3.258 \(\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx\)
Optimal. Leaf size=114 \[ \frac{a^4 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{4 a^4 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{6 a^4 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{4 a^4 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{a^4 \sin ^{n+5}(c+d x)}{d (n+5)} \]
[Out]
(a^4*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (4*a^4*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (6*a^4*Sin[c + d*x]^(3 + n
))/(d*(3 + n)) + (4*a^4*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (a^4*Sin[c + d*x]^(5 + n))/(d*(5 + n))
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Rubi [A] time = 0.115689, antiderivative size = 114, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used =
{2833, 43} \[ \frac{a^4 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{4 a^4 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{6 a^4 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{4 a^4 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{a^4 \sin ^{n+5}(c+d x)}{d (n+5)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^4,x]
[Out]
(a^4*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (4*a^4*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (6*a^4*Sin[c + d*x]^(3 + n
))/(d*(3 + n)) + (4*a^4*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (a^4*Sin[c + d*x]^(5 + n))/(d*(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 (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{x}{a}\right )^n (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^4 \left (\frac{x}{a}\right )^n+4 a^4 \left (\frac{x}{a}\right )^{1+n}+6 a^4 \left (\frac{x}{a}\right )^{2+n}+4 a^4 \left (\frac{x}{a}\right )^{3+n}+a^4 \left (\frac{x}{a}\right )^{4+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^4 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac{4 a^4 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac{6 a^4 \sin ^{3+n}(c+d x)}{d (3+n)}+\frac{4 a^4 \sin ^{4+n}(c+d x)}{d (4+n)}+\frac{a^4 \sin ^{5+n}(c+d x)}{d (5+n)}\\ \end{align*}
Mathematica [A] time = 0.257733, size = 80, normalized size = 0.7 \[ \frac{a^4 \sin ^{n+1}(c+d x) \left (\frac{\sin ^4(c+d x)}{n+5}+\frac{4 \sin ^3(c+d x)}{n+4}+\frac{6 \sin ^2(c+d x)}{n+3}+\frac{4 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right )}{d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^4,x]
[Out]
(a^4*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (4*Sin[c + d*x])/(2 + n) + (6*Sin[c + d*x]^2)/(3 + n) + (4*Sin[c + d
*x]^3)/(4 + n) + Sin[c + d*x]^4/(5 + n)))/d
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Maple [F] time = 3.113, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{n} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^4,x)
[Out]
int(cos(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^4,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(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.62567, size = 701, normalized size = 6.15 \begin{align*} \frac{{\left (8 \, a^{4} n^{4} + 96 \, a^{4} n^{3} + 400 \, a^{4} n^{2} + 672 \, a^{4} n + 4 \,{\left (a^{4} n^{4} + 11 \, a^{4} n^{3} + 41 \, a^{4} n^{2} + 61 \, a^{4} n + 30 \, a^{4}\right )} \cos \left (d x + c\right )^{4} + 360 \, a^{4} - 4 \,{\left (3 \, a^{4} n^{4} + 35 \, a^{4} n^{3} + 141 \, a^{4} n^{2} + 229 \, a^{4} n + 120 \, a^{4}\right )} \cos \left (d x + c\right )^{2} +{\left (8 \, a^{4} n^{4} + 96 \, a^{4} n^{3} + 400 \, a^{4} n^{2} + 672 \, a^{4} n +{\left (a^{4} n^{4} + 10 \, a^{4} n^{3} + 35 \, a^{4} n^{2} + 50 \, a^{4} n + 24 \, a^{4}\right )} \cos \left (d x + c\right )^{4} + 384 \, a^{4} - 4 \,{\left (2 \, a^{4} n^{4} + 23 \, a^{4} n^{3} + 91 \, a^{4} n^{2} + 142 \, a^{4} n + 72 \, a^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{5} + 15 \, d n^{4} + 85 \, d n^{3} + 225 \, d n^{2} + 274 \, d n + 120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^4,x, algorithm="fricas")
[Out]
(8*a^4*n^4 + 96*a^4*n^3 + 400*a^4*n^2 + 672*a^4*n + 4*(a^4*n^4 + 11*a^4*n^3 + 41*a^4*n^2 + 61*a^4*n + 30*a^4)*
cos(d*x + c)^4 + 360*a^4 - 4*(3*a^4*n^4 + 35*a^4*n^3 + 141*a^4*n^2 + 229*a^4*n + 120*a^4)*cos(d*x + c)^2 + (8*
a^4*n^4 + 96*a^4*n^3 + 400*a^4*n^2 + 672*a^4*n + (a^4*n^4 + 10*a^4*n^3 + 35*a^4*n^2 + 50*a^4*n + 24*a^4)*cos(d
