\(\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx\) [173]

Optimal result

Integrand size = 36, antiderivative size = 132 \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\frac {2^{\frac {1}{2}+n+\frac {p}{2}} (g \cos (e+f x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 n-p),\frac {1}{2} (1+2 m+p),\frac {1}{2} (3+2 m+p),\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac {1}{2} (-1-2 n-p)} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n}{f g (1+2 m+p)} \] Output:

2^(1/2+n+1/2*p)*(g*cos(f*x+e))^(p+1)*hypergeom([1/2-n-1/2*p, 1/2+m+1/2*p], 
[3/2+m+1/2*p],1/2+1/2*sin(f*x+e))*(1-sin(f*x+e))^(-1/2-n-1/2*p)*(a+a*sin(f 
*x+e))^m*(c-c*sin(f*x+e))^n/f/g/(1+2*m+p)
 

Mathematica [F]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx \] Input:

Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n 
,x]
 

Output:

Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n 
, x]
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.45, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3332, 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.

\(\displaystyle \int (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^p \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^pdx\)

\(\Big \downarrow \) 3332

\(\displaystyle (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m} \int (g \cos (e+f x))^{2 m+p} (c-c \sin (e+f x))^{n-m}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m} \int (g \cos (e+f x))^{2 m+p} (c-c \sin (e+f x))^{n-m}dx\)

\(\Big \downarrow \) 3168

\(\displaystyle \frac {c^2 (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1} (c-c \sin (e+f x))^{\frac {1}{2} (-2 m-p-1)+m} (c \sin (e+f x)+c)^{\frac {1}{2} (-2 m-p-1)} \int (c-c \sin (e+f x))^{\frac {1}{2} (2 n+p-1)} (\sin (e+f x) c+c)^{\frac {1}{2} (2 m+p-1)}d\sin (e+f x)}{f g}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {c^2 2^{\frac {1}{2} (2 n+p-1)} (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1} (1-\sin (e+f x))^{\frac {1}{2} (-2 n-p+1)} (c \sin (e+f x)+c)^{\frac {1}{2} (-2 m-p-1)} (c-c \sin (e+f x))^{\frac {1}{2} (-2 m-p-1)+m+\frac {1}{2} (2 n+p-1)} \int \left (\frac {1}{2}-\frac {1}{2} \sin (e+f x)\right )^{\frac {1}{2} (2 n+p-1)} (\sin (e+f x) c+c)^{\frac {1}{2} (2 m+p-1)}d\sin (e+f x)}{f g}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {c 2^{\frac {1}{2} (2 n+p-1)+1} (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1} (1-\sin (e+f x))^{\frac {1}{2} (-2 n-p+1)} (c \sin (e+f x)+c)^{\frac {1}{2} (-2 m-p-1)+\frac {1}{2} (2 m+p+1)} (c-c \sin (e+f x))^{\frac {1}{2} (-2 m-p-1)+m+\frac {1}{2} (2 n+p-1)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 n-p+1),\frac {1}{2} (2 m+p+1),\frac {1}{2} (2 m+p+3),\frac {1}{2} (\sin (e+f x)+1)\right )}{f g (2 m+p+1)}\)

Input:

Int[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]
 

Output:

(2^(1 + (-1 + 2*n + p)/2)*c*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1 
- 2*n - p)/2, (1 + 2*m + p)/2, (3 + 2*m + p)/2, (1 + Sin[e + f*x])/2]*(1 - 
 Sin[e + f*x])^((1 - 2*n - p)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x 
])^(m + (-1 - 2*m - p)/2 + (-1 + 2*n + p)/2)*(c + c*Sin[e + f*x])^((-1 - 2 
*m - p)/2 + (1 + 2*m + p)/2))/(f*g*(1 + 2*m + p))
 

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 3332

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

Maple [F]

\[\int \left (g \cos \left (f x +e \right )\right )^{p} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{n}d x\]

Input:

int((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)
 

Output:

int((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)
 

Fricas [F]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \] Input:

integrate((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algori 
thm="fricas")
 

Output:

integral((g*cos(f*x + e))^p*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^n 
, x)
 

Sympy [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\text {Timed out} \] Input:

integrate((g*cos(f*x+e))**p*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**n,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \] Input:

integrate((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algori 
thm="maxima")
 

Output:

integrate((g*cos(f*x + e))^p*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^ 
n, x)
 

Giac [F]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \] Input:

integrate((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algori 
thm="giac")
 

Output:

integrate((g*cos(f*x + e))^p*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^ 
n, x)
 

Mupad [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^p\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^n \,d x \] Input:

int((g*cos(e + f*x))^p*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n,x)
 

Output:

int((g*cos(e + f*x))^p*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x)
 

Reduce [F]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx=g^{p} \left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (-\sin \left (f x +e \right ) c +c \right )^{n} \cos \left (f x +e \right )^{p}d x \right ) \] Input:

int((g*cos(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)
 

Output:

g**p*int((sin(e + f*x)*a + a)**m*( - sin(e + f*x)*c + c)**n*cos(e + f*x)** 
p,x)