^4*b^3 + 3*a^2*b^5 - b^7)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^8 + (384*a^7 - 1089*a^6*b
- 1848*a^5*b^2 + 3267*a^4*b^3 + 2240*a^3*b^4 - 3267*a^2*b^5 - 840*a*b^6 + 1089*b^7 - 2688*a^7*(cos(d*x + c) -
1)/(cos(d*x + c) + 1) + 8463*a^6*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 12096*a^5*b^2*(cos(d*x + c) - 1)/(c
os(d*x + c) + 1) - 24549*a^4*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 14000*a^3*b^4*(cos(d*x + c) - 1)/(cos
(d*x + c) + 1) + 23709*a^2*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 5040*a*b^6*(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) - 7623*b^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 8064*a^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^
2 - 28749*a^6*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 32088*a^5*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) +
1)^2 + 78687*a^4*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 35280*a^3*b^4*(cos(d*x + c) - 1)^2/(cos(d*x
+ c) + 1)^2 - 72807*a^2*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 12600*a*b^6*(cos(d*x + c) - 1)^2/(cos(
d*x + c) + 1)^2 + 22869*b^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 13440*a^7*(cos(d*x + c) - 1)^3/(cos(d*
x + c) + 1)^3 + 56035*a^6*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 40320*a^5*b^2*(cos(d*x + c) - 1)^3/(co
s(d*x + c) + 1)^3 - 136185*a^4*b^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 45920*a^3*b^4*(cos(d*x + c) - 1
)^3/(cos(d*x + c) + 1)^3 + 122745*a^2*b^5*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 16800*a*b^6*(cos(d*x + c
) - 1)^3/(cos(d*x + c) + 1)^3 - 38115*b^7*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 56035*a^6*b*(cos(d*x + c
) - 1)^4/(cos(d*x + c) + 1)^4 - 24360*a^5*b^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 136185*a^4*b^3*(cos(
d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 32480*a^3*b^4*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 122745*a^2*b^
5*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 12600*a*b^6*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 38115*b^
7*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 28749*a^6*b*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 6720*a^5
*b^2*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 78687*a^4*b^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 117
60*a^3*b^4*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 72807*a^2*b^5*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5
+ 5040*a*b^6*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 22869*b^7*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5
- 8463*a^6*b*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 840*a^5*b^2*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6
+ 24549*a^4*b^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 1680*a^3*b^4*(cos(d*x + c) - 1)^6/(cos(d*x + c) +
1)^6 - 23709*a^2*b^5*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 840*a*b^6*(cos(d*x + c) - 1)^6/(cos(d*x + c)
+ 1)^6 + 7623*b^7*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 1089*a^6*b*(cos(d*x + c) - 1)^7/(cos(d*x + c) +
1)^7 - 3267*a^4*b^3*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 3267*a^2*b^5*(cos(d*x + c) - 1)^7/(cos(d*x +
c) + 1)^7 - 1089*b^7*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7)/(a^8*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) -
1)^7))/d