3.928 \(\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx\)

Optimal. Leaf size=134 \[ \frac {(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}-\frac {4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac {6 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac {4 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}+\frac {(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]

[Out]

(a+a*sin(d*x+c))^(1+m)/a/d/(1+m)-4*(a+a*sin(d*x+c))^(2+m)/a^2/d/(2+m)+6*(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)

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Rubi [A]  time = 0.12, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ -\frac {4 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}+\frac {6 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac {4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac {(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac {(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^m,x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

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

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Mathematica [A]  time = 1.40, size = 150, normalized size = 1.12 \[ \frac {(a (\sin (c+d x)+1))^{m+1} \left (\frac {3 \left (-2 \left (m^2+3 m+2\right ) \cos (2 (c+d x))-8 (m+1) \sin (c+d x)+m^2+m+6\right )}{(m+1) (m+2) (m+3)}+\frac {16 (\sin (c+d x)+1)^4}{m+5}-\frac {64 (\sin (c+d x)+1)^3}{m+4}+\frac {84 (\sin (c+d x)+1)^2}{m+3}-\frac {40 (\sin (c+d x)+1)}{m+2}+\frac {7}{m+1}\right )}{16 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^m,x]

[Out]

((a*(1 + Sin[c + d*x]))^(1 + m)*(7/(1 + m) - (40*(1 + Sin[c + d*x]))/(2 + m) + (84*(1 + Sin[c + d*x])^2)/(3 +
m) - (64*(1 + Sin[c + d*x])^3)/(4 + m) + (16*(1 + Sin[c + d*x])^4)/(5 + m) + (3*(6 + m + m^2 - 2*(2 + 3*m + m^
2)*Cos[2*(c + d*x)] - 8*(1 + m)*Sin[c + d*x]))/((1 + m)*(2 + m)*(3 + m))))/(16*a*d)

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fricas [A]  time = 0.48, size = 197, normalized size = 1.47 \[ \frac {{\left ({\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} \cos \left (d x + c\right )^{4} + m^{4} + 6 \, m^{3} - 2 \, {\left (m^{4} + 6 \, m^{3} + 17 \, m^{2} + 12 \, m\right )} \cos \left (d x + c\right )^{2} + 23 \, m^{2} + {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} \cos \left (d x + c\right )^{4} + m^{4} + 6 \, m^{3} - 2 \, {\left (m^{4} + 8 \, m^{3} + 29 \, m^{2} + 46 \, m + 24\right )} \cos \left (d x + c\right )^{2} + 23 \, m^{2} + 18 \, m + 24\right )} \sin \left (d x + c\right ) + 18 \, m + 24\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{5} + 15 \, d m^{4} + 85 \, d m^{3} + 225 \, d m^{2} + 274 \, d m + 120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^4*(a+a*sin(d*x+c))^m,x, algorithm="fricas")

[Out]

((m^4 + 6*m^3 + 11*m^2 + 6*m)*cos(d*x + c)^4 + m^4 + 6*m^3 - 2*(m^4 + 6*m^3 + 17*m^2 + 12*m)*cos(d*x + c)^2 +
23*m^2 + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*cos(d*x + c)^4 + m^4 + 6*m^3 - 2*(m^4 + 8*m^3 + 29*m^2 + 46*m +
24)*cos(d*x + c)^2 + 23*m^2 + 18*m + 24)*sin(d*x + c) + 18*m + 24)*(a*sin(d*x + c) + a)^m/(d*m^5 + 15*d*m^4 +
85*d*m^3 + 225*d*m^2 + 274*d*m + 120*d)

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giac [B]  time = 0.27, size = 402, normalized size = 3.00 \[ \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{4} \sin \left (d x + c\right )^{5} + {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{4} \sin \left (d x + c\right )^{4} + 10 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{3} \sin \left (d x + c\right )^{5} + 6 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{3} \sin \left (d x + c\right )^{4} + 35 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{2} \sin \left (d x + c\right )^{5} - 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{3} \sin \left (d x + c\right )^{3} + 11 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{2} \sin \left (d x + c\right )^{4} + 50 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m \sin \left (d x + c\right )^{5} - 12 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{2} \sin \left (d x + c\right )^{3} + 6 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m \sin \left (d x + c\right )^{4} + 24 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{5} + 12 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{2} \sin \left (d x + c\right )^{2} - 8 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m \sin \left (d x + c\right )^{3} + 12 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m \sin \left (d x + c\right )^{2} - 24 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m \sin \left (d x + c\right ) + 24 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^4*(a+a*sin(d*x+c))^m,x, algorithm="giac")

