3.461 \(\int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=207 \[ \frac {\left (a^2-b^2\right )^3}{7 b^7 d (a+b \sin (c+d x))^7}-\frac {a \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))^6}+\frac {5 a^2-b^2}{b^7 d (a+b \sin (c+d x))^3}-\frac {a \left (5 a^2-3 b^2\right )}{b^7 d (a+b \sin (c+d x))^4}+\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right )}{5 b^7 d (a+b \sin (c+d x))^5}+\frac {1}{b^7 d (a+b \sin (c+d x))}-\frac {3 a}{b^7 d (a+b \sin (c+d x))^2} \]

[Out]

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

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Rubi [A]  time = 0.17, antiderivative size = 207, normalized size of antiderivative = 1.00, 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 = {2668, 697} \[ \frac {\left (a^2-b^2\right )^3}{7 b^7 d (a+b \sin (c+d x))^7}-\frac {a \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))^6}+\frac {5 a^2-b^2}{b^7 d (a+b \sin (c+d x))^3}-\frac {a \left (5 a^2-3 b^2\right )}{b^7 d (a+b \sin (c+d x))^4}+\frac {3 \left (-6 a^2 b^2+5 a^4+b^4\right )}{5 b^7 d (a+b \sin (c+d x))^5}+\frac {1}{b^7 d (a+b \sin (c+d x))}-\frac {3 a}{b^7 d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7/(a + b*Sin[c + d*x])^8,x]

[Out]

(a^2 - b^2)^3/(7*b^7*d*(a + b*Sin[c + d*x])^7) - (a*(a^2 - b^2)^2)/(b^7*d*(a + b*Sin[c + d*x])^6) + (3*(5*a^4
- 6*a^2*b^2 + b^4))/(5*b^7*d*(a + b*Sin[c + d*x])^5) - (a*(5*a^2 - 3*b^2))/(b^7*d*(a + b*Sin[c + d*x])^4) + (5
*a^2 - b^2)/(b^7*d*(a + b*Sin[c + d*x])^3) - (3*a)/(b^7*d*(a + b*Sin[c + d*x])^2) + 1/(b^7*d*(a + b*Sin[c + d*
x]))

Rule 697

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

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

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

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Mathematica [A]  time = 1.14, size = 171, normalized size = 0.83 \[ \frac {\frac {\left (a^2-b^2\right )^3}{7 (a+b \sin (c+d x))^7}-\frac {a \left (a^2-b^2\right )^2}{(a+b \sin (c+d x))^6}+\frac {5 a^2-b^2}{(a+b \sin (c+d x))^3}-\frac {a \left (5 a^2-3 b^2\right )}{(a+b \sin (c+d x))^4}+\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right )}{5 (a+b \sin (c+d x))^5}+\frac {1}{a+b \sin (c+d x)}-\frac {3 a}{(a+b \sin (c+d x))^2}}{b^7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7/(a + b*Sin[c + d*x])^8,x]

[Out]

((a^2 - b^2)^3/(7*(a + b*Sin[c + d*x])^7) - (a*(a^2 - b^2)^2)/(a + b*Sin[c + d*x])^6 + (3*(5*a^4 - 6*a^2*b^2 +
b^4))/(5*(a + b*Sin[c + d*x])^5) - (a*(5*a^2 - 3*b^2))/(a + b*Sin[c + d*x])^4 + (5*a^2 - b^2)/(a + b*Sin[c +
d*x])^3 - (3*a)/(a + b*Sin[c + d*x])^2 + (a + b*Sin[c + d*x])^(-1))/(b^7*d)

