3.286 \(\int \frac{\tan ^9(c+d x)}{a+b \sec (c+d x)} \, dx\)

Optimal. Leaf size=250 \[ \frac{\left (a^2-4 b^2\right ) \sec ^5(c+d x)}{5 b^3 d}-\frac{a \left (a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}+\frac{\left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^3(c+d x)}{3 b^5 d}-\frac{a \left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}+\frac{\left (-4 a^4 b^2+6 a^2 b^4+a^6-4 b^6\right ) \sec (c+d x)}{b^7 d}-\frac{\left (a^2-b^2\right )^4 \log (a+b \sec (c+d x))}{a b^8 d}-\frac{a \sec ^6(c+d x)}{6 b^2 d}-\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^7(c+d x)}{7 b d} \]

[Out]

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

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Rubi [A]  time = 0.196866, antiderivative size = 250, 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 = {3885, 894} \[ \frac{\left (a^2-4 b^2\right ) \sec ^5(c+d x)}{5 b^3 d}-\frac{a \left (a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}+\frac{\left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^3(c+d x)}{3 b^5 d}-\frac{a \left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}+\frac{\left (-4 a^4 b^2+6 a^2 b^4+a^6-4 b^6\right ) \sec (c+d x)}{b^7 d}-\frac{\left (a^2-b^2\right )^4 \log (a+b \sec (c+d x))}{a b^8 d}-\frac{a \sec ^6(c+d x)}{6 b^2 d}-\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^7(c+d x)}{7 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3885

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

Rule 894

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] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{\tan ^9(c+d x)}{a+b \sec (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^4}{x (a+x)} \, dx,x,b \sec (c+d x)\right )}{b^8 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^6 \left (1+\frac{-4 a^4 b^2+6 a^2 b^4-4 b^6}{a^6}\right )+\frac{b^8}{a x}-a \left (a^4-4 a^2 b^2+6 b^4\right ) x+\left (a^4-4 a^2 b^2+6 b^4\right ) x^2-a \left (a^2-4 b^2\right ) x^3+\left (a^2-4 b^2\right ) x^4-a x^5+x^6-\frac{\left (a^2-b^2\right )^4}{a (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^8 d}\\ &=-\frac{\log (\cos (c+d x))}{a d}-\frac{\left (a^2-b^2\right )^4 \log (a+b \sec (c+d x))}{a b^8 d}+\frac{\left (a^6-4 a^4 b^2+6 a^2 b^4-4 b^6\right ) \sec (c+d x)}{b^7 d}-\frac{a \left (a^4-4 a^2 b^2+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}+\frac{\left (a^4-4 a^2 b^2+6 b^4\right ) \sec ^3(c+d x)}{3 b^5 d}-\frac{a \left (a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}+\frac{\left (a^2-4 b^2\right ) \sec ^5(c+d x)}{5 b^3 d}-\frac{a \sec ^6(c+d x)}{6 b^2 d}+\frac{\sec ^7(c+d x)}{7 b d}\\ \end{align*}

