3.344 \(\int \cos ^5(c+d x) (a+a \sin (c+d x))^m \, dx\)
Optimal. Leaf size=81 \[ \frac {(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}-\frac {4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac {4 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)} \]
[Out]
4*(a+a*sin(d*x+c))^(3+m)/a^3/d/(3+m)-4*(a+a*sin(d*x+c))^(4+m)/a^4/d/(4+m)+(a+a*sin(d*x+c))^(5+m)/a^5/d/(5+m)
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 81, 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 =
{2667, 43} \[ \frac {4 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac {4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac {(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^m,x]
[Out]
(4*(a + a*Sin[c + d*x])^(3 + m))/(a^3*d*(3 + m)) - (4*(a + a*Sin[c + d*x])^(4 + m))/(a^4*d*(4 + m)) + (a + a*S
in[c + d*x])^(5 + m)/(a^5*d*(5 + m))
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rule 2667
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 + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/2])
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^{2+m} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (a+x)^{2+m}-4 a (a+x)^{3+m}+(a+x)^{4+m}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {4 (a+a \sin (c+d x))^{3+m}}{a^3 d (3+m)}-\frac {4 (a+a \sin (c+d x))^{4+m}}{a^4 d (4+m)}+\frac {(a+a \sin (c+d x))^{5+m}}{a^5 d (5+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.32, size = 68, normalized size = 0.84 \[ \frac {(a (\sin (c+d x)+1))^{m+3} \left (-\frac {4 a^2 (\sin (c+d x)+1)}{m+4}+\frac {4 a^2}{m+3}+\frac {(a \sin (c+d x)+a)^2}{m+5}\right )}{a^5 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^m,x]
[Out]
((a*(1 + Sin[c + d*x]))^(3 + m)*((4*a^2)/(3 + m) - (4*a^2*(1 + Sin[c + d*x]))/(4 + m) + (a + a*Sin[c + d*x])^2
/(5 + m)))/(a^5*d)
________________________________________________________________________________________
fricas [A] time = 0.47, size = 102, normalized size = 1.26 \[ \frac {{\left ({\left (m^{2} + 3 \, m\right )} \cos \left (d x + c\right )^{4} + 8 \, m \cos \left (d x + c\right )^{2} + {\left ({\left (m^{2} + 7 \, m + 12\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (m + 2\right )} \cos \left (d x + c\right )^{2} + 32\right )} \sin \left (d x + c\right ) + 32\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{3} + 12 \, d m^{2} + 47 \, d m + 60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^m,x, algorithm="fricas")
[Out]
((m^2 + 3*m)*cos(d*x + c)^4 + 8*m*cos(d*x + c)^2 + ((m^2 + 7*m + 12)*cos(d*x + c)^4 + 8*(m + 2)*cos(d*x + c)^2
+ 32)*sin(d*x + c) + 32)*(a*sin(d*x + c) + a)^m/(d*m^3 + 12*d*m^2 + 47*d*m + 60*d)
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^m,x, algorithm="giac")
[Out]
Timed out
________________________________________________________________________________________
maple [F] time = 3.22, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{5}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^5*(a+a*sin(d*x+c))^m,x)
[Out]
int(cos(d*x+c)^5*(a+a*sin(d*x+c))^m,x)