[Out]

((a*sin(d*x + c) + a)^m*m^4*sin(d*x + c)^5 + (a*sin(d*x + c) + a)^m*m^4*sin(d*x + c)^4 + 10*(a*sin(d*x + c) +
a)^m*m^3*sin(d*x + c)^5 + 6*(a*sin(d*x + c) + a)^m*m^3*sin(d*x + c)^4 + 35*(a*sin(d*x + c) + a)^m*m^2*sin(d*x
+ c)^5 - 4*(a*sin(d*x + c) + a)^m*m^3*sin(d*x + c)^3 + 11*(a*sin(d*x + c) + a)^m*m^2*sin(d*x + c)^4 + 50*(a*si
n(d*x + c) + a)^m*m*sin(d*x + c)^5 - 12*(a*sin(d*x + c) + a)^m*m^2*sin(d*x + c)^3 + 6*(a*sin(d*x + c) + a)^m*m
*sin(d*x + c)^4 + 24*(a*sin(d*x + c) + a)^m*sin(d*x + c)^5 + 12*(a*sin(d*x + c) + a)^m*m^2*sin(d*x + c)^2 - 8*
(a*sin(d*x + c) + a)^m*m*sin(d*x + c)^3 + 12*(a*sin(d*x + c) + a)^m*m*sin(d*x + c)^2 - 24*(a*sin(d*x + c) + a)
^m*m*sin(d*x + c) + 24*(a*sin(d*x + c) + a)^m)/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*d)

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maple [F]  time = 6.76, size = 0, normalized size = 0.00 \[ \int \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*sin(d*x+c)^4*(a+a*sin(d*x+c))^m,x)

[Out]

int(cos(d*x+c)*sin(d*x+c)^4*(a+a*sin(d*x+c))^m,x)

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maxima [A]  time = 0.91, size = 159, normalized size = 1.19 \[ \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (d x + c\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (d x + c\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 24 \, a^{m} m \sin \left (d x + c\right ) + 24 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^4*(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)*(si
n(d*x + c) + 1)^m/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*d)

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mupad [B]  time = 12.19, size = 349, normalized size = 2.60 \[ \frac {{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^m\,\left (132\,m+240\,\sin \left (c+d\,x\right )-120\,\sin \left (3\,c+3\,d\,x\right )+24\,\sin \left (5\,c+5\,d\,x\right )+20\,m\,\sin \left (c+d\,x\right )-144\,m\,\cos \left (2\,c+2\,d\,x\right )+12\,m\,\cos \left (4\,c+4\,d\,x\right )-218\,m\,\sin \left (3\,c+3\,d\,x\right )+50\,m\,\sin \left (5\,c+5\,d\,x\right )+206\,m^2\,\sin \left (c+d\,x\right )+52\,m^3\,\sin \left (c+d\,x\right )+10\,m^4\,\sin \left (c+d\,x\right )+162\,m^2+36\,m^3+6\,m^4-184\,m^2\,\cos \left (2\,c+2\,d\,x\right )-48\,m^3\,\cos \left (2\,c+2\,d\,x\right )-8\,m^4\,\cos \left (2\,c+2\,d\,x\right )+22\,m^2\,\cos \left (4\,c+4\,d\,x\right )+12\,m^3\,\cos \left (4\,c+4\,d\,x\right )+2\,m^4\,\cos \left (4\,c+4\,d\,x\right )-127\,m^2\,\sin \left (3\,c+3\,d\,x\right )-34\,m^3\,\sin \left (3\,c+3\,d\,x\right )-5\,m^4\,\sin \left (3\,c+3\,d\,x\right )+35\,m^2\,\sin \left (5\,c+5\,d\,x\right )+10\,m^3\,\sin \left (5\,c+5\,d\,x\right )+m^4\,\sin \left (5\,c+5\,d\,x\right )+384\right )}{16\,d\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)*sin(c + d*x)^4*(a + a*sin(c + d*x))^m,x)

