3.630 \(\int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx\)
Optimal. Leaf size=254 \[ -\frac{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{m+1}}{b^7 d (m+1)}+\frac{6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{m+2}}{b^7 d (m+2)}-\frac{3 \left (-6 a^2 b^2+5 a^4+b^4\right ) (a+b \sin (c+d x))^{m+3}}{b^7 d (m+3)}+\frac{4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{m+4}}{b^7 d (m+4)}-\frac{3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{m+5}}{b^7 d (m+5)}+\frac{6 a (a+b \sin (c+d x))^{m+6}}{b^7 d (m+6)}-\frac{(a+b \sin (c+d x))^{m+7}}{b^7 d (m+7)} \]
[Out]
-(((a^2 - b^2)^3*(a + b*Sin[c + d*x])^(1 + m))/(b^7*d*(1 + m))) + (6*a*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(2 +
m))/(b^7*d*(2 + m)) - (3*(5*a^4 - 6*a^2*b^2 + b^4)*(a + b*Sin[c + d*x])^(3 + m))/(b^7*d*(3 + m)) + (4*a*(5*a^
2 - 3*b^2)*(a + b*Sin[c + d*x])^(4 + m))/(b^7*d*(4 + m)) - (3*(5*a^2 - b^2)*(a + b*Sin[c + d*x])^(5 + m))/(b^7
*d*(5 + m)) + (6*a*(a + b*Sin[c + d*x])^(6 + m))/(b^7*d*(6 + m)) - (a + b*Sin[c + d*x])^(7 + m)/(b^7*d*(7 + m)
)
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Rubi [A] time = 0.16399, antiderivative size = 254, 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 =
{2668, 697} \[ -\frac{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{m+1}}{b^7 d (m+1)}+\frac{6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{m+2}}{b^7 d (m+2)}-\frac{3 \left (-6 a^2 b^2+5 a^4+b^4\right ) (a+b \sin (c+d x))^{m+3}}{b^7 d (m+3)}+\frac{4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{m+4}}{b^7 d (m+4)}-\frac{3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{m+5}}{b^7 d (m+5)}+\frac{6 a (a+b \sin (c+d x))^{m+6}}{b^7 d (m+6)}-\frac{(a+b \sin (c+d x))^{m+7}}{b^7 d (m+7)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^7*(a + b*Sin[c + d*x])^m,x]
[Out]
-(((a^2 - b^2)^3*(a + b*Sin[c + d*x])^(1 + m))/(b^7*d*(1 + m))) + (6*a*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(2 +
m))/(b^7*d*(2 + m)) - (3*(5*a^4 - 6*a^2*b^2 + b^4)*(a + b*Sin[c + d*x])^(3 + m))/(b^7*d*(3 + m)) + (4*a*(5*a^
2 - 3*b^2)*(a + b*Sin[c + d*x])^(4 + m))/(b^7*d*(4 + m)) - (3*(5*a^2 - b^2)*(a + b*Sin[c + d*x])^(5 + m))/(b^7
*d*(5 + m)) + (6*a*(a + b*Sin[c + d*x])^(6 + m))/(b^7*d*(6 + m)) - (a + b*Sin[c + d*x])^(7 + m)/(b^7*d*(7 + m)
)
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]
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]
Rubi steps
\begin{align*} \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^m \left (b^2-x^2\right )^3 \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\left (a^2-b^2\right )^3 (a+x)^m+6 a \left (a^2-b^2\right )^2 (a+x)^{1+m}-3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+x)^{2+m}+4 a \left (5 a^2-3 b^2\right ) (a+x)^{3+m}-3 \left (5 a^2-b^2\right ) (a+x)^{4+m}+6 a (a+x)^{5+m}-(a+x)^{6+m}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=-\frac{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{1+m}}{b^7 d (1+m)}+\frac{6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{2+m}}{b^7 d (2+m)}-\frac{3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+b \sin (c+d x))^{3+m}}{b^7 d (3+m)}+\frac{4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{4+m}}{b^7 d (4+m)}-\frac{3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{5+m}}{b^7 d (5+m)}+\frac{6 a (a+b \sin (c+d x))^{6+m}}{b^7 d (6+m)}-\frac{(a+b \sin (c+d x))^{7+m}}{b^7 d (7+m)}\\ \end{align*}