________________________________________________________________________________________
maxima [B] time = 0.82, size = 266, normalized size = 3.28 \[ \frac {\frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (d x + c\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (d x + c\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 24 \, a^{m} m \sin \left (d x + c\right ) + 24 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} - \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (d x + c\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 2 \, a^{m} m \sin \left (d x + c\right ) + 2 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m + 1}}{a {\left (m + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^m,x, algorithm="maxima")
[Out]
(((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*a^m*sin(d*x + c)^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(d*x + c)^4 - 4
*(m^3 + 3*m^2 + 2*m)*a^m*sin(d*x + c)^3 + 12*(m^2 + m)*a^m*sin(d*x + c)^2 - 24*a^m*m*sin(d*x + c) + 24*a^m)*(s
in(d*x + c) + 1)^m/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) - 2*((m^2 + 3*m + 2)*a^m*sin(d*x + c)^3 + (
m^2 + m)*a^m*sin(d*x + c)^2 - 2*a^m*m*sin(d*x + c) + 2*a^m)*(sin(d*x + c) + 1)^m/(m^3 + 6*m^2 + 11*m + 6) + (a
*sin(d*x + c) + a)^(m + 1)/(a*(m + 1)))/d
________________________________________________________________________________________
mupad [B] time = 1.97, size = 195, normalized size = 2.41 \[ \frac {{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^m\,\left (82\,m+600\,\sin \left (c+d\,x\right )+100\,\sin \left (3\,c+3\,d\,x\right )+12\,\sin \left (5\,c+5\,d\,x\right )+46\,m\,\sin \left (c+d\,x\right )+88\,m\,\cos \left (2\,c+2\,d\,x\right )+6\,m\,\cos \left (4\,c+4\,d\,x\right )+53\,m\,\sin \left (3\,c+3\,d\,x\right )+7\,m\,\sin \left (5\,c+5\,d\,x\right )+2\,m^2\,\sin \left (c+d\,x\right )+6\,m^2+8\,m^2\,\cos \left (2\,c+2\,d\,x\right )+2\,m^2\,\cos \left (4\,c+4\,d\,x\right )+3\,m^2\,\sin \left (3\,c+3\,d\,x\right )+m^2\,\sin \left (5\,c+5\,d\,x\right )+512\right )}{16\,d\,\left (m^3+12\,m^2+47\,m+60\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(c + d*x)^5*(a + a*sin(c + d*x))^m,x)
[Out]
((a*(sin(c + d*x) + 1))^m*(82*m + 600*sin(c + d*x) + 100*sin(3*c + 3*d*x) + 12*sin(5*c + 5*d*x) + 46*m*sin(c +
d*x) + 88*m*cos(2*c + 2*d*x) + 6*m*cos(4*c + 4*d*x) + 53*m*sin(3*c + 3*d*x) + 7*m*sin(5*c + 5*d*x) + 2*m^2*si
n(c + d*x) + 6*m^2 + 8*m^2*cos(2*c + 2*d*x) + 2*m^2*cos(4*c + 4*d*x) + 3*m^2*sin(3*c + 3*d*x) + m^2*sin(5*c +
5*d*x) + 512))/(16*d*(47*m + 12*m^2 + m^3 + 60))
________________________________________________________________________________________
sympy [A] time = 172.27, size = 5534, normalized size = 68.32 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**5*(a+a*sin(d*x+c))**m,x)
[Out]
Piecewise((x*(a*sin(c) + a)**m*cos(c)**5, Eq(d, 0)), (12*log(sin(c + d*x) + 1)*sin(c + d*x)**4/(12*a**5*d*sin(
c + d*x)**4 + 48*a**5*d*sin(c + d*x)**3 + 72*a**5*d*sin(c + d*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**5*d) + 48
*log(sin(c + d*x) + 1)*sin(c + d*x)**3/(12*a**5*d*sin(c + d*x)**4 + 48*a**5*d*sin(c + d*x)**3 + 72*a**5*d*sin(
c + d*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**5*d) + 72*log(sin(c + d*x) + 1)*sin(c + d*x)**2/(12*a**5*d*sin(c
+ d*x)**4 + 48*a**5*d*sin(c + d*x)**3 + 72*a**5*d*sin(c + d*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**5*d) + 48*l
