3.913 \(\int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx\)
Optimal. Leaf size=175 \[ \frac{a^4 (c-d)^4 (c+d \sin (e+f x))^{n+1}}{d^5 f (n+1)}-\frac{4 a^4 (c-d)^3 (c+d \sin (e+f x))^{n+2}}{d^5 f (n+2)}+\frac{6 a^4 (c-d)^2 (c+d \sin (e+f x))^{n+3}}{d^5 f (n+3)}-\frac{4 a^4 (c-d) (c+d \sin (e+f x))^{n+4}}{d^5 f (n+4)}+\frac{a^4 (c+d \sin (e+f x))^{n+5}}{d^5 f (n+5)} \]
[Out]
(a^4*(c - d)^4*(c + d*Sin[e + f*x])^(1 + n))/(d^5*f*(1 + n)) - (4*a^4*(c - d)^3*(c + d*Sin[e + f*x])^(2 + n))/
(d^5*f*(2 + n)) + (6*a^4*(c - d)^2*(c + d*Sin[e + f*x])^(3 + n))/(d^5*f*(3 + n)) - (4*a^4*(c - d)*(c + d*Sin[e
+ f*x])^(4 + n))/(d^5*f*(4 + n)) + (a^4*(c + d*Sin[e + f*x])^(5 + n))/(d^5*f*(5 + n))
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Rubi [A] time = 0.205881, antiderivative size = 175, 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} \[ \frac{a^4 (c-d)^4 (c+d \sin (e+f x))^{n+1}}{d^5 f (n+1)}-\frac{4 a^4 (c-d)^3 (c+d \sin (e+f x))^{n+2}}{d^5 f (n+2)}+\frac{6 a^4 (c-d)^2 (c+d \sin (e+f x))^{n+3}}{d^5 f (n+3)}-\frac{4 a^4 (c-d) (c+d \sin (e+f x))^{n+4}}{d^5 f (n+4)}+\frac{a^4 (c+d \sin (e+f x))^{n+5}}{d^5 f (n+5)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^4*(c + d*Sin[e + f*x])^n,x]
[Out]
(a^4*(c - d)^4*(c + d*Sin[e + f*x])^(1 + n))/(d^5*f*(1 + n)) - (4*a^4*(c - d)^3*(c + d*Sin[e + f*x])^(2 + n))/
(d^5*f*(2 + n)) + (6*a^4*(c - d)^2*(c + d*Sin[e + f*x])^(3 + n))/(d^5*f*(3 + n)) - (4*a^4*(c - d)*(c + d*Sin[e
+ f*x])^(4 + n))/(d^5*f*(4 + n)) + (a^4*(c + d*Sin[e + f*x])^(5 + n))/(d^5*f*(5 + n))
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))^4 (c+d \sin (e+f x))^n \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^4 \left (c+\frac{d x}{a}\right )^n \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^4 (c-d)^4 \left (c+\frac{d x}{a}\right )^n}{d^4}-\frac{4 a^4 (c-d)^3 \left (c+\frac{d x}{a}\right )^{1+n}}{d^4}+\frac{6 a^4 (c-d)^2 \left (c+\frac{d x}{a}\right )^{2+n}}{d^4}-\frac{4 a^4 (c-d) \left (c+\frac{d x}{a}\right )^{3+n}}{d^4}+\frac{a^4 \left (c+\frac{d x}{a}\right )^{4+n}}{d^4}\right ) \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{a^4 (c-d)^4 (c+d \sin (e+f x))^{1+n}}{d^5 f (1+n)}-\frac{4 a^4 (c-d)^3 (c+d \sin (e+f x))^{2+n}}{d^5 f (2+n)}+\frac{6 a^4 (c-d)^2 (c+d \sin (e+f x))^{3+n}}{d^5 f (3+n)}-\frac{4 a^4 (c-d) (c+d \sin (e+f x))^{4+n}}{d^5 f (4+n)}+\frac{a^4 (c+d \sin (e+f x))^{5+n}}{d^5 f (5+n)}\\ \end{align*}
Mathematica [A] time = 0.595756, size = 130, normalized size = 0.74 \[ \frac{a^4 (c+d \sin (e+f x))^{n+1} \left (-\frac{4 (c-d)^3 (c+d \sin (e+f x))}{n+2}+\frac{6 (c-d)^2 (c+d \sin (e+f x))^2}{n+3}-\frac{4 (c-d) (c+d \sin (e+f x))^3}{n+4}+\frac{(c+d \sin (e+f x))^4}{n+5}+\frac{(c-d)^4}{n+1}\right )}{d^5 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^4*(c + d*Sin[e + f*x])^n,x]
[Out]
(a^4*(c + d*Sin[e + f*x])^(1 + n)*((c - d)^4/(1 + n) - (4*(c - d)^3*(c + d*Sin[e + f*x]))/(2 + n) + (6*(c - d)
^2*(c + d*Sin[e + f*x])^2)/(3 + n) - (4*(c - d)*(c + d*Sin[e + f*x])^3)/(4 + n) + (c + d*Sin[e + f*x])^4/(5 +
n)))/(d^5*f)
