3.259 \(\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx\)
Optimal. Leaf size=91 \[ \frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{3 a^3 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{a^3 \sin ^{n+4}(c+d x)}{d (n+4)} \]
[Out]
(a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (3*a^3*Sin[c + d*x]^(3 + n
))/(d*(3 + n)) + (a^3*Sin[c + d*x]^(4 + n))/(d*(4 + n))
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Rubi [A] time = 0.0953491, antiderivative size = 91, 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} \[ \frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{3 a^3 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{a^3 \sin ^{n+4}(c+d x)}{d (n+4)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]
[Out]
(a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (3*a^3*Sin[c + d*x]^(3 + n
))/(d*(3 + n)) + (a^3*Sin[c + d*x]^(4 + n))/(d*(4 + 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))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{x}{a}\right )^n (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3 \left (\frac{x}{a}\right )^n+3 a^3 \left (\frac{x}{a}\right )^{1+n}+3 a^3 \left (\frac{x}{a}\right )^{2+n}+a^3 \left (\frac{x}{a}\right )^{3+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^3 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac{3 a^3 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac{3 a^3 \sin ^{3+n}(c+d x)}{d (3+n)}+\frac{a^3 \sin ^{4+n}(c+d x)}{d (4+n)}\\ \end{align*}
Mathematica [A] time = 0.148885, size = 65, normalized size = 0.71 \[ \frac{a^3 \sin ^{n+1}(c+d x) \left (\frac{\sin ^3(c+d x)}{n+4}+\frac{3 \sin ^2(c+d x)}{n+3}+\frac{3 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right )}{d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]
[Out]
(a^3*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (3*Sin[c + d*x])/(2 + n) + (3*Sin[c + d*x]^2)/(3 + n) + Sin[c + d*x]
^3/(4 + n)))/d
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Maple [F] time = 2.313, 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 ) ^{3}\, 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))^3,x)
[Out]
int(cos(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,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))^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.85824, size = 468, normalized size = 5.14 \begin{align*} \frac{{\left (4 \, a^{3} n^{3} + 30 \, a^{3} n^{2} +{\left (a^{3} n^{3} + 6 \, a^{3} n^{2} + 11 \, a^{3} n + 6 \, a^{3}\right )} \cos \left (d x + c\right )^{4} + 68 \, a^{3} n + 42 \, a^{3} -{\left (5 \, a^{3} n^{3} + 36 \, a^{3} n^{2} + 79 \, a^{3} n + 48 \, a^{3}\right )} \cos \left (d x + c\right )^{2} +{\left (4 \, a^{3} n^{3} + 30 \, a^{3} n^{2} + 68 \, a^{3} n + 48 \, a^{3} - 3 \,{\left (a^{3} n^{3} + 7 \, a^{3} n^{2} + 14 \, a^{3} n + 8 \, a^{3}\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^{4} + 10 \, d n^{3} + 35 \, d n^{2} + 50 \, d n + 24 \, 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))^3,x, algorithm="fricas")
[Out]
(4*a^3*n^3 + 30*a^3*n^2 + (a^3*n^3 + 6*a^3*n^2 + 11*a^3*n + 6*a^3)*cos(d*x + c)^4 + 68*a^3*n + 42*a^3 - (5*a^3
*n^3 + 36*a^3*n^2 + 79*a^3*n + 48*a^3)*cos(d*x + c)^2 + (4*a^3*n^3 + 30*a^3*n^2 + 68*a^3*n + 48*a^3 - 3*(a^3*n
^3 + 7*a^3*n^2 + 14*a^3*n + 8*a^3)*cos(d*x + c)^2)*sin(d*x + c))*sin(d*x + c)^n/(d*n^4 + 10*d*n^3 + 35*d*n^2 +
50*d*n + 24*d)
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Sympy [A] time = 77.423, size = 1061, normalized size = 11.66 \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))**3,x)
[Out]
Piecewise((x*(a*sin(c) + a)**3*sin(c)**n*cos(c), Eq(d, 0)), (a**3*log(sin(c + d*x))/d - 3*a**3/(d*sin(c + d*x)
) - 3*a**3/(2*d*sin(c + d*x)**2) - a**3/(3*d*sin(c + d*x)**3), Eq(n, -4)), (3*a**3*log(sin(c + d*x))/d + a**3*
sin(c + d*x)/d - 3*a**3/(d*sin(c + d*x)) - a**3/(2*d*sin(c + d*x)**2), Eq(n, -3)), (3*a**3*log(sin(c + d*x))/d
+ 3*a**3*sin(c + d*x)/d - a**3*cos(c + d*x)**2/(2*d) - a**3/(d*sin(c + d*x)), Eq(n, -2)), (a**3*log(sin(c + d
*x))/d + a**3*sin(c + d*x)**3/(3*d) + 3*a**3*sin(c + d*x)/d - 3*a**3*cos(c + d*x)**2/(2*d), Eq(n, -1)), (a**3*
n**3*sin(c + d*x)**4*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + 3*a**3*n**3*sin(c + d*
x)**3*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + 3*a**3*n**3*sin(c + d*x)**2*sin(c + d
*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + a**3*n**3*sin(c + d*x)*sin(c + d*x)**n/(d*n**4 + 10*
d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + 6*a**3*n**2*sin(c + d*x)**4*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n
**2 + 50*d*n + 24*d) + 21*a**3*n**2*sin(c + d*x)**3*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n +
24*d) + 24*a**3*n**2*sin(c + d*x)**2*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + 9*a**
3*n**2*sin(c + d*x)*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + 11*a**3*n*sin(c + d*x)*
*4*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + 42*a**3*n*sin(c + d*x)**3*sin(c + d*x)**
n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + 57*a**3*n*sin(c + d*x)**2*sin(c + d*x)**n/(d*n**4 + 10*d*
n**3 + 35*d*n**2 + 50*d*n + 24*d) + 26*a**3*n*sin(c + d*x)*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 5
0*d*n + 24*d) + 6*a**3*sin(c + d*x)**4*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + 24*a
**3*sin(c + d*x)**3*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + 36*a**3*sin(c + d*x)**2
*sin(c + d*x)**n/(d*n**4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d) + 24*a**3*sin(c + d*x)*sin(c + d*x)**n/(d*n*
*4 + 10*d*n**3 + 35*d*n**2 + 50*d*n + 24*d), True))
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Giac [A] time = 1.18669, size = 136, normalized size = 1.49 \begin{align*} \frac{\frac{a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4}}{n + 4} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}}{n + 2} + \frac{a^{3} \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))^3,x, algorithm="giac")
[Out]
(a^3*sin(d*x + c)^n*sin(d*x + c)^4/(n + 4) + 3*a^3*sin(d*x + c)^n*sin(d*x + c)^3/(n + 3) + 3*a^3*sin(d*x + c)^
n*sin(d*x + c)^2/(n + 2) + a^3*sin(d*x + c)^(n + 1)/(n + 1))/d