Mathematica [A] time = 6.11701, size = 459, normalized size = 1.81 \[ \frac{\frac{6 \left (\left (b^2-a^2\right ) \left (\frac{4 \left (\left (b^2-a^2\right ) \left (-\frac{\left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+1}}{m+1}+\frac{2 a (a+b \sin (c+d x))^{m+2}}{m+2}-\frac{(a+b \sin (c+d x))^{m+3}}{m+3}\right )+a \left (-\frac{\left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+2}}{m+2}+\frac{2 a (a+b \sin (c+d x))^{m+3}}{m+3}-\frac{(a+b \sin (c+d x))^{m+4}}{m+4}\right )\right )}{m+5}+\frac{b^4 \cos ^4(c+d x) (a+b \sin (c+d x))^{m+1}}{m+5}\right )+a \left (\frac{4 \left (\left (b^2-a^2\right ) \left (-\frac{\left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+2}}{m+2}+\frac{2 a (a+b \sin (c+d x))^{m+3}}{m+3}-\frac{(a+b \sin (c+d x))^{m+4}}{m+4}\right )+a \left (-\frac{\left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+3}}{m+3}+\frac{2 a (a+b \sin (c+d x))^{m+4}}{m+4}-\frac{(a+b \sin (c+d x))^{m+5}}{m+5}\right )\right )}{m+6}+\frac{b^4 \cos ^4(c+d x) (a+b \sin (c+d x))^{m+2}}{m+6}\right )\right )}{m+7}+\frac{b^6 \cos ^6(c+d x) (a+b \sin (c+d x))^{m+1}}{m+7}}{b^7 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^7*(a + b*Sin[c + d*x])^m,x]
[Out]
((b^6*Cos[c + d*x]^6*(a + b*Sin[c + d*x])^(1 + m))/(7 + m) + (6*((-a^2 + b^2)*((b^4*Cos[c + d*x]^4*(a + b*Sin[
c + d*x])^(1 + m))/(5 + m) + (4*((-a^2 + b^2)*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(1 + m))/(1 + m)) + (2*a*(a
+ b*Sin[c + d*x])^(2 + m))/(2 + m) - (a + b*Sin[c + d*x])^(3 + m)/(3 + m)) + a*(-(((a^2 - b^2)*(a + b*Sin[c +
d*x])^(2 + m))/(2 + m)) + (2*a*(a + b*Sin[c + d*x])^(3 + m))/(3 + m) - (a + b*Sin[c + d*x])^(4 + m)/(4 + m)))
)/(5 + m)) + a*((b^4*Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(2 + m))/(6 + m) + (4*((-a^2 + b^2)*(-(((a^2 - b^2)*(
a + b*Sin[c + d*x])^(2 + m))/(2 + m)) + (2*a*(a + b*Sin[c + d*x])^(3 + m))/(3 + m) - (a + b*Sin[c + d*x])^(4 +
m)/(4 + m)) + a*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(3 + m))/(3 + m)) + (2*a*(a + b*Sin[c + d*x])^(4 + m))/(
4 + m) - (a + b*Sin[c + d*x])^(5 + m)/(5 + m))))/(6 + m))))/(7 + m))/(b^7*d)
________________________________________________________________________________________
Maple [F] time = 0.3, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{7} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x)
[Out]
int(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 2.6926, size = 1801, normalized size = 7.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x, algorithm="fricas")
[Out]
-(720*a^7 - 3024*a^5*b^2 + 5040*a^3*b^4 - 5040*a*b^6 - (a*b^6*m^6 + 15*a*b^6*m^5 + 85*a*b^6*m^4 + 225*a*b^6*m^
3 + 274*a*b^6*m^2 + 120*a*b^6*m)*cos(d*x + c)^6 - 6*(2*a*b^6*m^5 - (5*a^3*b^4 - 23*a*b^6)*m^4 - 2*(15*a^3*b^4
- 44*a*b^6)*m^3 - (55*a^3*b^4 - 133*a*b^6)*m^2 - 6*(5*a^3*b^4 - 11*a*b^6)*m)*cos(d*x + c)^4 - 192*(a^3*b^4 + a
*b^6)*m^3 + 288*(a^5*b^2 - 2*a^3*b^4 - 7*a*b^6)*m^2 - 24*((a^3*b^4 + 3*a*b^6)*m^4 - 6*(a^3*b^4 - 5*a*b^6)*m^3
+ (15*a^5*b^2 - 55*a^3*b^4 + 84*a*b^6)*m^2 + 3*(5*a^5*b^2 - 16*a^3*b^4 + 19*a*b^6)*m)*cos(d*x + c)^2 - 192*(3*
a^5*b^2 - 13*a^3*b^4 + 32*a*b^6)*m - (2304*b^7 + (b^7*m^6 + 21*b^7*m^5 + 175*b^7*m^4 + 735*b^7*m^3 + 1624*b^7*
m^2 + 1764*b^7*m + 720*b^7)*cos(d*x + c)^6 + 6*(144*b^7 + (a^2*b^5 + b^7)*m^5 + 2*(5*a^2*b^5 + 8*b^7)*m^4 + 5*
(7*a^2*b^5 + 19*b^7)*m^3 + 10*(5*a^2*b^5 + 26*b^7)*m^2 + 12*(2*a^2*b^5 + 27*b^7)*m)*cos(d*x + c)^4 + 48*(a^4*b
^3 + 6*a^2*b^5 + b^7)*m^3 - 576*(a^4*b^3 - 4*a^2*b^5 - b^7)*m^2 + 24*(48*b^7 + (3*a^2*b^5 + b^7)*m^4 - (5*a^4*
b^3 - 24*a^2*b^5 - 13*b^7)*m^3 - (15*a^4*b^3 - 51*a^2*b^5 - 56*b^7)*m^2 - 2*(5*a^4*b^3 - 15*a^2*b^5 - 46*b^7)*
m)*cos(d*x + c)^2 + 48*(15*a^6*b - 58*a^4*b^3 + 87*a^2*b^5 + 44*b^7)*m)*sin(d*x + c))*(b*sin(d*x + c) + a)^m/(
b^7*d*m^7 + 28*b^7*d*m^6 + 322*b^7*d*m^5 + 1960*b^7*d*m^4 + 6769*b^7*d*m^3 + 13132*b^7*d*m^2 + 13068*b^7*d*m +
5040*b^7*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(cos(d*x+c)**7*(a+b*sin(d*x+c))**m,x)
[Out]
Timed out
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Giac [B] time = 1.17609, size = 4832, normalized size = 19.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x, algorithm="giac")
[Out]
