3.260 \(\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=68 \[ \frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{a^2 \sin ^{n+3}(c+d x)}{d (n+3)} \]

[Out]

(a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a^2*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (a^2*Sin[c + d*x]^(3 + n))
/(d*(3 + n))

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Rubi [A]  time = 0.0862026, antiderivative size = 68, 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} \[ \frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{a^2 \sin ^{n+3}(c+d x)}{d (n+3)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]

[Out]

(a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a^2*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (a^2*Sin[c + d*x]^(3 + n))
/(d*(3 + 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))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{x}{a}\right )^n (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 \left (\frac{x}{a}\right )^n+2 a^2 \left (\frac{x}{a}\right )^{1+n}+a^2 \left (\frac{x}{a}\right )^{2+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^2 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac{2 a^2 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac{a^2 \sin ^{3+n}(c+d x)}{d (3+n)}\\ \end{align*}

Mathematica [A]  time = 0.201845, size = 50, normalized size = 0.74 \[ \frac{a^2 \sin ^{n+1}(c+d x) \left (\frac{\sin ^2(c+d x)}{n+3}+\frac{2 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]

[Out]

(a^2*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (2*Sin[c + d*x])/(2 + n) + Sin[c + d*x]^2/(3 + n)))/d

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Maple [F]  time = 2.069, 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 ) ^{2}\, 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))^2,x)

[Out]

int(cos(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,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))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.744, size = 292, normalized size = 4.29 \begin{align*} \frac{{\left (2 \, a^{2} n^{2} + 8 \, a^{2} n - 2 \,{\left (a^{2} n^{2} + 4 \, a^{2} n + 3 \, a^{2}\right )} \cos \left (d x + c\right )^{2} + 6 \, a^{2} +{\left (2 \, a^{2} n^{2} + 8 \, a^{2} n -{\left (a^{2} n^{2} + 3 \, a^{2} n + 2 \, a^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{3} + 6 \, d n^{2} + 11 \, d n + 6 \, 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))^2,x, algorithm="fricas")

[Out]

(2*a^2*n^2 + 8*a^2*n - 2*(a^2*n^2 + 4*a^2*n + 3*a^2)*cos(d*x + c)^2 + 6*a^2 + (2*a^2*n^2 + 8*a^2*n - (a^2*n^2
+ 3*a^2*n + 2*a^2)*cos(d*x + c)^2 + 8*a^2)*sin(d*x + c))*sin(d*x + c)^n/(d*n^3 + 6*d*n^2 + 11*d*n + 6*d)

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Sympy [A]  time = 18.3099, size = 530, normalized size = 7.79 \begin{align*} \begin{cases} x \left (a \sin{\left (c \right )} + a\right )^{2} \sin ^{n}{\left (c \right )} \cos{\left (c \right )} & \text{for}\: d = 0 \\\frac{a^{2} \log{\left (\sin{\left (c + d x \right )} \right )}}{d} - \frac{2 a^{2}}{d \sin{\left (c + d x \right )}} - \frac{a^{2}}{2 d \sin ^{2}{\left (c + d x \right )}} & \text{for}\: n = -3 \\\frac{2 a^{2} \log{\left (\sin{\left (c + d x \right )} \right )}}{d} + \frac{a^{2} \sin{\left (c + d x \right )}}{d} - \frac{a^{2}}{d \sin{\left (c + d x \right )}} & \text{for}\: n = -2 \\\frac{a^{2} \log{\left (\sin{\left (c + d x \right )} \right )}}{d} + \frac{2 a^{2} \sin{\left (c + d x \right )}}{d} - \frac{a^{2} \cos ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: n = -1 \\\frac{a^{2} n^{2} \sin ^{3}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac{2 a^{2} n^{2} \sin ^{2}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac{a^{2} n^{2} \sin{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac{3 a^{2} n \sin ^{3}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac{8 a^{2} n \sin ^{2}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac{5 a^{2} n \sin{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac{2 a^{2} \sin ^{3}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac{6 a^{2} \sin ^{2}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} + \frac{6 a^{2} \sin{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{3} + 6 d n^{2} + 11 d n + 6 d} & \text{otherwise} \end{cases} \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))**2,x)

[Out]

Piecewise((x*(a*sin(c) + a)**2*sin(c)**n*cos(c), Eq(d, 0)), (a**2*log(sin(c + d*x))/d - 2*a**2/(d*sin(c + d*x)
) - a**2/(2*d*sin(c + d*x)**2), Eq(n, -3)), (2*a**2*log(sin(c + d*x))/d + a**2*sin(c + d*x)/d - a**2/(d*sin(c
+ d*x)), Eq(n, -2)), (a**2*log(sin(c + d*x))/d + 2*a**2*sin(c + d*x)/d - a**2*cos(c + d*x)**2/(2*d), Eq(n, -1)
), (a**2*n**2*sin(c + d*x)**3*sin(c + d*x)**n/(d*n**3 + 6*d*n**2 + 11*d*n + 6*d) + 2*a**2*n**2*sin(c + d*x)**2
*sin(c + d*x)**n/(d*n**3 + 6*d*n**2 + 11*d*n + 6*d) + a**2*n**2*sin(c + d*x)*sin(c + d*x)**n/(d*n**3 + 6*d*n**
2 + 11*d*n + 6*d) + 3*a**2*n*sin(c + d*x)**3*sin(c + d*x)**n/(d*n**3 + 6*d*n**2 + 11*d*n + 6*d) + 8*a**2*n*sin
(c + d*x)**2*sin(c + d*x)**n/(d*n**3 + 6*d*n**2 + 11*d*n + 6*d) + 5*a**2*n*sin(c + d*x)*sin(c + d*x)**n/(d*n**
3 + 6*d*n**2 + 11*d*n + 6*d) + 2*a**2*sin(c + d*x)**3*sin(c + d*x)**n/(d*n**3 + 6*d*n**2 + 11*d*n + 6*d) + 6*a
**2*sin(c + d*x)**2*sin(c + d*x)**n/(d*n**3 + 6*d*n**2 + 11*d*n + 6*d) + 6*a**2*sin(c + d*x)*sin(c + d*x)**n/(
d*n**3 + 6*d*n**2 + 11*d*n + 6*d), True))

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Giac [A]  time = 1.16253, size = 101, normalized size = 1.49 \begin{align*} \frac{\frac{a^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} + \frac{2 \, a^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}}{n + 2} + \frac{a^{2} \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))^2,x, algorithm="giac")

[Out]

(a^2*sin(d*x + c)^n*sin(d*x + c)^3/(n + 3) + 2*a^2*sin(d*x + c)^n*sin(d*x + c)^2/(n + 2) + a^2*sin(d*x + c)^(n
+ 1)/(n + 1))/d