3.915 \(\int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx\)
Optimal. Leaf size=101 \[ \frac{a^2 (c-d)^2 (c+d \sin (e+f x))^{n+1}}{d^3 f (n+1)}-\frac{2 a^2 (c-d) (c+d \sin (e+f x))^{n+2}}{d^3 f (n+2)}+\frac{a^2 (c+d \sin (e+f x))^{n+3}}{d^3 f (n+3)} \]
[Out]
(a^2*(c - d)^2*(c + d*Sin[e + f*x])^(1 + n))/(d^3*f*(1 + n)) - (2*a^2*(c - d)*(c + d*Sin[e + f*x])^(2 + n))/(d
^3*f*(2 + n)) + (a^2*(c + d*Sin[e + f*x])^(3 + n))/(d^3*f*(3 + n))
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Rubi [A] time = 0.149943, antiderivative size = 101, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used =
{2833, 43} \[ \frac{a^2 (c-d)^2 (c+d \sin (e+f x))^{n+1}}{d^3 f (n+1)}-\frac{2 a^2 (c-d) (c+d \sin (e+f x))^{n+2}}{d^3 f (n+2)}+\frac{a^2 (c+d \sin (e+f x))^{n+3}}{d^3 f (n+3)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]
[Out]
(a^2*(c - d)^2*(c + d*Sin[e + f*x])^(1 + n))/(d^3*f*(1 + n)) - (2*a^2*(c - d)*(c + d*Sin[e + f*x])^(2 + n))/(d
^3*f*(2 + n)) + (a^2*(c + d*Sin[e + f*x])^(3 + n))/(d^3*f*(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 (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^2 \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^2 (c-d)^2 \left (c+\frac{d x}{a}\right )^n}{d^2}-\frac{2 a^2 (c-d) \left (c+\frac{d x}{a}\right )^{1+n}}{d^2}+\frac{a^2 \left (c+\frac{d x}{a}\right )^{2+n}}{d^2}\right ) \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{a^2 (c-d)^2 (c+d \sin (e+f x))^{1+n}}{d^3 f (1+n)}-\frac{2 a^2 (c-d) (c+d \sin (e+f x))^{2+n}}{d^3 f (2+n)}+\frac{a^2 (c+d \sin (e+f x))^{3+n}}{d^3 f (3+n)}\\ \end{align*}
Mathematica [A] time = 0.359379, size = 78, normalized size = 0.77 \[ \frac{a^2 (c+d \sin (e+f x))^{n+1} \left (-\frac{2 (c-d) (c+d \sin (e+f x))}{n+2}+\frac{(c+d \sin (e+f x))^2}{n+3}+\frac{(c-d)^2}{n+1}\right )}{d^3 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]
[Out]
(a^2*(c + d*Sin[e + f*x])^(1 + n)*((c - d)^2/(1 + n) - (2*(c - d)*(c + d*Sin[e + f*x]))/(2 + n) + (c + d*Sin[e
+ f*x])^2/(3 + n)))/(d^3*f)
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Maple [F] time = 0.466, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( fx+e \right ) \left ( a+a\sin \left ( fx+e \right ) \right ) ^{2} \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))^2*(c+d*sin(f*x+e))^n,x)
[Out]
int(cos(f*x+e)*(a+a*sin(f*x+e))^2*(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))^2*(c+d*sin(f*x+e))^n,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.01878, size = 599, normalized size = 5.93 \begin{align*} \frac{{\left (2 \, a^{2} c^{3} - 6 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} + 6 \, a^{2} d^{3} + 2 \,{\left (a^{2} c d^{2} + a^{2} d^{3}\right )} n^{2} -{\left (6 \, a^{2} d^{3} +{\left (a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} n^{2} +{\left (a^{2} c d^{2} + 8 \, a^{2} d^{3}\right )} n\right )} \cos \left (f x + e\right )^{2} - 2 \,{\left (a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 4 \, a^{2} d^{3}\right )} n +{\left (8 \, a^{2} d^{3} + 2 \,{\left (a^{2} c d^{2} + a^{2} d^{3}\right )} n^{2} -{\left (a^{2} d^{3} n^{2} + 3 \, a^{2} d^{3} n + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} - 2 \,{\left (a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 4 \, a^{2} d^{3}\right )} n\right )} \sin \left (f x + e\right )\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{d^{3} f n^{3} + 6 \, d^{3} f n^{2} + 11 \, d^{3} f n + 6 \, d^{3} 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))^2*(c+d*sin(f*x+e))^n,x, algorithm="fricas")