-((b*sin(d*x + c) + a)^7*(b*sin(d*x + c) + a)^m*m^6 - 6*(b*sin(d*x + c) + a)^6*(b*sin(d*x + c) + a)^m*a*m^6 +
15*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*a^2*m^6 - 20*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a^
3*m^6 + 15*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^4*m^6 - 6*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) +
a)^m*a^5*m^6 + (b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^6*m^6 - 3*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c)
+ a)^m*b^2*m^6 + 12*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a*b^2*m^6 - 18*(b*sin(d*x + c) + a)^3*(b*sin
(d*x + c) + a)^m*a^2*b^2*m^6 + 12*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a^3*b^2*m^6 - 3*(b*sin(d*x + c
) + a)*(b*sin(d*x + c) + a)^m*a^4*b^2*m^6 + 3*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*b^4*m^6 - 6*(b*sin
(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a*b^4*m^6 + 3*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^2*b^4*m^6
- (b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*b^6*m^6 + 21*(b*sin(d*x + c) + a)^7*(b*sin(d*x + c) + a)^m*m^5 -
132*(b*sin(d*x + c) + a)^6*(b*sin(d*x + c) + a)^m*a*m^5 + 345*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*a
^2*m^5 - 480*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a^3*m^5 + 375*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c
) + a)^m*a^4*m^5 - 156*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a^5*m^5 + 27*(b*sin(d*x + c) + a)*(b*sin(
d*x + c) + a)^m*a^6*m^5 - 69*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*b^2*m^5 + 288*(b*sin(d*x + c) + a)^
4*(b*sin(d*x + c) + a)^m*a*b^2*m^5 - 450*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^2*b^2*m^5 + 312*(b*si
n(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a^3*b^2*m^5 - 81*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^4*b^2*
m^5 + 75*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*b^4*m^5 - 156*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) +
a)^m*a*b^4*m^5 + 81*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^2*b^4*m^5 - 27*(b*sin(d*x + c) + a)*(b*sin(d
*x + c) + a)^m*b^6*m^5 + 175*(b*sin(d*x + c) + a)^7*(b*sin(d*x + c) + a)^m*m^4 - 1140*(b*sin(d*x + c) + a)^6*(
b*sin(d*x + c) + a)^m*a*m^4 + 3105*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*a^2*m^4 - 4520*(b*sin(d*x + c
) + a)^4*(b*sin(d*x + c) + a)^m*a^3*m^4 + 3705*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^4*m^4 - 1620*(b
*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a^5*m^4 + 295*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^6*m^4
- 621*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*b^2*m^4 + 2712*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)
^m*a*b^2*m^4 - 4446*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^2*b^2*m^4 + 3240*(b*sin(d*x + c) + a)^2*(b
*sin(d*x + c) + a)^m*a^3*b^2*m^4 - 885*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^4*b^2*m^4 + 741*(b*sin(d*
x + c) + a)^3*(b*sin(d*x + c) + a)^m*b^4*m^4 - 1620*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a*b^4*m^4 +
885*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^2*b^4*m^4 - 295*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*
b^6*m^4 + 735*(b*sin(d*x + c) + a)^7*(b*sin(d*x + c) + a)^m*m^3 - 4920*(b*sin(d*x + c) + a)^6*(b*sin(d*x + c)
+ a)^m*a*m^3 + 13875*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*a^2*m^3 - 21120*(b*sin(d*x + c) + a)^4*(b*s
in(d*x + c) + a)^m*a^3*m^3 + 18285*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^4*m^3 - 8520*(b*sin(d*x + c
) + a)^2*(b*sin(d*x + c) + a)^m*a^5*m^3 + 1665*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^6*m^3 - 2775*(b*s
in(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*b^2*m^3 + 12672*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a*b^2*