og(sin(c + d*x) + 1)*sin(c + d*x)/(12*a**5*d*sin(c + d*x)**4 + 48*a**5*d*sin(c + d*x)**3 + 72*a**5*d*sin(c + d
*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**5*d) + 12*log(sin(c + d*x) + 1)/(12*a**5*d*sin(c + d*x)**4 + 48*a**5*d
*sin(c + d*x)**3 + 72*a**5*d*sin(c + d*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**5*d) + 20*sin(c + d*x)**3/(12*a*
*5*d*sin(c + d*x)**4 + 48*a**5*d*sin(c + d*x)**3 + 72*a**5*d*sin(c + d*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**
5*d) + 6*sin(c + d*x)**2*cos(c + d*x)**2/(12*a**5*d*sin(c + d*x)**4 + 48*a**5*d*sin(c + d*x)**3 + 72*a**5*d*si
n(c + d*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**5*d) + 56*sin(c + d*x)**2/(12*a**5*d*sin(c + d*x)**4 + 48*a**5*
d*sin(c + d*x)**3 + 72*a**5*d*sin(c + d*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**5*d) + 8*sin(c + d*x)*cos(c + d
*x)**2/(12*a**5*d*sin(c + d*x)**4 + 48*a**5*d*sin(c + d*x)**3 + 72*a**5*d*sin(c + d*x)**2 + 48*a**5*d*sin(c +
d*x) + 12*a**5*d) + 52*sin(c + d*x)/(12*a**5*d*sin(c + d*x)**4 + 48*a**5*d*sin(c + d*x)**3 + 72*a**5*d*sin(c +
d*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**5*d) - 3*cos(c + d*x)**4/(12*a**5*d*sin(c + d*x)**4 + 48*a**5*d*sin(
c + d*x)**3 + 72*a**5*d*sin(c + d*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**5*d) + 2*cos(c + d*x)**2/(12*a**5*d*s
in(c + d*x)**4 + 48*a**5*d*sin(c + d*x)**3 + 72*a**5*d*sin(c + d*x)**2 + 48*a**5*d*sin(c + d*x) + 12*a**5*d) +
16/(12*a**5*d*sin(c + d*x)**4 + 48*a**5*d*sin(c + d*x)**3 + 72*a**5*d*sin(c + d*x)**2 + 48*a**5*d*sin(c + d*x
) + 12*a**5*d), Eq(m, -5)), (-12*log(sin(c + d*x) + 1)*sin(c + d*x)**3/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*si
n(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a**4*d) - 36*log(sin(c + d*x) + 1)*sin(c + d*x)**2/(3*a**4*d*sin(c +
d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a**4*d) - 36*log(sin(c + d*x) + 1)*sin(c + d*x
)/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a**4*d) - 12*log(sin(c + d*
x) + 1)/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a**4*d) + 8*sin(c + d
*x)**4/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a**4*d) + 4*sin(c + d*
x)**2*cos(c + d*x)**2/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a**4*d)
- 52*sin(c + d*x)**2/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a**4*d)
+ 6*sin(c + d*x)*cos(c + d*x)**2/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x)
+ 3*a**4*d) - 72*sin(c + d*x)/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) +
3*a**4*d) - cos(c + d*x)**4/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a
**4*d) + 2*cos(c + d*x)**2/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a*
*4*d) - 28/(3*a**4*d*sin(c + d*x)**3 + 9*a**4*d*sin(c + d*x)**2 + 9*a**4*d*sin(c + d*x) + 3*a**4*d), Eq(m, -4)
), (8*log(tan(c/2 + d*x/2) + 1)*tan(c/2 + d*x/2)**4/(a**3*d*tan(c/2 + d*x/2)**4 + 2*a**3*d*tan(c/2 + d*x/2)**2
+ a**3*d) + 16*log(tan(c/2 + d*x/2) + 1)*tan(c/2 + d*x/2)**2/(a**3*d*tan(c/2 + d*x/2)**4 + 2*a**3*d*tan(c/2 +
d*x/2)**2 + a**3*d) + 8*log(tan(c/2 + d*x/2) + 1)/(a**3*d*tan(c/2 + d*x/2)**4 + 2*a**3*d*tan(c/2 + d*x/2)**2
+ a**3*d) - 4*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**4/(a**3*d*tan(c/2 + d*x/2)**4 + 2*a**3*d*tan(c/2
