\(\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\) [1025]

Optimal result

Integrand size = 31, antiderivative size = 124 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {B \cos ^3(e+f x) (a+a \sin (e+f x))^m}{f (3+m)}-\frac {2^{\frac {3}{2}+m} (B m+A (3+m)) \cos ^3(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {1}{2}-m,\frac {5}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {3}{2}-m} (a+a \sin (e+f x))^m}{3 f (3+m)} \] Output:

-B*cos(f*x+e)^3*(a+a*sin(f*x+e))^m/f/(3+m)-1/3*2^(3/2+m)*(B*m+A*(3+m))*cos 
(f*x+e)^3*hypergeom([3/2, -1/2-m],[5/2],1/2-1/2*sin(f*x+e))*(1+sin(f*x+e)) 
^(-3/2-m)*(a+a*sin(f*x+e))^m/f/(3+m)
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {\cos ^3(e+f x) (1+\sin (e+f x))^{-\frac {3}{2}-m} (a (1+\sin (e+f x)))^m \left (2^{\frac {3}{2}+m} (B m+A (3+m)) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {1}{2}-m,\frac {5}{2},\frac {1}{2} (1-\sin (e+f x))\right )+3 B (1+\sin (e+f x))^{\frac {3}{2}+m}\right )}{3 f (3+m)} \] Input:

Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]
 

Output:

-1/3*(Cos[e + f*x]^3*(1 + Sin[e + f*x])^(-3/2 - m)*(a*(1 + Sin[e + f*x]))^ 
m*(2^(3/2 + m)*(B*m + A*(3 + m))*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (1 
- Sin[e + f*x])/2] + 3*B*(1 + Sin[e + f*x])^(3/2 + m)))/(f*(3 + m))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3042, 3339, 3042, 3168, 80, 79}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \cos ^2(e+f x) (a \sin (e+f x)+a)^m (A+B \sin (e+f x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (e+f x)^2 (a \sin (e+f x)+a)^m (A+B \sin (e+f x))dx\)

\(\Big \downarrow \) 3339

\(\displaystyle \left (A+\frac {B m}{m+3}\right ) \int \cos ^2(e+f x) (\sin (e+f x) a+a)^mdx-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^m}{f (m+3)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \left (A+\frac {B m}{m+3}\right ) \int \cos (e+f x)^2 (\sin (e+f x) a+a)^mdx-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^m}{f (m+3)}\)

\(\Big \downarrow \) 3168

\(\displaystyle \frac {a^2 \left (A+\frac {B m}{m+3}\right ) \cos ^3(e+f x) \int \sqrt {a-a \sin (e+f x)} (\sin (e+f x) a+a)^{m+\frac {1}{2}}d\sin (e+f x)}{f (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{3/2}}-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^m}{f (m+3)}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {a^2 2^{m+\frac {1}{2}} \left (A+\frac {B m}{m+3}\right ) \cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^{m-1} \int \left (\frac {1}{2} \sin (e+f x)+\frac {1}{2}\right )^{m+\frac {1}{2}} \sqrt {a-a \sin (e+f x)}d\sin (e+f x)}{f (a-a \sin (e+f x))^{3/2}}-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^m}{f (m+3)}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {a 2^{m+\frac {3}{2}} \left (A+\frac {B m}{m+3}\right ) \cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^{m-1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-m-\frac {1}{2},\frac {5}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{3 f}-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^m}{f (m+3)}\)

Input:

Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]
 

Output:

-1/3*(2^(3/2 + m)*a*(A + (B*m)/(3 + m))*Cos[e + f*x]^3*Hypergeometric2F1[3 
/2, -1/2 - m, 5/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a 
+ a*Sin[e + f*x])^(-1 + m))/f - (B*Cos[e + f*x]^3*(a + a*Sin[e + f*x])^m)/ 
(f*(3 + m))
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3168

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.), x_Symbol] :> Simp[a^2*((g*Cos[e + f*x])^(p + 1)/(f*g*(a + b*Sin 
[e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2)))   Subst[Int[(a + 
b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; Fre 
eQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m]
 

rule 3339

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* 
(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S 
imp[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1))   Int[(g*Cos[e + f*x])^p*(a + 
 b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[ 
a^2 - b^2, 0] && NeQ[m + p + 1, 0]
 

Maple [F]

\[\int \cos \left (f x +e \right )^{2} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )d x\]

Input:

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)
 

Output:

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)
 

Fricas [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \] Input:

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="f 
ricas")
 

Output:

integral((B*cos(f*x + e)^2*sin(f*x + e) + A*cos(f*x + e)^2)*(a*sin(f*x + e 
) + a)^m, x)
 

Sympy [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + B \sin {\left (e + f x \right )}\right ) \cos ^{2}{\left (e + f x \right )}\, dx \] Input:

integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**m*(A+B*sin(f*x+e)),x)
 

Output:

Integral((a*(sin(e + f*x) + 1))**m*(A + B*sin(e + f*x))*cos(e + f*x)**2, x 
)
 

Maxima [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \] Input:

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="m 
axima")
 

Output:

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m*cos(f*x + e)^2, x)
 

Giac [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \] Input:

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="g 
iac")
 

Output:

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m*cos(f*x + e)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int {\cos \left (e+f\,x\right )}^2\,\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \] Input:

int(cos(e + f*x)^2*(A + B*sin(e + f*x))*(a + a*sin(e + f*x))^m,x)
 

Output:

int(cos(e + f*x)^2*(A + B*sin(e + f*x))*(a + a*sin(e + f*x))^m, x)
 

Reduce [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )d x \right ) b +\left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \cos \left (f x +e \right )^{2}d x \right ) a \] Input:

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)
 

Output:

int((sin(e + f*x)*a + a)**m*cos(e + f*x)**2*sin(e + f*x),x)*b + int((sin(e 
 + f*x)*a + a)**m*cos(e + f*x)**2,x)*a