m^3 - 21942*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^2*b^2*m^3 + 17040*(b*sin(d*x + c) + a)^2*(b*sin(d*
x + c) + a)^m*a^3*b^2*m^3 - 4995*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^4*b^2*m^3 + 3657*(b*sin(d*x + c
) + a)^3*(b*sin(d*x + c) + a)^m*b^4*m^3 - 8520*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a*b^4*m^3 + 4995*
(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^2*b^4*m^3 - 1665*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*b^6
*m^3 + 1624*(b*sin(d*x + c) + a)^7*(b*sin(d*x + c) + a)^m*m^2 - 11094*(b*sin(d*x + c) + a)^6*(b*sin(d*x + c) +
a)^m*a*m^2 + 32160*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*a^2*m^2 - 50900*(b*sin(d*x + c) + a)^4*(b*si
n(d*x + c) + a)^m*a^3*m^2 + 46680*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^4*m^2 - 23574*(b*sin(d*x + c
) + a)^2*(b*sin(d*x + c) + a)^m*a^5*m^2 + 5104*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^6*m^2 - 6432*(b*s
in(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*b^2*m^2 + 30540*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a*b^2*
m^2 - 56016*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^2*b^2*m^2 + 47148*(b*sin(d*x + c) + a)^2*(b*sin(d*
x + c) + a)^m*a^3*b^2*m^2 - 15312*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^4*b^2*m^2 + 9336*(b*sin(d*x +
c) + a)^3*(b*sin(d*x + c) + a)^m*b^4*m^2 - 23574*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a*b^4*m^2 + 153
12*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^2*b^4*m^2 - 5104*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*
b^6*m^2 + 1764*(b*sin(d*x + c) + a)^7*(b*sin(d*x + c) + a)^m*m - 12228*(b*sin(d*x + c) + a)^6*(b*sin(d*x + c)
+ a)^m*a*m + 36180*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*a^2*m - 59040*(b*sin(d*x + c) + a)^4*(b*sin(d
*x + c) + a)^m*a^3*m + 56940*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^4*m - 31644*(b*sin(d*x + c) + a)^
2*(b*sin(d*x + c) + a)^m*a^5*m + 8028*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^6*m - 7236*(b*sin(d*x + c)
+ a)^5*(b*sin(d*x + c) + a)^m*b^2*m + 35424*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a*b^2*m - 68328*(b*
sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^2*b^2*m + 63288*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a^3
*b^2*m - 24084*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^4*b^2*m + 11388*(b*sin(d*x + c) + a)^3*(b*sin(d*x
+ c) + a)^m*b^4*m - 31644*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a*b^4*m + 24084*(b*sin(d*x + c) + a)*
(b*sin(d*x + c) + a)^m*a^2*b^4*m - 8028*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*b^6*m + 720*(b*sin(d*x + c
) + a)^7*(b*sin(d*x + c) + a)^m - 5040*(b*sin(d*x + c) + a)^6*(b*sin(d*x + c) + a)^m*a + 15120*(b*sin(d*x + c)
+ a)^5*(b*sin(d*x + c) + a)^m*a^2 - 25200*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a^3 + 25200*(b*sin(d*
x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^4 - 15120*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a^5 + 5040*(b*s
in(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^6 - 3024*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*b^2 + 15120*(
b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a*b^2 - 30240*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^2*b
^2 + 30240*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a^3*b^2 - 15120*(b*sin(d*x + c) + a)*(b*sin(d*x + c)
+ a)^m*a^4*b^2 + 5040*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*b^4 - 15120*(b*sin(d*x + c) + a)^2*(b*sin(
d*x + c) + a)^m*a*b^4 + 15120*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^2*b^4 - 5040*(b*sin(d*x + c) + a)*
(b*sin(d*x + c) + a)^m*b^6)/((b^6*m^7 + 28*b^6*m^6 + 322*b^6*m^5 + 1960*b^6*m^4 + 6769*b^6*m^3 + 13132*b^6*m^2
+ 13068*b^6*m + 5040*b^6)*b*d)