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fricas [A]  time = 0.84, size = 382, normalized size = 1.85 \[ \frac {35 \, b^{6} \cos \left (d x + c\right )^{6} - 5 \, a^{6} - 104 \, a^{4} b^{2} - 155 \, a^{2} b^{4} - 16 \, b^{6} - 35 \, {\left (5 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 7 \, {\left (15 \, a^{4} b^{2} + 47 \, a^{2} b^{4} + 8 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 7 \, {\left (15 \, a b^{5} \cos \left (d x + c\right )^{4} + 5 \, a^{5} b + 24 \, a^{3} b^{3} + 11 \, a b^{5} - 25 \, {\left (a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{35 \, {\left (7 \, a b^{13} d \cos \left (d x + c\right )^{6} - 7 \, {\left (5 \, a^{3} b^{11} + 3 \, a b^{13}\right )} d \cos \left (d x + c\right )^{4} + 7 \, {\left (3 \, a^{5} b^{9} + 10 \, a^{3} b^{11} + 3 \, a b^{13}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{7} b^{7} + 21 \, a^{5} b^{9} + 35 \, a^{3} b^{11} + 7 \, a b^{13}\right )} d + {\left (b^{14} d \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, a^{2} b^{12} + b^{14}\right )} d \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{10} + 42 \, a^{2} b^{12} + 3 \, b^{14}\right )} d \cos \left (d x + c\right )^{2} - {\left (7 \, a^{6} b^{8} + 35 \, a^{4} b^{10} + 21 \, a^{2} b^{12} + b^{14}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/35*(35*b^6*cos(d*x + c)^6 - 5*a^6 - 104*a^4*b^2 - 155*a^2*b^4 - 16*b^6 - 35*(5*a^2*b^4 + 2*b^6)*cos(d*x + c)
^4 + 7*(15*a^4*b^2 + 47*a^2*b^4 + 8*b^6)*cos(d*x + c)^2 - 7*(15*a*b^5*cos(d*x + c)^4 + 5*a^5*b + 24*a^3*b^3 +
11*a*b^5 - 25*(a^3*b^3 + a*b^5)*cos(d*x + c)^2)*sin(d*x + c))/(7*a*b^13*d*cos(d*x + c)^6 - 7*(5*a^3*b^11 + 3*a
*b^13)*d*cos(d*x + c)^4 + 7*(3*a^5*b^9 + 10*a^3*b^11 + 3*a*b^13)*d*cos(d*x + c)^2 - (a^7*b^7 + 21*a^5*b^9 + 35
*a^3*b^11 + 7*a*b^13)*d + (b^14*d*cos(d*x + c)^6 - 3*(7*a^2*b^12 + b^14)*d*cos(d*x + c)^4 + (35*a^4*b^10 + 42*
a^2*b^12 + 3*b^14)*d*cos(d*x + c)^2 - (7*a^6*b^8 + 35*a^4*b^10 + 21*a^2*b^12 + b^14)*d)*sin(d*x + c))

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giac [A]  time = 3.88, size = 215, normalized size = 1.04 \[ \frac {35 \, b^{6} \sin \left (d x + c\right )^{6} + 105 \, a b^{5} \sin \left (d x + c\right )^{5} + 175 \, a^{2} b^{4} \sin \left (d x + c\right )^{4} - 35 \, b^{6} \sin \left (d x + c\right )^{4} + 175 \, a^{3} b^{3} \sin \left (d x + c\right )^{3} - 35 \, a b^{5} \sin \left (d x + c\right )^{3} + 105 \, a^{4} b^{2} \sin \left (d x + c\right )^{2} - 21 \, a^{2} b^{4} \sin \left (d x + c\right )^{2} + 21 \, b^{6} \sin \left (d x + c\right )^{2} + 35 \, a^{5} b \sin \left (d x + c\right ) - 7 \, a^{3} b^{3} \sin \left (d x + c\right ) + 7 \, a b^{5} \sin \left (d x + c\right ) + 5 \, a^{6} - a^{4} b^{2} + a^{2} b^{4} - 5 \, b^{6}}{35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{7} b^{7} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/35*(35*b^6*sin(d*x + c)^6 + 105*a*b^5*sin(d*x + c)^5 + 175*a^2*b^4*sin(d*x + c)^4 - 35*b^6*sin(d*x + c)^4 +
175*a^3*b^3*sin(d*x + c)^3 - 35*a*b^5*sin(d*x + c)^3 + 105*a^4*b^2*sin(d*x + c)^2 - 21*a^2*b^4*sin(d*x + c)^2
+ 21*b^6*sin(d*x + c)^2 + 35*a^5*b*sin(d*x + c) - 7*a^3*b^3*sin(d*x + c) + 7*a*b^5*sin(d*x + c) + 5*a^6 - a^4*
b^2 + a^2*b^4 - 5*b^6)/((b*sin(d*x + c) + a)^7*b^7*d)

