\(\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx\) [928]
Optimal result
Integrand size = 27, antiderivative size = 134 \[
\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {(a+a \sin (c+d x))^{1+m}}{a d (1+m)}-\frac {4 (a+a \sin (c+d x))^{2+m}}{a^2 d (2+m)}+\frac {6 (a+a \sin (c+d x))^{3+m}}{a^3 d (3+m)}-\frac {4 (a+a \sin (c+d x))^{4+m}}{a^4 d (4+m)}+\frac {(a+a \sin (c+d x))^{5+m}}{a^5 d (5+m)}
\]
[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)
Rubi [A] (verified)
Time = 0.09 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number
of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 12, 45}
\[
\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}-\frac {4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac {6 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac {4 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}+\frac {(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}
\]
[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 45
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 2912
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/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4 (a+x)^m}{a^4} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int x^4 (a+x)^m \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {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*}
Mathematica [A] (verified)
Time = 0.97 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.12
\[
\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {(a (1+\sin (c+d x)))^{1+m} \left (\frac {7}{1+m}-\frac {40 (1+\sin (c+d x))}{2+m}+\frac {84 (1+\sin (c+d x))^2}{3+m}-\frac {64 (1+\sin (c+d x))^3}{4+m}+\frac {16 (1+\sin (c+d x))^4}{5+m}+\frac {3 \left (6+m+m^2-2 \left (2+3 m+m^2\right ) \cos (2 (c+d x))-8 (1+m) \sin (c+d x)\right )}{(1+m) (2+m) (3+m)}\right )}{16 a d}
\]
[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)
Maple [A] (verified)
Time = 2.22 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.54
| | |
| method | result | size |
| | |
| parallelrisch |
\(\frac {\left (a \left (1+\sin \left (d x +c \right )\right )\right )^{m} \left (\left (-5 m^{4}-34 m^{3}-127 m^{2}-218 m -120\right ) \sin \left (3 d x +3 c \right )+\left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right ) \sin \left (5 d x +5 c \right )+\left (-8 m^{4}-48 m^{3}-184 m^{2}-144 m \right ) \cos \left (2 d x +2 c \right )+\left (2 m^{4}+12 m^{3}+22 m^{2}+12 m \right ) \cos \left (4 d x +4 c \right )+\left (10 m^{4}+52 m^{3}+206 m^{2}+20 m +240\right ) \sin \left (d x +c \right )+6 m^{4}+36 m^{3}+162 m^{2}+132 m +384\right )}{16 d \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\) |
\(206\) |
| derivativedivides |
\(\frac {\left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{d \left (5+m \right )}+\frac {m \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{\left (m^{2}+9 m +20\right ) d}+\frac {24 \,{\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{d \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}-\frac {4 m \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{d \left (m^{3}+12 m^{2}+47 m +60\right )}-\frac {24 m \sin \left (d x +c \right ) {\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{d \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {12 m \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{\left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right ) d}\) |
\(256\) |
| default |
\(\frac {\left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{d \left (5+m \right )}+\frac {m \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{\left (m^{2}+9 m +20\right ) d}+\frac {24 \,{\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{d \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}-\frac {4 m \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{d \left (m^{3}+12 m^{2}+47 m +60\right )}-\frac {24 m \sin \left (d x +c \right ) {\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{d \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {12 m \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (d x +c \right )\right )}}{\left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right ) d}\) |
\(256\) |
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[In]
int(cos(d*x+c)*sin(d*x+c)^4*(a+a*sin(d*x+c))^m,x,method=_RETURNVERBOSE)
[Out]
1/16*(a*(1+sin(d*x+c)))^m*((-5*m^4-34*m^3-127*m^2-218*m-120)*sin(3*d*x+3*c)+(m^4+10*m^3+35*m^2+50*m+24)*sin(5*
d*x+5*c)+(-8*m^4-48*m^3-184*m^2-144*m)*cos(2*d*x+2*c)+(2*m^4+12*m^3+22*m^2+12*m)*cos(4*d*x+4*c)+(10*m^4+52*m^3
+206*m^2+20*m+240)*sin(d*x+c)+6*m^4+36*m^3+162*m^2+132*m+384)/d/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)
Fricas [A] (verification not implemented)
none
Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.47
\[
\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {{\left ({\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} \cos \left (d x + c\right )^{4} + m^{4} + 6 \, m^{3} - 2 \, {\left (m^{4} + 6 \, m^{3} + 17 \, m^{2} + 12 \, m\right )} \cos \left (d x + c\right )^{2} + 23 \, m^{2} + {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} \cos \left (d x + c\right )^{4} + m^{4} + 6 \, m^{3} - 2 \, {\left (m^{4} + 8 \, m^{3} + 29 \, m^{2} + 46 \, m + 24\right )} \cos \left (d x + c\right )^{2} + 23 \, m^{2} + 18 \, m + 24\right )} \sin \left (d x + c\right ) + 18 \, m + 24\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{5} + 15 \, d m^{4} + 85 \, d m^{3} + 225 \, d m^{2} + 274 \, d m + 120 \, d}
\]
[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)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2747 vs. \(2 (112) = 224\).