+ d*x/2)**2 + a**3*d) - 8*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**2/(a**3*d*tan(c/2 + d*x/2)**4 + 2*a**
3*d*tan(c/2 + d*x/2)**2 + a**3*d) - 4*log(tan(c/2 + d*x/2)**2 + 1)/(a**3*d*tan(c/2 + d*x/2)**4 + 2*a**3*d*tan(
c/2 + d*x/2)**2 + a**3*d) - 6*tan(c/2 + d*x/2)**3/(a**3*d*tan(c/2 + d*x/2)**4 + 2*a**3*d*tan(c/2 + d*x/2)**2 +
a**3*d) + 2*tan(c/2 + d*x/2)**2/(a**3*d*tan(c/2 + d*x/2)**4 + 2*a**3*d*tan(c/2 + d*x/2)**2 + a**3*d) - 6*tan(
c/2 + d*x/2)/(a**3*d*tan(c/2 + d*x/2)**4 + 2*a**3*d*tan(c/2 + d*x/2)**2 + a**3*d), Eq(m, -3)), (6*tan(c/2 + d*
x/2)**5/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d
) - 12*tan(c/2 + d*x/2)**4/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d
*x/2)**2 + 3*a**2*d) + 20*tan(c/2 + d*x/2)**3/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9
*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d) - 12*tan(c/2 + d*x/2)**2/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(
c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d) + 6*tan(c/2 + d*x/2)/(3*a**2*d*tan(c/2 + d*x/2)**6
+ 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d), Eq(m, -2)), (6*tan(c/2 + d*x/2)**7/
(3*a*d*tan(c/2 + d*x/2)**8 + 12*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 12*a*d*tan(c/2 + d*x/2)
**2 + 3*a*d) - 6*tan(c/2 + d*x/2)**6/(3*a*d*tan(c/2 + d*x/2)**8 + 12*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2
+ d*x/2)**4 + 12*a*d*tan(c/2 + d*x/2)**2 + 3*a*d) + 10*tan(c/2 + d*x/2)**5/(3*a*d*tan(c/2 + d*x/2)**8 + 12*a*d
*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 12*a*d*tan(c/2 + d*x/2)**2 + 3*a*d) + 10*tan(c/2 + d*x/2)*
*3/(3*a*d*tan(c/2 + d*x/2)**8 + 12*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 12*a*d*tan(c/2 + d*x
/2)**2 + 3*a*d) - 6*tan(c/2 + d*x/2)**2/(3*a*d*tan(c/2 + d*x/2)**8 + 12*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c
/2 + d*x/2)**4 + 12*a*d*tan(c/2 + d*x/2)**2 + 3*a*d) + 6*tan(c/2 + d*x/2)/(3*a*d*tan(c/2 + d*x/2)**8 + 12*a*d*
tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 12*a*d*tan(c/2 + d*x/2)**2 + 3*a*d), Eq(m, -1)), (m**4*(a*s
in(c + d*x) + a)**m*sin(c + d*x)*cos(c + d*x)**4/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*
d) + m**4*(a*sin(c + d*x) + a)**m*cos(c + d*x)**4/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120
*d) + 4*m**3*(a*sin(c + d*x) + a)**m*sin(c + d*x)**3*cos(c + d*x)**2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m
**2 + 274*d*m + 120*d) + 8*m**3*(a*sin(c + d*x) + a)**m*sin(c + d*x)**2*cos(c + d*x)**2/(d*m**5 + 15*d*m**4 +
85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 14*m**3*(a*sin(c + d*x) + a)**m*sin(c + d*x)*cos(c + d*x)**4/(d*m*
*5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 4*m**3*(a*sin(c + d*x) + a)**m*sin(c + d*x)*cos(c
+ d*x)**2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 14*m**3*(a*sin(c + d*x) + a)**m*c
os(c + d*x)**4/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 8*m**2*(a*sin(c + d*x) + a)**
m*sin(c + d*x)**5/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 24*m**2*(a*sin(c + d*x) +
a)**m*sin(c + d*x)**4/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 44*m**2*(a*sin(c + d*x
) + a)**m*sin(c + d*x)**3*cos(c + d*x)**2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 24