[Out]
(2*a^2*c^3 - 6*a^2*c^2*d + 6*a^2*c*d^2 + 6*a^2*d^3 + 2*(a^2*c*d^2 + a^2*d^3)*n^2 - (6*a^2*d^3 + (a^2*c*d^2 + 2
*a^2*d^3)*n^2 + (a^2*c*d^2 + 8*a^2*d^3)*n)*cos(f*x + e)^2 - 2*(a^2*c^2*d - 3*a^2*c*d^2 - 4*a^2*d^3)*n + (8*a^2
*d^3 + 2*(a^2*c*d^2 + a^2*d^3)*n^2 - (a^2*d^3*n^2 + 3*a^2*d^3*n + 2*a^2*d^3)*cos(f*x + e)^2 - 2*(a^2*c^2*d - 3
*a^2*c*d^2 - 4*a^2*d^3)*n)*sin(f*x + e))*(d*sin(f*x + e) + c)^n/(d^3*f*n^3 + 6*d^3*f*n^2 + 11*d^3*f*n + 6*d^3*
f)
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Sympy [A] time = 29.7327, size = 2278, normalized size = 22.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(f*x+e)*(a+a*sin(f*x+e))**2*(c+d*sin(f*x+e))**n,x)
[Out]
Piecewise((c**n*(a**2*sin(e + f*x)**3/(3*f) + a**2*sin(e + f*x)/f - a**2*cos(e + f*x)**2/f), Eq(d, 0)), (x*(c
+ d*sin(e))**n*(a*sin(e) + a)**2*cos(e), Eq(f, 0)), (2*a**2*c**2*log(c/d + sin(e + f*x))/(2*c**2*d**3*f + 4*c*
d**4*f*sin(e + f*x) + 2*d**5*f*sin(e + f*x)**2) + 3*a**2*c**2/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**
5*f*sin(e + f*x)**2) + 4*a**2*c*d*log(c/d + sin(e + f*x))*sin(e + f*x)/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x
) + 2*d**5*f*sin(e + f*x)**2) + 4*a**2*c*d*sin(e + f*x)/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**5*f*si
n(e + f*x)**2) - 2*a**2*c*d/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**5*f*sin(e + f*x)**2) + 2*a**2*d**2
*log(c/d + sin(e + f*x))*sin(e + f*x)**2/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**5*f*sin(e + f*x)**2)
- 4*a**2*d**2*sin(e + f*x)/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**5*f*sin(e + f*x)**2) - a**2*d**2/(2
*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**5*f*sin(e + f*x)**2), Eq(n, -3)), (-2*a**2*c**3*log(c/d + sin(e
+ f*x))/(c**2*d**3*f + c*d**4*f*sin(e + f*x)) - 2*a**2*c**3/(c**2*d**3*f + c*d**4*f*sin(e + f*x)) - 2*a**2*c**
2*d*log(c/d + sin(e + f*x))*sin(e + f*x)/(c**2*d**3*f + c*d**4*f*sin(e + f*x)) + 2*a**2*c**2*d*log(c/d + sin(e
+ f*x))/(c**2*d**3*f + c*d**4*f*sin(e + f*x)) + 2*a**2*c**2*d/(c**2*d**3*f + c*d**4*f*sin(e + f*x)) + 2*a**2*
c*d**2*log(c/d + sin(e + f*x))*sin(e + f*x)/(c**2*d**3*f + c*d**4*f*sin(e + f*x)) - a**2*c*d**2*cos(e + f*x)**
2/(c**2*d**3*f + c*d**4*f*sin(e + f*x)) - a**2*c*d**2/(c**2*d**3*f + c*d**4*f*sin(e + f*x)) - a**2*d**3*sin(e
+ f*x)**3/(c**2*d**3*f + c*d**4*f*sin(e + f*x)) - a**2*d**3*sin(e + f*x)*cos(e + f*x)**2/(c**2*d**3*f + c*d**4
*f*sin(e + f*x)), Eq(n, -2)), (a**2*c**2*log(c/d + sin(e + f*x))/(d**3*f) - 2*a**2*c*log(c/d + sin(e + f*x))/(
d**2*f) - a**2*c*sin(e + f*x)/(d**2*f) + a**2*log(c/d + sin(e + f*x))/(d*f) + 2*a**2*sin(e + f*x)/(d*f) - a**2
*cos(e + f*x)**2/(2*d*f), Eq(n, -1)), (2*a**2*c**3*(c + d*sin(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d
**3*f*n + 6*d**3*f) - 2*a**2*c**2*d*n*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d
**3*f*n + 6*d**3*f) - 2*a**2*c**2*d*n*(c + d*sin(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d
**3*f) - 6*a**2*c**2*d*(c + d*sin(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + a**2*c
*d**2*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 2*
a**2*c*d**2*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) +
a**2*c*d**2*n**2*(c + d*sin(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + a**2*c*d**2
*n*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 6*a**2*c*d