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Maple [F] time = 0.565, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( fx+e \right ) \left ( a+a\sin \left ( fx+e \right ) \right ) ^{4} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}\, 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))^4*(c+d*sin(f*x+e))^n,x)
[Out]
int(cos(f*x+e)*(a+a*sin(f*x+e))^4*(c+d*sin(f*x+e))^n,x)
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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))^4*(c+d*sin(f*x+e))^n,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.7658, size = 1897, normalized size = 10.84 \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))^4*(c+d*sin(f*x+e))^n,x, algorithm="fricas")
[Out]
(24*a^4*c^5 - 120*a^4*c^4*d + 240*a^4*c^3*d^2 - 240*a^4*c^2*d^3 + 120*a^4*c*d^4 + 360*a^4*d^5 + 8*(a^4*c*d^4 +
a^4*d^5)*n^4 + (120*a^4*d^5 + (a^4*c*d^4 + 4*a^4*d^5)*n^4 + 2*(3*a^4*c*d^4 + 22*a^4*d^5)*n^3 + (11*a^4*c*d^4
+ 164*a^4*d^5)*n^2 + 2*(3*a^4*c*d^4 + 122*a^4*d^5)*n)*cos(f*x + e)^4 - 16*(a^4*c^2*d^3 - 5*a^4*c*d^4 - 6*a^4*d
^5)*n^3 + 8*(3*a^4*c^3*d^2 - 15*a^4*c^2*d^3 + 32*a^4*c*d^4 + 50*a^4*d^5)*n^2 - 4*(120*a^4*d^5 + (2*a^4*c*d^4 +
3*a^4*d^5)*n^4 - (3*a^4*c^2*d^3 - 18*a^4*c*d^4 - 35*a^4*d^5)*n^3 + (3*a^4*c^3*d^2 - 18*a^4*c^2*d^3 + 49*a^4*c
*d^4 + 141*a^4*d^5)*n^2 + (3*a^4*c^3*d^2 - 15*a^4*c^2*d^3 + 33*a^4*c*d^4 + 229*a^4*d^5)*n)*cos(f*x + e)^2 - 8*
(3*a^4*c^4*d - 15*a^4*c^3*d^2 + 31*a^4*c^2*d^3 - 35*a^4*c*d^4 - 84*a^4*d^5)*n + (384*a^4*d^5 + 8*(a^4*c*d^4 +
a^4*d^5)*n^4 + (a^4*d^5*n^4 + 10*a^4*d^5*n^3 + 35*a^4*d^5*n^2 + 50*a^4*d^5*n + 24*a^4*d^5)*cos(f*x + e)^4 - 16
*(a^4*c^2*d^3 - 5*a^4*c*d^4 - 6*a^4*d^5)*n^3 + 8*(3*a^4*c^3*d^2 - 15*a^4*c^2*d^3 + 32*a^4*c*d^4 + 50*a^4*d^5)*
n^2 - 4*(72*a^4*d^5 + (a^4*c*d^4 + 2*a^4*d^5)*n^4 - (a^4*c^2*d^3 - 8*a^4*c*d^4 - 23*a^4*d^5)*n^3 - (3*a^4*c^2*
d^3 - 17*a^4*c*d^4 - 91*a^4*d^5)*n^2 - 2*(a^4*c^2*d^3 - 5*a^4*c*d^4 - 71*a^4*d^5)*n)*cos(f*x + e)^2 - 8*(3*a^4
*c^4*d - 15*a^4*c^3*d^2 + 31*a^4*c^2*d^3 - 35*a^4*c*d^4 - 84*a^4*d^5)*n)*sin(f*x + e))*(d*sin(f*x + e) + c)^n/
(d^5*f*n^5 + 15*d^5*f*n^4 + 85*d^5*f*n^3 + 225*d^5*f*n^2 + 274*d^5*f*n + 120*d^5*f)
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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))**4*(c+d*sin(f*x+e))**n,x)
[Out]
Timed out
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Giac [B] time = 1.21482, size = 2484, normalized size = 14.19 \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))^4*(c+d*sin(f*x+e))^n,x, algorithm="giac")
[Out]
(((d*sin(f*x + e) + c)^5*(d*sin(f*x + e) + c)^n*n^4 - 4*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)^n*c*n^4 +
6*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c^2*n^4 - 4*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c^3*
n^4 + (d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^4*n^4 + 10*(d*sin(f*x + e) + c)^5*(d*sin(f*x + e) + c)^n*n
^3 - 44*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)^n*c*n^3 + 72*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n
*c^2*n^3 - 52*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c^3*n^3 + 14*(d*sin(f*x + e) + c)*(d*sin(f*x + e)