*m**2*(a*sin(c + d*x) + a)**m*sin(c + d*x)**3/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d)
+ 84*m**2*(a*sin(c + d*x) + a)**m*sin(c + d*x)**2*cos(c + d*x)**2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2
+ 274*d*m + 120*d) + 8*m**2*(a*sin(c + d*x) + a)**m*sin(c + d*x)**2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m
**2 + 274*d*m + 120*d) + 71*m**2*(a*sin(c + d*x) + a)**m*sin(c + d*x)*cos(c + d*x)**4/(d*m**5 + 15*d*m**4 + 85
*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 36*m**2*(a*sin(c + d*x) + a)**m*sin(c + d*x)*cos(c + d*x)**2/(d*m**5
+ 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 71*m**2*(a*sin(c + d*x) + a)**m*cos(c + d*x)**4/(d*
m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) - 4*m**2*(a*sin(c + d*x) + a)**m*cos(c + d*x)**2/
(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 48*m*(a*sin(c + d*x) + a)**m*sin(c + d*x)**5
/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 120*m*(a*sin(c + d*x) + a)**m*sin(c + d*x)*
*4/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 152*m*(a*sin(c + d*x) + a)**m*sin(c + d*x
)**3*cos(c + d*x)**2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 72*m*(a*sin(c + d*x) +
a)**m*sin(c + d*x)**3/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 268*m*(a*sin(c + d*x)
+ a)**m*sin(c + d*x)**2*cos(c + d*x)**2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) - 24*m
*(a*sin(c + d*x) + a)**m*sin(c + d*x)**2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 154
*m*(a*sin(c + d*x) + a)**m*sin(c + d*x)*cos(c + d*x)**4/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m
+ 120*d) + 80*m*(a*sin(c + d*x) + a)**m*sin(c + d*x)*cos(c + d*x)**2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*
m**2 + 274*d*m + 120*d) - 24*m*(a*sin(c + d*x) + a)**m*sin(c + d*x)/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m*
*2 + 274*d*m + 120*d) + 154*m*(a*sin(c + d*x) + a)**m*cos(c + d*x)**4/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*
m**2 + 274*d*m + 120*d) - 36*m*(a*sin(c + d*x) + a)**m*cos(c + d*x)**2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d
*m**2 + 274*d*m + 120*d) + 64*(a*sin(c + d*x) + a)**m*sin(c + d*x)**5/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*
m**2 + 274*d*m + 120*d) + 120*(a*sin(c + d*x) + a)**m*sin(c + d*x)**4/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*
m**2 + 274*d*m + 120*d) + 160*(a*sin(c + d*x) + a)**m*sin(c + d*x)**3*cos(c + d*x)**2/(d*m**5 + 15*d*m**4 + 85
*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 240*(a*sin(c + d*x) + a)**m*sin(c + d*x)**2*cos(c + d*x)**2/(d*m**5
+ 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) - 80*(a*sin(c + d*x) + a)**m*sin(c + d*x)**2/(d*m**5 +
15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 120*(a*sin(c + d*x) + a)**m*sin(c + d*x)*cos(c + d*x)
**4/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 120*(a*sin(c + d*x) + a)**m*cos(c + d*x)
**4/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) - 80*(a*sin(c + d*x) + a)**m*cos(c + d*x)*
*2/(d*m**5 + 15*d*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d) + 24*(a*sin(c + d*x) + a)**m/(d*m**5 + 15*d
*m**4 + 85*d*m**3 + 225*d*m**2 + 274*d*m + 120*d), True))
________________________________________________________________________________________