3.920 \(\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx\)
Optimal. Leaf size=170 \[ \frac{6 d^2 (c-d)^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{4 d^3 (c-d) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac{4 d (c-d)^3 (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{d^4 (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac{(c-d)^4 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)} \]
[Out]
((c - d)^4*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (4*(c - d)^3*d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(
2 + m)) + (6*(c - d)^2*d^2*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) + (4*(c - d)*d^3*(a + a*Sin[e + f*x])
^(4 + m))/(a^4*f*(4 + m)) + (d^4*(a + a*Sin[e + f*x])^(5 + m))/(a^5*f*(5 + m))
________________________________________________________________________________________
Rubi [A] time = 0.181892, antiderivative size = 170, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used =
{2833, 43} \[ \frac{6 d^2 (c-d)^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{4 d^3 (c-d) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac{4 d (c-d)^3 (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{d^4 (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac{(c-d)^4 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^4,x]
[Out]
((c - d)^4*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (4*(c - d)^3*d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(
2 + m)) + (6*(c - d)^2*d^2*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) + (4*(c - d)*d^3*(a + a*Sin[e + f*x])
^(4 + m))/(a^4*f*(4 + m)) + (d^4*(a + a*Sin[e + f*x])^(5 + m))/(a^5*f*(5 + m))
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]
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])
Rubi steps
\begin{align*} \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^m \left (c+\frac{d x}{a}\right )^4 \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{\operatorname{Subst}\left (\int \left ((c-d)^4 (a+x)^m+\frac{4 (c-d)^3 d (a+x)^{1+m}}{a}+\frac{6 (c-d)^2 d^2 (a+x)^{2+m}}{a^2}+\frac{4 (c-d) d^3 (a+x)^{3+m}}{a^3}+\frac{d^4 (a+x)^{4+m}}{a^4}\right ) \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{(c-d)^4 (a+a \sin (e+f x))^{1+m}}{a f (1+m)}+\frac{4 (c-d)^3 d (a+a \sin (e+f x))^{2+m}}{a^2 f (2+m)}+\frac{6 (c-d)^2 d^2 (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)}+\frac{4 (c-d) d^3 (a+a \sin (e+f x))^{4+m}}{a^4 f (4+m)}+\frac{d^4 (a+a \sin (e+f x))^{5+m}}{a^5 f (5+m)}\\ \end{align*}
Mathematica [A] time = 0.65945, size = 143, normalized size = 0.84 \[ \frac{(a (\sin (e+f x)+1))^{m+1} \left (\frac{4 a^4 d^3 (c-d) (\sin (e+f x)+1)^3}{m+4}+\frac{6 a^4 d^2 (c-d)^2 (\sin (e+f x)+1)^2}{m+3}+\frac{4 a^4 d (c-d)^3 (\sin (e+f x)+1)}{m+2}+\frac{a^4 (c-d)^4}{m+1}+\frac{d^4 (a \sin (e+f x)+a)^4}{m+5}\right )}{a^5 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^4,x]
[Out]
((a*(1 + Sin[e + f*x]))^(1 + m)*((a^4*(c - d)^4)/(1 + m) + (4*a^4*(c - d)^3*d*(1 + Sin[e + f*x]))/(2 + m) + (6
*a^4*(c - d)^2*d^2*(1 + Sin[e + f*x])^2)/(3 + m) + (4*a^4*(c - d)*d^3*(1 + Sin[e + f*x])^3)/(4 + m) + (d^4*(a
+ a*Sin[e + f*x])^4)/(5 + m)))/(a^5*f)
________________________________________________________________________________________
Maple [F] time = 7.776, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( fx+e \right ) \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^4,x)
[Out]
int(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^4,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(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 2.47408, size = 1706, normalized size = 10.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^4,x, algorithm="fricas")
[Out]
((c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*m^4 + ((4*c*d^3 + d^4)*m^4 + 120*c*d^3 + 2*(22*c*d^3 + 3*d^4)*m^3
+ (164*c*d^3 + 11*d^4)*m^2 + 2*(122*c*d^3 + 3*d^4)*m)*cos(f*x + e)^4 + 120*c^4 + 240*c^2*d^2 + 24*d^4 + 2*(7*
c^4 + 24*c^3*d + 30*c^2*d^2 + 16*c*d^3 + 3*d^4)*m^3 + (71*c^4 + 188*c^3*d + 186*c^2*d^2 + 92*c*d^3 + 23*d^4)*m
^2 - 2*((2*c^3*d + 3*c^2*d^2 + 4*c*d^3 + d^4)*m^4 + 120*c^3*d + 120*c*d^3 + 2*(13*c^3*d + 15*c^2*d^2 + 19*c*d^
3 + 3*d^4)*m^3 + (118*c^3*d + 87*c^2*d^2 + 128*c*d^3 + 17*d^4)*m^2 + 2*(107*c^3*d + 30*c^2*d^2 + 107*c*d^3 + 6
*d^4)*m)*cos(f*x + e)^2 + 2*(77*c^4 + 120*c^3*d + 114*c^2*d^2 + 80*c*d^3 + 9*d^4)*m + ((c^4 + 4*c^3*d + 6*c^2*
d^2 + 4*c*d^3 + d^4)*m^4 + (d^4*m^4 + 10*d^4*m^3 + 35*d^4*m^2 + 50*d^4*m + 24*d^4)*cos(f*x + e)^4 + 120*c^4 +