**2*n*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 5*a**2*c*d
**2*n*(c + d*sin(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 6*a**2*c*d**2*(c + d*si
n(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + a**2*d**3*n**2*(c + d*sin(e + f*x))**n
*sin(e + f*x)**3/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 2*a**2*d**3*n**2*(c + d*sin(e + f*x)
)**n*sin(e + f*x)**2/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + a**2*d**3*n**2*(c + d*sin(e + f*
x))**n*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 3*a**2*d**3*n*(c + d*sin(e + f*x)
)**n*sin(e + f*x)**3/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 8*a**2*d**3*n*(c + d*sin(e + f*x
))**n*sin(e + f*x)**2/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 5*a**2*d**3*n*(c + d*sin(e + f*
x))**n*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 2*a**2*d**3*(c + d*sin(e + f*x))*
*n*sin(e + f*x)**3/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 6*a**2*d**3*(c + d*sin(e + f*x))**
n*sin(e + f*x)**2/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 6*a**2*d**3*(c + d*sin(e + f*x))**n
*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f), True))
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Giac [B] time = 1.13679, size = 625, normalized size = 6.19 \begin{align*} \frac{\frac{{\left ({\left (d \sin \left (f x + e\right ) + c\right )}^{3}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} n^{2} - 2 \,{\left (d \sin \left (f x + e\right ) + c\right )}^{2}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} c n^{2} +{\left (d \sin \left (f x + e\right ) + c\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} c^{2} n^{2} + 3 \,{\left (d \sin \left (f x + e\right ) + c\right )}^{3}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} n - 8 \,{\left (d \sin \left (f x + e\right ) + c\right )}^{2}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} c n + 5 \,{\left (d \sin \left (f x + e\right ) + c\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} c^{2} n + 2 \,{\left (d \sin \left (f x + e\right ) + c\right )}^{3}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} - 6 \,{\left (d \sin \left (f x + e\right ) + c\right )}^{2}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} c + 6 \,{\left (d \sin \left (f x + e\right ) + c\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} c^{2}\right )} a^{2}}{d^{2} n^{3} + 6 \, d^{2} n^{2} + 11 \, d^{2} n + 6 \, d^{2}} + \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n + 1} a^{2}}{n + 1} + \frac{2 \,{\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^{2}}{{\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))^2*(c+d*sin(f*x+e))^n,x, algorithm="giac")
[Out]
(((d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*n^2 - 2*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c*n^2 +
(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^2*n^2 + 3*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*n - 8*(d
*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c*n + 5*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^2*n + 2*(d*s
in(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n - 6*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c + 6*(d*sin(f*x +
e) + c)*(d*sin(f*x + e) + c)^n*c^2)*a^2/(d^2*n^3 + 6*d^2*n^2 + 11*d^2*n + 6*d^2) + (d*sin(f*x + e) + c)^(n +
1)*a^2/(n + 1) + 2*((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^2
/((n^2 + 3*n + 2)*d))/(d*f)