3.922 \(\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx\)
Optimal. Leaf size=96 \[ \frac{2 d (c-d) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{d^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{(c-d)^2 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)} \]
[Out]
((c - d)^2*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (2*(c - d)*d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2
+ m)) + (d^2*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m))
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Rubi [A] time = 0.111723, antiderivative size = 96, 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} \[ \frac{2 d (c-d) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{d^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{(c-d)^2 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2,x]
[Out]
((c - d)^2*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (2*(c - d)*d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2
+ m)) + (d^2*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m))
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))^m (c+d \sin (e+f x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^m \left (c+\frac{d x}{a}\right )^2 \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{\operatorname{Subst}\left (\int \left ((c-d)^2 (a+x)^m+\frac{2 (c-d) d (a+x)^{1+m}}{a}+\frac{d^2 (a+x)^{2+m}}{a^2}\right ) \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{(c-d)^2 (a+a \sin (e+f x))^{1+m}}{a f (1+m)}+\frac{2 (c-d) d (a+a \sin (e+f x))^{2+m}}{a^2 f (2+m)}+\frac{d^2 (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)}\\ \end{align*}
Mathematica [A] time = 0.393096, size = 83, normalized size = 0.86 \[ \frac{(a (\sin (e+f x)+1))^{m+1} \left (\frac{2 a^2 d (c-d) (\sin (e+f x)+1)}{m+2}+\frac{a^2 (c-d)^2}{m+1}+\frac{d^2 (a \sin (e+f x)+a)^2}{m+3}\right )}{a^3 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2,x]
[Out]
((a*(1 + Sin[e + f*x]))^(1 + m)*((a^2*(c - d)^2)/(1 + m) + (2*a^2*(c - d)*d*(1 + Sin[e + f*x]))/(2 + m) + (d^2
*(a + a*Sin[e + f*x])^2)/(3 + m)))/(a^3*f)
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Maple [F] time = 2.322, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( fx+e \right ) \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{2}\, 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))^m*(c+d*sin(f*x+e))^2,x)
[Out]
int(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^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(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.27003, size = 427, normalized size = 4.45 \begin{align*} \frac{{\left ({\left (c^{2} + 2 \, c d + d^{2}\right )} m^{2} -{\left ({\left (2 \, c d + d^{2}\right )} m^{2} + 6 \, c d +{\left (8 \, c d + d^{2}\right )} m\right )} \cos \left (f x + e\right )^{2} + 6 \, c^{2} + 2 \, d^{2} +{\left (5 \, c^{2} + 6 \, c d + d^{2}\right )} m +{\left ({\left (c^{2} + 2 \, c d + d^{2}\right )} m^{2} -{\left (d^{2} m^{2} + 3 \, d^{2} m + 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 6 \, c^{2} + 2 \, d^{2} +{\left (5 \, c^{2} + 6 \, c d + d^{2}\right )} m\right )} \sin \left (f x + e\right )\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f m^{3} + 6 \, f m^{2} + 11 \, f m + 6 \, 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))^m*(c+d*sin(f*x+e))^2,x, algorithm="fricas")
[Out]
((c^2 + 2*c*d + d^2)*m^2 - ((2*c*d + d^2)*m^2 + 6*c*d + (8*c*d + d^2)*m)*cos(f*x + e)^2 + 6*c^2 + 2*d^2 + (5*c
^2 + 6*c*d + d^2)*m + ((c^2 + 2*c*d + d^2)*m^2 - (d^2*m^2 + 3*d^2*m + 2*d^2)*cos(f*x + e)^2 + 6*c^2 + 2*d^2 +
(5*c^2 + 6*c*d + d^2)*m)*sin(f*x + e))*(a*sin(f*x + e) + a)^m/(f*m^3 + 6*f*m^2 + 11*f*m + 6*f)
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Sympy [A] time = 28.4177, size = 1686, normalized size = 17.56 \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))**m*(c+d*sin(f*x+e))**2,x)
[Out]
Piecewise((x*(c + d*sin(e))**2*(a*sin(e) + a)**m*cos(e), Eq(f, 0)), (-c**2/(2*a**3*f*sin(e + f*x)**2 + 4*a**3*
f*sin(e + f*x) + 2*a**3*f) - 4*c*d*sin(e + f*x)/(2*a**3*f*sin(e + f*x)**2 + 4*a**3*f*sin(e + f*x) + 2*a**3*f)
- 2*c*d/(2*a**3*f*sin(e + f*x)**2 + 4*a**3*f*sin(e + f*x) + 2*a**3*f) + 2*d**2*log(sin(e + f*x) + 1)*sin(e + f
*x)**2/(2*a**3*f*sin(e + f*x)**2 + 4*a**3*f*sin(e + f*x) + 2*a**3*f) + 4*d**2*log(sin(e + f*x) + 1)*sin(e + f*
x)/(2*a**3*f*sin(e + f*x)**2 + 4*a**3*f*sin(e + f*x) + 2*a**3*f) + 2*d**2*log(sin(e + f*x) + 1)/(2*a**3*f*sin(
e + f*x)**2 + 4*a**3*f*sin(e + f*x) + 2*a**3*f) + 4*d**2*sin(e + f*x)/(2*a**3*f*sin(e + f*x)**2 + 4*a**3*f*sin
(e + f*x) + 2*a**3*f) + 3*d**2/(2*a**3*f*sin(e + f*x)**2 + 4*a**3*f*sin(e + f*x) + 2*a**3*f), Eq(m, -3)), (-c*
*2/(a**2*f*sin(e + f*x) + a**2*f) + 2*c*d*log(sin(e + f*x) + 1)*sin(e + f*x)/(a**2*f*sin(e + f*x) + a**2*f) +
2*c*d*log(sin(e + f*x) + 1)/(a**2*f*sin(e + f*x) + a**2*f) + 2*c*d/(a**2*f*sin(e + f*x) + a**2*f) - 2*d**2*log