240*c^2*d^2 + 24*d^4 + 2*(7*c^4 + 24*c^3*d + 30*c^2*d^2 + 16*c*d^3 + 3*d^4)*m^3 + (71*c^4 + 188*c^3*d + 186*c^
2*d^2 + 92*c*d^3 + 23*d^4)*m^2 - 2*((3*c^2*d^2 + 2*c*d^3 + d^4)*m^4 + 120*c^2*d^2 + 24*d^4 + 4*(9*c^2*d^2 + 4*
c*d^3 + 2*d^4)*m^3 + (147*c^2*d^2 + 34*c*d^3 + 29*d^4)*m^2 + 2*(117*c^2*d^2 + 10*c*d^3 + 23*d^4)*m)*cos(f*x +
e)^2 + 2*(77*c^4 + 120*c^3*d + 114*c^2*d^2 + 80*c*d^3 + 9*d^4)*m)*sin(f*x + e))*(a*sin(f*x + e) + a)^m/(f*m^5
+ 15*f*m^4 + 85*f*m^3 + 225*f*m^2 + 274*f*m + 120*f)
________________________________________________________________________________________
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(f*x+e)*(a+a*sin(f*x+e))**m*(c+d*sin(f*x+e))**4,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.32179, size = 2491, normalized size = 14.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^4,x, algorithm="giac")
[Out]
(6*((a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*m^2 - 2*(a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a*m^2
+ (a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^2*m^2 + 3*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*m - 8*
(a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a*m + 5*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^2*m + 2*(a
*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m - 6*(a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a + 6*(a*sin(f*x
+ e) + a)*(a*sin(f*x + e) + a)^m*a^2)*c^2*d^2/(a^2*m^3 + 6*a^2*m^2 + 11*a^2*m + 6*a^2) + 4*((a*sin(f*x + e) +
a)^4*(a*sin(f*x + e) + a)^m*m^3 - 3*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a*m^3 + 3*(a*sin(f*x + e) +
a)^2*(a*sin(f*x + e) + a)^m*a^2*m^3 - (a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^3*m^3 + 6*(a*sin(f*x + e)
+ a)^4*(a*sin(f*x + e) + a)^m*m^2 - 21*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a*m^2 + 24*(a*sin(f*x +
e) + a)^2*(a*sin(f*x + e) + a)^m*a^2*m^2 - 9*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^3*m^2 + 11*(a*sin(f
*x + e) + a)^4*(a*sin(f*x + e) + a)^m*m - 42*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a*m + 57*(a*sin(f*x
+ e) + a)^2*(a*sin(f*x + e) + a)^m*a^2*m - 26*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^3*m + 6*(a*sin(f*
x + e) + a)^4*(a*sin(f*x + e) + a)^m - 24*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a + 36*(a*sin(f*x + e)
+ a)^2*(a*sin(f*x + e) + a)^m*a^2 - 24*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^3)*c*d^3/(a^3*m^4 + 10*a
^3*m^3 + 35*a^3*m^2 + 50*a^3*m + 24*a^3) + ((a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*m^4 - 4*(a*sin(f*x +
e) + a)^4*(a*sin(f*x + e) + a)^m*a*m^4 + 6*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^2*m^4 - 4*(a*sin(f
*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a^3*m^4 + (a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^4*m^4 + 10*(a*si
n(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*m^3 - 44*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a*m^3 + 72*(a*
sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^2*m^3 - 52*(a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a^3*m^3
+ 14*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^4*m^3 + 35*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*m^
2 - 164*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a*m^2 + 294*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^
m*a^2*m^2 - 236*(a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a^3*m^2 + 71*(a*sin(f*x + e) + a)*(a*sin(f*x + e
) + a)^m*a^4*m^2 + 50*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*m - 244*(a*sin(f*x + e) + a)^4*(a*sin(f*x
+ e) + a)^m*a*m + 468*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^2*m - 428*(a*sin(f*x + e) + a)^2*(a*sin(
f*x + e) + a)^m*a^3*m + 154*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^4*m + 24*(a*sin(f*x + e) + a)^5*(a*s
in(f*x + e) + a)^m - 120*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a + 240*(a*sin(f*x + e) + a)^3*(a*sin(f
*x + e) + a)^m*a^2 - 240*(a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a^3 + 120*(a*sin(f*x + e) + a)*(a*sin(f
*x + e) + a)^m*a^4)*d^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*sin(f*x +
e) + a)^(m + 1)*c^4/(m + 1) + 4*((a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*m - (a*sin(f*x + e) + a)*(a*si
n(f*x + e) + a)^m*a*m + (a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m - 2*(a*sin(f*x + e) + a)*(a*sin(f*x + e)
+ a)^m*a)*c^3*d/((m^2 + 3*m + 2)*a))/(a*f)