Mathematica [B]  time = 6.22729, size = 520, normalized size = 2.08 \[ \frac{\left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^4(c+d x) (a \cos (c+d x)+b)}{3 b^5 d (a+b \sec (c+d x))}-\frac{a \left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^3(c+d x) (a \cos (c+d x)+b)}{2 b^6 d (a+b \sec (c+d x))}-\frac{\left (2 b^2-a^2\right ) \left (-2 a^2 b^2+a^4+2 b^4\right ) \sec ^2(c+d x) (a \cos (c+d x)+b)}{b^7 d (a+b \sec (c+d x))}+\frac{\left (-4 a^5 b^2+6 a^3 b^4+a^7-4 a b^6\right ) \sec (c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b)}{b^8 d (a+b \sec (c+d x))}+\frac{\left (4 a^6 b^2-6 a^4 b^4+4 a^2 b^6-a^8-b^8\right ) \sec (c+d x) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a b^8 d (a+b \sec (c+d x))}-\frac{a \sec ^7(c+d x) (a \cos (c+d x)+b)}{6 b^2 d (a+b \sec (c+d x))}-\frac{(2 b-a) (a+2 b) \sec ^6(c+d x) (a \cos (c+d x)+b)}{5 b^3 d (a+b \sec (c+d x))}+\frac{a (2 b-a) (a+2 b) \sec ^5(c+d x) (a \cos (c+d x)+b)}{4 b^4 d (a+b \sec (c+d x))}+\frac{\sec ^8(c+d x) (a \cos (c+d x)+b)}{7 b d (a+b \sec (c+d x))} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.064, size = 460, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{db \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{4}{5\,db \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{ad}}+{\frac{1}{7\,db \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}-4\,{\frac{1}{db\cos \left ( dx+c \right ) }}+{\frac{{a}^{4}}{3\,d{b}^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{4\,{a}^{2}}{3\,d{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{a}^{7}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{8}}}-4\,{\frac{{a}^{5}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{6}}}+6\,{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{4}}}-4\,{\frac{a\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{2}}}-{\frac{{a}^{7}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{8}}}+{\frac{{a}^{6}}{d{b}^{7}\cos \left ( dx+c \right ) }}-4\,{\frac{{a}^{4}}{d{b}^{5}\cos \left ( dx+c \right ) }}+6\,{\frac{{a}^{2}}{d{b}^{3}\cos \left ( dx+c \right ) }}-{\frac{{a}^{3}}{4\,d{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a}{d{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{5}}{2\,d{b}^{6} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{{a}^{3}}{d{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-3\,{\frac{a}{d{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{{a}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{6}}}-6\,{\frac{{a}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{4}}}+4\,{\frac{a\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{2}}}-{\frac{a}{6\,d{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{2}}{5\,d{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00017, size = 362, normalized size = 1.45 \begin{align*} \frac{\frac{420 \,{\left (a^{7} - 4 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 4 \, a b^{6}\right )} \log \left (\cos \left (d x + c\right )\right )}{b^{8}} - \frac{420 \,{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a b^{8}} - \frac{70 \, a b^{5} \cos \left (d x + c\right ) - 420 \,{\left (a^{6} - 4 \, a^{4} b^{2} + 6 \, a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - 60 \, b^{6} + 210 \,{\left (a^{5} b - 4 \, a^{3} b^{3} + 6 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 140 \,{\left (a^{4} b^{2} - 4 \, a^{2} b^{4} + 6 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 105 \,{\left (a^{3} b^{3} - 4 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 84 \,{\left (a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{2}}{b^{7} \cos \left (d x + c\right )^{7}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.13327, size = 674, normalized size = 2.7 \begin{align*} -\frac{70 \, a^{2} b^{6} \cos \left (d x + c\right ) + 420 \,{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{7} \log \left (a \cos \left (d x + c\right ) + b\right ) - 420 \,{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 60 \, a b^{7} - 420 \,{\left (a^{7} b - 4 \, a^{5} b^{3} + 6 \, a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )^{6} + 210 \,{\left (a^{6} b^{2} - 4 \, a^{4} b^{4} + 6 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{5} - 140 \,{\left (a^{5} b^{3} - 4 \, a^{3} b^{5} + 6 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 105 \,{\left (a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} - 84 \,{\left (a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )^{2}}{420 \, a b^{8} d \cos \left (d x + c\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(tan(d*x+c)**9/(a+b*sec(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 11.2655, size = 2267, normalized size = 9.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/420*(420*(a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9
)*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^2*b^8
- a*b^9) - 420*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a - 420*(a^7 - 4*a^5*b^2 + 6*a^3*b^4 - 4*
a*b^6)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1))/b^8 + (1089*a^7 - 840*a^6*b - 4356*a^5*b^2 + 3080*
a^4*b^3 + 6534*a^3*b^4 - 4088*a^2*b^5 - 4356*a*b^6 + 2232*b^7 + 7623*a^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)
- 5040*a^6*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 31332*a^5*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 19
040*a^4*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 48258*a^3*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2609
6*a^2*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 33012*a*b^6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 14784*b^
7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 22869*a^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 12600*a^6*b*(c
os(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 95676*a^5*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 47880*a^4*
b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 151494*a^3*b^4*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 673
68*a^2*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 107436*a*b^6*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2
+ 40152*b^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 38115*a^7*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 -
16800*a^6*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 160860*a^5*b^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)
^3 + 62720*a^4*b^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 258930*a^3*b^4*(cos(d*x + c) - 1)^3/(cos(d*x +
c) + 1)^3 - 86240*a^2*b^5*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 192220*a*b^6*(cos(d*x + c) - 1)^3/(cos(d
*x + c) + 1)^3 + 53760*b^7*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 38115*a^7*(cos(d*x + c) - 1)^4/(cos(d*x
+ c) + 1)^4 - 12600*a^6*b*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 160860*a^5*b^2*(cos(d*x + c) - 1)^4/(co
s(d*x + c) + 1)^4 + 45080*a^4*b^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 258930*a^3*b^4*(cos(d*x + c) - 1
)^4/(cos(d*x + c) + 1)^4 - 56840*a^2*b^5*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 192220*a*b^6*(cos(d*x + c
) - 1)^4/(cos(d*x + c) + 1)^4 + 24360*b^7*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 22869*a^7*(cos(d*x + c)
- 1)^5/(cos(d*x + c) + 1)^5 - 5040*a^6*b*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 95676*a^5*b^2*(cos(d*x +
c) - 1)^5/(cos(d*x + c) + 1)^5 + 16800*a^4*b^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 151494*a^3*b^4*(cos
(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 18480*a^2*b^5*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 107436*a*b^6
*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 6720*b^7*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 7623*a^7*(co
s(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 840*a^6*b*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 31332*a^5*b^2*(
cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 2520*a^4*b^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 48258*a^3*
b^4*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 2520*a^2*b^5*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 33012
*a*b^6*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 840*b^7*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 1089*a^
7*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 - 4356*a^5*b^2*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 6534*a^
3*b^4*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 - 4356*a*b^6*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7)/(b^8*(
(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^7))/d