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maple [A]  time = 0.36, size = 208, normalized size = 1.00 \[ \frac {-\frac {a \left (5 a^{2}-3 b^{2}\right )}{b^{7} \left (a +b \sin \left (d x +c \right )\right )^{4}}-\frac {-a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}+b^{6}}{7 b^{7} \left (a +b \sin \left (d x +c \right )\right )^{7}}+\frac {1}{b^{7} \left (a +b \sin \left (d x +c \right )\right )}-\frac {-15 a^{4}+18 a^{2} b^{2}-3 b^{4}}{5 b^{7} \left (a +b \sin \left (d x +c \right )\right )^{5}}-\frac {a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}{b^{7} \left (a +b \sin \left (d x +c \right )\right )^{6}}-\frac {3 a}{b^{7} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {-15 a^{2}+3 b^{2}}{3 b^{7} \left (a +b \sin \left (d x +c \right )\right )^{3}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7/(a+b*sin(d*x+c))^8,x)

[Out]

1/d*(-a*(5*a^2-3*b^2)/b^7/(a+b*sin(d*x+c))^4-1/7*(-a^6+3*a^4*b^2-3*a^2*b^4+b^6)/b^7/(a+b*sin(d*x+c))^7+1/b^7/(
a+b*sin(d*x+c))-1/5*(-15*a^4+18*a^2*b^2-3*b^4)/b^7/(a+b*sin(d*x+c))^5-a*(a^4-2*a^2*b^2+b^4)/b^7/(a+b*sin(d*x+c
))^6-3*a/b^7/(a+b*sin(d*x+c))^2-1/3*(-15*a^2+3*b^2)/b^7/(a+b*sin(d*x+c))^3)

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maxima [A]  time = 0.33, size = 279, normalized size = 1.35 \[ \frac {35 \, b^{6} \sin \left (d x + c\right )^{6} + 105 \, a b^{5} \sin \left (d x + c\right )^{5} + 5 \, a^{6} - a^{4} b^{2} + a^{2} b^{4} - 5 \, b^{6} + 35 \, {\left (5 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{4} + 35 \, {\left (5 \, a^{3} b^{3} - a b^{5}\right )} \sin \left (d x + c\right )^{3} + 21 \, {\left (5 \, a^{4} b^{2} - a^{2} b^{4} + b^{6}\right )} \sin \left (d x + c\right )^{2} + 7 \, {\left (5 \, a^{5} b - a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )}{35 \, {\left (b^{14} \sin \left (d x + c\right )^{7} + 7 \, a b^{13} \sin \left (d x + c\right )^{6} + 21 \, a^{2} b^{12} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{11} \sin \left (d x + c\right )^{4} + 35 \, a^{4} b^{10} \sin \left (d x + c\right )^{3} + 21 \, a^{5} b^{9} \sin \left (d x + c\right )^{2} + 7 \, a^{6} b^{8} \sin \left (d x + c\right ) + a^{7} b^{7}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

1/35*(35*b^6*sin(d*x + c)^6 + 105*a*b^5*sin(d*x + c)^5 + 5*a^6 - a^4*b^2 + a^2*b^4 - 5*b^6 + 35*(5*a^2*b^4 - b
^6)*sin(d*x + c)^4 + 35*(5*a^3*b^3 - a*b^5)*sin(d*x + c)^3 + 21*(5*a^4*b^2 - a^2*b^4 + b^6)*sin(d*x + c)^2 + 7
*(5*a^5*b - a^3*b^3 + a*b^5)*sin(d*x + c))/((b^14*sin(d*x + c)^7 + 7*a*b^13*sin(d*x + c)^6 + 21*a^2*b^12*sin(d
*x + c)^5 + 35*a^3*b^11*sin(d*x + c)^4 + 35*a^4*b^10*sin(d*x + c)^3 + 21*a^5*b^9*sin(d*x + c)^2 + 7*a^6*b^8*si
n(d*x + c) + a^7*b^7)*d)

