3.916 \(\int \cos (e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx\)

Optimal. Leaf size=61 \[ \frac{a (c+d \sin (e+f x))^{n+2}}{d^2 f (n+2)}-\frac{a (c-d) (c+d \sin (e+f x))^{n+1}}{d^2 f (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.09559, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2833, 43} \[ \frac{a (c+d \sin (e+f x))^{n+2}}{d^2 f (n+2)}-\frac{a (c-d) (c+d \sin (e+f x))^{n+1}}{d^2 f (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.486797, size = 52, normalized size = 0.85 \[ \frac{a (c+d \sin (e+f x))^{n+1} (-c+d (n+1) \sin (e+f x)+d (n+2))}{d^2 f (n+1) (n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.269, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( fx+e \right ) \left ( a+a\sin \left ( fx+e \right ) \right ) \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))*(c+d*sin(f*x+e))^n,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.94721, size = 250, normalized size = 4.1 \begin{align*} -\frac{{\left (a c^{2} - 2 \, a c d - a d^{2} +{\left (a d^{2} n + a d^{2}\right )} \cos \left (f x + e\right )^{2} -{\left (a c d + a d^{2}\right )} n -{\left (2 \, a d^{2} +{\left (a c d + a d^{2}\right )} n\right )} \sin \left (f x + e\right )\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{d^{2} f n^{2} + 3 \, d^{2} f n + 2 \, d^{2} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))*(c+d*sin(f*x+e))^n,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 10.7143, size = 586, normalized size = 9.61 \begin{align*} \begin{cases} c^{n} \left (\frac{a \sin{\left (e + f x \right )}}{f} - \frac{a \cos ^{2}{\left (e + f x \right )}}{2 f}\right ) & \text{for}\: d = 0 \\x \left (c + d \sin{\left (e \right )}\right )^{n} \left (a \sin{\left (e \right )} + a\right ) \cos{\left (e \right )} & \text{for}\: f = 0 \\\frac{a c \log{\left (\frac{c}{d} + \sin{\left (e + f x \right )} \right )}}{c d^{2} f + d^{3} f \sin{\left (e + f x \right )}} + \frac{a c}{c d^{2} f + d^{3} f \sin{\left (e + f x \right )}} + \frac{a d \log{\left (\frac{c}{d} + \sin{\left (e + f x \right )} \right )} \sin{\left (e + f x \right )}}{c d^{2} f + d^{3} f \sin{\left (e + f x \right )}} - \frac{a d}{c d^{2} f + d^{3} f \sin{\left (e + f x \right )}} & \text{for}\: n = -2 \\- \frac{a c \log{\left (\frac{c}{d} + \sin{\left (e + f x \right )} \right )}}{d^{2} f} + \frac{a \log{\left (\frac{c}{d} + \sin{\left (e + f x \right )} \right )}}{d f} + \frac{a \sin{\left (e + f x \right )}}{d f} & \text{for}\: n = -1 \\- \frac{a c^{2} \left (c + d \sin{\left (e + f x \right )}\right )^{n}}{d^{2} f n^{2} + 3 d^{2} f n + 2 d^{2} f} + \frac{a c d n \left (c + d \sin{\left (e + f x \right )}\right )^{n} \sin{\left (e + f x \right )}}{d^{2} f n^{2} + 3 d^{2} f n + 2 d^{2} f} + \frac{a c d n \left (c + d \sin{\left (e + f x \right )}\right )^{n}}{d^{2} f n^{2} + 3 d^{2} f n + 2 d^{2} f} + \frac{2 a c d \left (c + d \sin{\left (e + f x \right )}\right )^{n}}{d^{2} f n^{2} + 3 d^{2} f n + 2 d^{2} f} + \frac{a d^{2} n \left (c + d \sin{\left (e + f x \right )}\right )^{n} \sin ^{2}{\left (e + f x \right )}}{d^{2} f n^{2} + 3 d^{2} f n + 2 d^{2} f} + \frac{a d^{2} n \left (c + d \sin{\left (e + f x \right )}\right )^{n} \sin{\left (e + f x \right )}}{d^{2} f n^{2} + 3 d^{2} f n + 2 d^{2} f} + \frac{a d^{2} \left (c + d \sin{\left (e + f x \right )}\right )^{n} \sin ^{2}{\left (e + f x \right )}}{d^{2} f n^{2} + 3 d^{2} f n + 2 d^{2} f} + \frac{2 a d^{2} \left (c + d \sin{\left (e + f x \right )}\right )^{n} \sin{\left (e + f x \right )}}{d^{2} f n^{2} + 3 d^{2} f n + 2 d^{2} f} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))*(c+d*sin(f*x+e))**n,x)

[Out]

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

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Giac [B]  time = 1.26225, size = 211, normalized size = 3.46 \begin{align*} \frac{\frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n + 1} a}{n + 1} + \frac{{\left ({\left (d \sin \left (f x + e\right ) + c\right )}^{2}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} n -{\left (d \sin \left (f x + e\right ) + c\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} c n +{\left (d \sin \left (f x + e\right ) + c\right )}^{2}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} - 2 \,{\left (d \sin \left (f x + e\right ) + c\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} c\right )} a}{{\left (n^{2} + 3 \, n + 2\right )} d}}{d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))*(c+d*sin(f*x+e))^n,x, algorithm="giac")

[Out]

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