(sin(e + f*x) + 1)*sin(e + f*x)/(a**2*f*sin(e + f*x) + a**2*f) - 2*d**2*log(sin(e + f*x) + 1)/(a**2*f*sin(e +
f*x) + a**2*f) - d**2*sin(e + f*x)**3/(a**2*f*sin(e + f*x) + a**2*f) - d**2*sin(e + f*x)*cos(e + f*x)**2/(a**2
*f*sin(e + f*x) + a**2*f) - d**2*cos(e + f*x)**2/(a**2*f*sin(e + f*x) + a**2*f) - 2*d**2/(a**2*f*sin(e + f*x)
+ a**2*f), Eq(m, -2)), (c**2*log(sin(e + f*x) + 1)/(a*f) - 2*c*d*log(sin(e + f*x) + 1)/(a*f) + 2*c*d*sin(e + f
*x)/(a*f) + d**2*log(sin(e + f*x) + 1)/(a*f) - d**2*sin(e + f*x)/(a*f) - d**2*cos(e + f*x)**2/(2*a*f), Eq(m, -
1)), (c**2*m**2*(a*sin(e + f*x) + a)**m*sin(e + f*x)/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) + c**2*m**2*(a*sin(e +
f*x) + a)**m/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) + 5*c**2*m*(a*sin(e + f*x) + a)**m*sin(e + f*x)/(f*m**3 + 6*f
*m**2 + 11*f*m + 6*f) + 5*c**2*m*(a*sin(e + f*x) + a)**m/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) + 6*c**2*(a*sin(e
+ f*x) + a)**m*sin(e + f*x)/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) + 6*c**2*(a*sin(e + f*x) + a)**m/(f*m**3 + 6*f*
m**2 + 11*f*m + 6*f) + 2*c*d*m**2*(a*sin(e + f*x) + a)**m*sin(e + f*x)**2/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) +
2*c*d*m**2*(a*sin(e + f*x) + a)**m*sin(e + f*x)/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) + 8*c*d*m*(a*sin(e + f*x)
+ a)**m*sin(e + f*x)**2/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) + 6*c*d*m*(a*sin(e + f*x) + a)**m*sin(e + f*x)/(f*m
**3 + 6*f*m**2 + 11*f*m + 6*f) - 2*c*d*m*(a*sin(e + f*x) + a)**m/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) + 6*c*d*(a
*sin(e + f*x) + a)**m*sin(e + f*x)**2/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) - 6*c*d*(a*sin(e + f*x) + a)**m/(f*m*
*3 + 6*f*m**2 + 11*f*m + 6*f) + d**2*m**2*(a*sin(e + f*x) + a)**m*sin(e + f*x)**3/(f*m**3 + 6*f*m**2 + 11*f*m
+ 6*f) + d**2*m**2*(a*sin(e + f*x) + a)**m*sin(e + f*x)**2/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) + 3*d**2*m*(a*si
n(e + f*x) + a)**m*sin(e + f*x)**3/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) + d**2*m*(a*sin(e + f*x) + a)**m*sin(e +
f*x)**2/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) - 2*d**2*m*(a*sin(e + f*x) + a)**m*sin(e + f*x)/(f*m**3 + 6*f*m**2
+ 11*f*m + 6*f) + 2*d**2*(a*sin(e + f*x) + a)**m*sin(e + f*x)**3/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f) + 2*d**2*
(a*sin(e + f*x) + a)**m/(f*m**3 + 6*f*m**2 + 11*f*m + 6*f), True))
________________________________________________________________________________________
Giac [B] time = 1.23389, size = 624, normalized size = 6.5 \begin{align*} \frac{\frac{{\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} m^{2} - 2 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a m^{2} +{\left (a \sin \left (f x + e\right ) + a\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a^{2} m^{2} + 3 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} m - 8 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a m + 5 \,{\left (a \sin \left (f x + e\right ) + a\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a^{2} m + 2 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} - 6 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a + 6 \,{\left (a \sin \left (f x + e\right ) + a\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a^{2}\right )} d^{2}}{a^{2} m^{3} + 6 \, a^{2} m^{2} + 11 \, a^{2} m + 6 \, a^{2}} + \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{m + 1} c^{2}}{m + 1} + \frac{2 \,{\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} m -{\left (a \sin \left (f x + e\right ) + a\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a m +{\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} - 2 \,{\left (a \sin \left (f x + e\right ) + a\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a\right )} c d}{{\left (m^{2} + 3 \, m + 2\right )} a}}{a 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))^m*(c+d*sin(f*x+e))^2,x, algorithm="giac")
[Out]
(((a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*m^2 - 2*(a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a*m^2 +
(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^2*m^2 + 3*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*m - 8*(a
*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a*m + 5*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^2*m + 2*(a*s
in(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m - 6*(a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a + 6*(a*sin(f*x +
e) + a)*(a*sin(f*x + e) + a)^m*a^2)*d^2/(a^2*m^3 + 6*a^2*m^2 + 11*a^2*m + 6*a^2) + (a*sin(f*x + e) + a)^(m +
1)*c^2/(m + 1) + 2*((a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*m - (a*sin(f*x + e) + a)*(a*sin(f*x + e) + a
)^m*a*m + (a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m - 2*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a)*c*d
/((m^2 + 3*m + 2)*a))/(a*f)