+ c)^n*c^4*n^3 + 35*(d*sin(f*x + e) + c)^5*(d*sin(f*x + e) + c)^n*n^2 - 164*(d*sin(f*x + e) + c)^4*(d*sin(f*x
+ e) + c)^n*c*n^2 + 294*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c^2*n^2 - 236*(d*sin(f*x + e) + c)^2*(d*
sin(f*x + e) + c)^n*c^3*n^2 + 71*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^4*n^2 + 50*(d*sin(f*x + e) + c)
^5*(d*sin(f*x + e) + c)^n*n - 244*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)^n*c*n + 468*(d*sin(f*x + e) + c)
^3*(d*sin(f*x + e) + c)^n*c^2*n - 428*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c^3*n + 154*(d*sin(f*x + e
) + c)*(d*sin(f*x + e) + c)^n*c^4*n + 24*(d*sin(f*x + e) + c)^5*(d*sin(f*x + e) + c)^n - 120*(d*sin(f*x + e) +
c)^4*(d*sin(f*x + e) + c)^n*c + 240*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c^2 - 240*(d*sin(f*x + e) +
c)^2*(d*sin(f*x + e) + c)^n*c^3 + 120*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^4)*a^4/(d^4*n^5 + 15*d^4*
n^4 + 85*d^4*n^3 + 225*d^4*n^2 + 274*d^4*n + 120*d^4) + 4*((d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)^n*n^3 -
3*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c*n^3 + 3*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c^2*n
^3 - (d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^3*n^3 + 6*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)^n*n^2
- 21*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c*n^2 + 24*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c
^2*n^2 - 9*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^3*n^2 + 11*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c
)^n*n - 42*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c*n + 57*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^
n*c^2*n - 26*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^3*n + 6*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)
^n - 24*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c + 36*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c^2
- 24*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^3)*a^4/(d^3*n^4 + 10*d^3*n^3 + 35*d^3*n^2 + 50*d^3*n + 24*
d^3) + 6*((d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*n^2 - 2*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*
c*n^2 + (d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^2*n^2 + 3*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*
n - 8*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c*n + 5*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^2*n
+ 2*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n - 6*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c + 6*(d*s
in(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^2)*a^4/(d^2*n^3 + 6*d^2*n^2 + 11*d^2*n + 6*d^2) + (d*sin(f*x + e) +
c)^(n + 1)*a^4/(n + 1) + 4*((d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*n - (d*sin(f*x + e) + c)*(d*sin(f*x
+ e) + c)^n*c*n + (d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n - 2*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^
n*c)*a^4/((n^2 + 3*n + 2)*d))/(d*f)