[Out]

((a*(sin(c + d*x) + 1))^m*(132*m + 240*sin(c + d*x) - 120*sin(3*c + 3*d*x) + 24*sin(5*c + 5*d*x) + 20*m*sin(c
+ d*x) - 144*m*cos(2*c + 2*d*x) + 12*m*cos(4*c + 4*d*x) - 218*m*sin(3*c + 3*d*x) + 50*m*sin(5*c + 5*d*x) + 206
*m^2*sin(c + d*x) + 52*m^3*sin(c + d*x) + 10*m^4*sin(c + d*x) + 162*m^2 + 36*m^3 + 6*m^4 - 184*m^2*cos(2*c + 2
*d*x) - 48*m^3*cos(2*c + 2*d*x) - 8*m^4*cos(2*c + 2*d*x) + 22*m^2*cos(4*c + 4*d*x) + 12*m^3*cos(4*c + 4*d*x) +
2*m^4*cos(4*c + 4*d*x) - 127*m^2*sin(3*c + 3*d*x) - 34*m^3*sin(3*c + 3*d*x) - 5*m^4*sin(3*c + 3*d*x) + 35*m^2
*sin(5*c + 5*d*x) + 10*m^3*sin(5*c + 5*d*x) + m^4*sin(5*c + 5*d*x) + 384))/(16*d*(274*m + 225*m^2 + 85*m^3 + 1
5*m^4 + m^5 + 120))

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sympy [A]  time = 67.34, size = 2747, normalized size = 20.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)**4*(a+a*sin(d*x+c))**m,x)

[Out]

Piecewise((x*(a*sin(c) + a)**m*sin(c)**4*cos(c), 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*log(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*si
n(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) + 48*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) + 108*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) + 88*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) + 25/(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*lo
g(sin(c + d*x) + 1)*sin(c + d*x)**3/(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)**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) + 3*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) - 36*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) - 54*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) - 22/(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)), (12*log(sin(c + d*x) + 1)*sin(c + d*x)**2/(2*a**3*d*sin
(c + d*x)**2 + 4*a**3*d*sin(c + d*x) + 2*a**3*d) + 24*log(sin(c + d*x) + 1)*sin(c + d*x)/(2*a**3*d*sin(c + d*x
)**2 + 4*a**3*d*sin(c + d*x) + 2*a**3*d) + 12*log(sin(c + d*x) + 1)/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*sin(c
+ d*x) + 2*a**3*d) + sin(c + d*x)**4/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*sin(c + d*x) + 2*a**3*d) - 4*sin(c
+ d*x)**3/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*sin(c + d*x) + 2*a**3*d) + 24*sin(c + d*x)/(2*a**3*d*sin(c + d*
x)**2 + 4*a**3*d*sin(c + d*x) + 2*a**3*d) + 18/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*sin(c + d*x) + 2*a**3*d),
Eq(m, -3)), (-12*log(sin(c + d*x) + 1)*sin(c + d*x)/(3*a**2*d*sin(c + d*x) + 3*a**2*d) - 12*log(sin(c + d*x) +
1)/(3*a**2*d*sin(c + d*x) + 3*a**2*d) + sin(c + d*x)**4/(3*a**2*d*sin(c + d*x) + 3*a**2*d) - 2*sin(c + d*x)**
3/(3*a**2*d*sin(c + d*x) + 3*a**2*d) + 6*sin(c + d*x)**2/(3*a**2*d*sin(c + d*x) + 3*a**2*d) - 12/(3*a**2*d*sin
(c + d*x) + 3*a**2*d), Eq(m, -2)), (log(sin(c + d*x) + 1)/(a*d) + sin(c + d*x)**4/(4*a*d) - sin(c + d*x)**3/(3
*a*d) - sin(c + d*x)/(a*d) - cos(c + d*x)**2/(2*a*d), Eq(m, -1)), (m**4*(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) + m**4*(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) + 10*m**3*(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) + 6*m**3*(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) - 4*m**3*(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) + 35*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) + 11*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) - 12*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) + 12*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) + 50*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) + 6*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) - 8*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) + 12*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) - 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) + 24*(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*(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))

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