*x + c)^4 + 384*a^4 - 4*(2*a^4*n^4 + 23*a^4*n^3 + 91*a^4*n^2 + 142*a^4*n + 72*a^4)*cos(d*x + c)^2)*sin(d*x + c
))*sin(d*x + c)^n/(d*n^5 + 15*d*n^4 + 85*d*n^3 + 225*d*n^2 + 274*d*n + 120*d)
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Sympy [A] time = 134.487, size = 1856, normalized size = 16.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*sin(d*x+c)**n*(a+a*sin(d*x+c))**4,x)
[Out]
Piecewise((x*(a*sin(c) + a)**4*sin(c)**n*cos(c), Eq(d, 0)), (a**4*log(sin(c + d*x))/d - 4*a**4/(d*sin(c + d*x)
) - 3*a**4/(d*sin(c + d*x)**2) - 4*a**4/(3*d*sin(c + d*x)**3) - a**4/(4*d*sin(c + d*x)**4), Eq(n, -5)), (4*a**
4*log(sin(c + d*x))/d + a**4*sin(c + d*x)/d - 6*a**4/(d*sin(c + d*x)) - 2*a**4/(d*sin(c + d*x)**2) - a**4/(3*d
*sin(c + d*x)**3), Eq(n, -4)), (6*a**4*log(sin(c + d*x))/d + 4*a**4*sin(c + d*x)/d - a**4*cos(c + d*x)**2/(2*d
) - 4*a**4/(d*sin(c + d*x)) - a**4/(2*d*sin(c + d*x)**2), Eq(n, -3)), (4*a**4*log(sin(c + d*x))/d + a**4*sin(c
+ d*x)**3/(3*d) + 6*a**4*sin(c + d*x)/d - 2*a**4*cos(c + d*x)**2/d - a**4/(d*sin(c + d*x)), Eq(n, -2)), (a**4
*log(sin(c + d*x))/d + 4*a**4*sin(c + d*x)**3/(3*d) - a**4*sin(c + d*x)**2*cos(c + d*x)**2/(2*d) + 4*a**4*sin(
c + d*x)/d - a**4*cos(c + d*x)**4/(4*d) - 3*a**4*cos(c + d*x)**2/d, Eq(n, -1)), (a**4*n**4*sin(c + d*x)**5*sin
(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 4*a**4*n**4*sin(c + d*x)**4*sin
(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 6*a**4*n**4*sin(c + d*x)**3*sin
(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 4*a**4*n**4*sin(c + d*x)**2*sin
(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + a**4*n**4*sin(c + d*x)*sin(c +
d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 10*a**4*n**3*sin(c + d*x)**5*sin(c +
d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 44*a**4*n**3*sin(c + d*x)**4*sin(c
+ d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 72*a**4*n**3*sin(c + d*x)**3*sin(c
+ d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 52*a**4*n**3*sin(c + d*x)**2*sin(
c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 14*a**4*n**3*sin(c + d*x)*sin(c
+ d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 35*a**4*n**2*sin(c + d*x)**5*sin(c
+ d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 164*a**4*n**2*sin(c + d*x)**4*sin
(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 294*a**4*n**2*sin(c + d*x)**3*s
in(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 236*a**4*n**2*sin(c + d*x)**2
*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 71*a**4*n**2*sin(c + d*x)*s
in(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 50*a**4*n*sin(c + d*x)**5*sin
(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 244*a**4*n*sin(c + d*x)**4*sin(
c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 468*a**4*n*sin(c + d*x)**3*sin(c
+ d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 428*a**4*n*sin(c + d*x)**2*sin(c
+ d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 154*a**4*n*sin(c + d*x)*sin(c + d*
x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 24*a**4*sin(c + d*x)**5*sin(c + d*x)**
n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 120*a**4*sin(c + d*x)**4*sin(c + d*x)**n/(
d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 240*a**4*sin(c + d*x)**3*sin(c + d*x)**n/(d*n
**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 240*a**4*sin(c + d*x)**2*sin(c + d*x)**n/(d*n**5
+ 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d) + 120*a**4*sin(c + d*x)*sin(c + d*x)**n/(d*n**5 + 15*
d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 120*d), True))
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Giac [A] time = 1.2136, size = 171, normalized size = 1.5 \begin{align*} \frac{\frac{a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5}}{n + 5} + \frac{4 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4}}{n + 4} + \frac{6 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} + \frac{4 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}}{n + 2} + \frac{a^{4} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^4,x, algorithm="giac")
[Out]
(a^4*sin(d*x + c)^n*sin(d*x + c)^5/(n + 5) + 4*a^4*sin(d*x + c)^n*sin(d*x + c)^4/(n + 4) + 6*a^4*sin(d*x + c)^
n*sin(d*x + c)^3/(n + 3) + 4*a^4*sin(d*x + c)^n*sin(d*x + c)^2/(n + 2) + a^4*sin(d*x + c)^(n + 1)/(n + 1))/d