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mupad [B]  time = 0.24, size = 276, normalized size = 1.33 \[ \frac {\frac {5\,a^6-a^4\,b^2+a^2\,b^4-5\,b^6}{35\,b^7}+\frac {{\sin \left (c+d\,x\right )}^6}{b}+\frac {3\,{\sin \left (c+d\,x\right )}^2\,\left (5\,a^4-a^2\,b^2+b^4\right )}{5\,b^5}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{b^2}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (5\,a^2-b^2\right )}{b^3}+\frac {a\,\sin \left (c+d\,x\right )\,\left (5\,a^4-a^2\,b^2+b^4\right )}{5\,b^6}+\frac {a\,{\sin \left (c+d\,x\right )}^3\,\left (5\,a^2-b^2\right )}{b^4}}{d\,\left (a^7+7\,a^6\,b\,\sin \left (c+d\,x\right )+21\,a^5\,b^2\,{\sin \left (c+d\,x\right )}^2+35\,a^4\,b^3\,{\sin \left (c+d\,x\right )}^3+35\,a^3\,b^4\,{\sin \left (c+d\,x\right )}^4+21\,a^2\,b^5\,{\sin \left (c+d\,x\right )}^5+7\,a\,b^6\,{\sin \left (c+d\,x\right )}^6+b^7\,{\sin \left (c+d\,x\right )}^7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^7/(a + b*sin(c + d*x))^8,x)

[Out]

((5*a^6 - 5*b^6 + a^2*b^4 - a^4*b^2)/(35*b^7) + sin(c + d*x)^6/b + (3*sin(c + d*x)^2*(5*a^4 + b^4 - a^2*b^2))/
(5*b^5) + (3*a*sin(c + d*x)^5)/b^2 + (sin(c + d*x)^4*(5*a^2 - b^2))/b^3 + (a*sin(c + d*x)*(5*a^4 + b^4 - a^2*b
^2))/(5*b^6) + (a*sin(c + d*x)^3*(5*a^2 - b^2))/b^4)/(d*(a^7 + b^7*sin(c + d*x)^7 + 7*a*b^6*sin(c + d*x)^6 + 2
1*a^5*b^2*sin(c + d*x)^2 + 35*a^4*b^3*sin(c + d*x)^3 + 35*a^3*b^4*sin(c + d*x)^4 + 21*a^2*b^5*sin(c + d*x)^5 +
7*a^6*b*sin(c + d*x)))

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sympy [A]  time = 47.44, size = 2530, normalized size = 12.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7/(a+b*sin(d*x+c))**8,x)

[Out]

Piecewise((x*cos(c)**7/a**8, Eq(b, 0) & Eq(d, 0)), ((16*sin(c + d*x)**7/(35*d) + 8*sin(c + d*x)**5*cos(c + d*x
)**2/(5*d) + 2*sin(c + d*x)**3*cos(c + d*x)**4/d + sin(c + d*x)*cos(c + d*x)**6/d)/a**8, Eq(b, 0)), (x*cos(c)*
*7/(a + b*sin(c))**8, Eq(d, 0)), (5*a**6/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(
c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c +
d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) + 35*a**5*b*sin(c + d*x)/(35*a**7*b**7*
d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*
a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*s
in(c + d*x)**7) + 104*a**4*b**2*sin(c + d*x)**2/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9
*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*
sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) - a**4*b**2*cos(c + d*x)**2/(35*
a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)*
*3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35
*b**14*d*sin(c + d*x)**7) + 168*a**3*b**3*sin(c + d*x)**3/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735
*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**
2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) - 7*a**3*b**3*sin(c +
d*x)*cos(c + d*x)**2/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a
**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**
13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) + 155*a**2*b**4*sin(c + d*x)**4/(35*a**7*b**7*d + 245*a**6*
b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*
sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**
7) - 19*a**2*b**4*sin(c + d*x)**2*cos(c + d*x)**2/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b*
*9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*
d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) + a**2*b**4*cos(c + d*x)**4/(3
5*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x
)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 +
35*b**14*d*sin(c + d*x)**7) + 77*a*b**5*sin(c + d*x)**5/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a
**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*
b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) - 21*a*b**5*sin(c + d*x)
**3*cos(c + d*x)**2/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a*
*4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**1
3*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) + 7*a*b**5*sin(c + d*x)*cos(c + d*x)**4/(35*a**7*b**7*d + 24
5*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b
**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c +
d*x)**7) + 16*b**6*sin(c + d*x)**6/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d
*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)
**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) - 8*b**6*sin(c + d*x)**4*cos(c + d*x)**2/(35
*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)
**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 3
5*b**14*d*sin(c + d*x)**7) + 6*b**6*sin(c + d*x)**2*cos(c + d*x)**4/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c +
d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4
+ 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) - 5*b**6*cos
(c + d*x)**6/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**1
0*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin
(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7), True))

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