Time = 11.03 (sec) , antiderivative size = 2747, normalized size of antiderivative = 20.50
\[
\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\text {Too large to display}
\]
[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)**2/(2*a*d) - sin(c + d*x)/(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))
Maxima [A] (verification not implemented)
none
Time = 0.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.19
\[
\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (d x + c\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (d x + c\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 24 \, a^{m} m \sin \left (d x + c\right ) + 24 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} d}
\]
[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)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 791 vs. \(2 (134) = 268\).
Time = 0.43 (sec) , antiderivative size = 791, normalized size of antiderivative = 5.90
\[
\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\text {Too large to display}
\]
[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)^5*(a*sin(d*x + c) + a)^m*m^4 - 4*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a*m^4 + 6
*(a*sin(d*x + c) + a)^3*(a*sin(d*x + c) + a)^m*a^2*m^4 - 4*(a*sin(d*x + c) + a)^2*(a*sin(d*x + c) + a)^m*a^3*m
^4 + (a*sin(d*x + c) + a)*(a*sin(d*x + c) + a)^m*a^4*m^4 + 10*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*m^
3 - 44*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a*m^3 + 72*(a*sin(d*x + c) + a)^3*(a*sin(d*x + c) + a)^m*
a^2*m^3 - 52*(a*sin(d*x + c) + a)^2*(a*sin(d*x + c) + a)^m*a^3*m^3 + 14*(a*sin(d*x + c) + a)*(a*sin(d*x + c) +
a)^m*a^4*m^3 + 35*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*m^2 - 164*(a*sin(d*x + c) + a)^4*(a*sin(d*x +
c) + a)^m*a*m^2 + 294*(a*sin(d*x + c) + a)^3*(a*sin(d*x + c) + a)^m*a^2*m^2 - 236*(a*sin(d*x + c) + a)^2*(a*s
in(d*x + c) + a)^m*a^3*m^2 + 71*(a*sin(d*x + c) + a)*(a*sin(d*x + c) + a)^m*a^4*m^2 + 50*(a*sin(d*x + c) + a)^
5*(a*sin(d*x + c) + a)^m*m - 244*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a*m + 468*(a*sin(d*x + c) + a)^
3*(a*sin(d*x + c) + a)^m*a^2*m - 428*(a*sin(d*x + c) + a)^2*(a*sin(d*x + c) + a)^m*a^3*m + 154*(a*sin(d*x + c)
+ a)*(a*sin(d*x + c) + a)^m*a^4*m + 24*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m - 120*(a*sin(d*x + c) +
a)^4*(a*sin(d*x + c) + a)^m*a + 240*(a*sin(d*x + c) + a)^3*(a*sin(d*x + c) + a)^m*a^2 - 240*(a*sin(d*x + c) +
a)^2*(a*sin(d*x + c) + a)^m*a^3 + 120*(a*sin(d*x + c) + a)*(a*sin(d*x + c) + a)^m*a^4)/((a^4*m^5 + 15*a^4*m^4
+ 85*a^4*m^3 + 225*a^4*m^2 + 274*a^4*m + 120*a^4)*a*d)
Mupad [B] (verification not implemented)
Time = 13.29 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.60
\[
\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^m\,\left (132\,m+240\,\sin \left (c+d\,x\right )-120\,\sin \left (3\,c+3\,d\,x\right )+24\,\sin \left (5\,c+5\,d\,x\right )+20\,m\,\sin \left (c+d\,x\right )-144\,m\,\cos \left (2\,c+2\,d\,x\right )+12\,m\,\cos \left (4\,c+4\,d\,x\right )-218\,m\,\sin \left (3\,c+3\,d\,x\right )+50\,m\,\sin \left (5\,c+5\,d\,x\right )+206\,m^2\,\sin \left (c+d\,x\right )+52\,m^3\,\sin \left (c+d\,x\right )+10\,m^4\,\sin \left (c+d\,x\right )+162\,m^2+36\,m^3+6\,m^4-184\,m^2\,\cos \left (2\,c+2\,d\,x\right )-48\,m^3\,\cos \left (2\,c+2\,d\,x\right )-8\,m^4\,\cos \left (2\,c+2\,d\,x\right )+22\,m^2\,\cos \left (4\,c+4\,d\,x\right )+12\,m^3\,\cos \left (4\,c+4\,d\,x\right )+2\,m^4\,\cos \left (4\,c+4\,d\,x\right )-127\,m^2\,\sin \left (3\,c+3\,d\,x\right )-34\,m^3\,\sin \left (3\,c+3\,d\,x\right )-5\,m^4\,\sin \left (3\,c+3\,d\,x\right )+35\,m^2\,\sin \left (5\,c+5\,d\,x\right )+10\,m^3\,\sin \left (5\,c+5\,d\,x\right )+m^4\,\sin \left (5\,c+5\,d\,x\right )+384\right )}{16\,d\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}
\]
[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))