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

Optimal. Leaf size=41 \[ \frac{a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{a \sin ^{n+2}(c+d x)}{d (n+2)} \]

[Out]

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

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Rubi [A]  time = 0.0524414, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2833, 43} \[ \frac{a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{a \sin ^{n+2}(c+d x)}{d (n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.268006, size = 38, normalized size = 0.93 \[ \frac{a \sin ^{n+1}(c+d x) ((n+1) \sin (c+d x)+n+2)}{d (n+1) (n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.553, 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 ) \, 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)),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.71689, size = 139, normalized size = 3.39 \begin{align*} -\frac{{\left ({\left (a n + a\right )} \cos \left (d x + c\right )^{2} - a n -{\left (a n + 2 \, a\right )} \sin \left (d x + c\right ) - a\right )} \sin \left (d x + c\right )^{n}}{d n^{2} + 3 \, d n + 2 \, 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)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.2123, size = 190, normalized size = 4.63 \begin{align*} \begin{cases} x \left (a \sin{\left (c \right )} + a\right ) \sin ^{n}{\left (c \right )} \cos{\left (c \right )} & \text{for}\: d = 0 \\\frac{a \log{\left (\sin{\left (c + d x \right )} \right )}}{d} - \frac{a}{d \sin{\left (c + d x \right )}} & \text{for}\: n = -2 \\\frac{a \log{\left (\sin{\left (c + d x \right )} \right )}}{d} + \frac{a \sin{\left (c + d x \right )}}{d} & \text{for}\: n = -1 \\\frac{a n \sin ^{2}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{2} + 3 d n + 2 d} + \frac{a n \sin{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{2} + 3 d n + 2 d} + \frac{a \sin ^{2}{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{2} + 3 d n + 2 d} + \frac{2 a \sin{\left (c + d x \right )} \sin ^{n}{\left (c + d x \right )}}{d n^{2} + 3 d n + 2 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)),x)

[Out]

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

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Giac [A]  time = 1.25171, size = 61, normalized size = 1.49 \begin{align*} \frac{\frac{a \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}}{n + 2} + \frac{a \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)),x, algorithm="giac")

[Out]

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