\(\int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx\) [362]

Optimal result

Integrand size = 27, antiderivative size = 106 \[ \int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx=-\frac {2^{1+\frac {m}{2}} (e \cos (c+d x))^{2-m} \operatorname {Hypergeometric2F1}\left (\frac {2-m}{2},-\frac {m}{2},\frac {4-m}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{2} (-2-m)} (a+a \sin (c+d x))^m}{d e (2-m)} \] Output:

-2^(1+1/2*m)*(e*cos(d*x+c))^(2-m)*hypergeom([-1/2*m, 1-1/2*m],[2-1/2*m],1/ 
2-1/2*sin(d*x+c))*(1+sin(d*x+c))^(-1-1/2*m)*(a+a*sin(d*x+c))^m/d/e/(2-m)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92 \[ \int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx=\frac {2^{1+\frac {m}{2}} (e \cos (c+d x))^{2-m} \operatorname {Hypergeometric2F1}\left (1-\frac {m}{2},-\frac {m}{2},2-\frac {m}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-1-\frac {m}{2}} (a (1+\sin (c+d x)))^m}{d e (-2+m)} \] Input:

Integrate[(e*Cos[c + d*x])^(1 - m)*(a + a*Sin[c + d*x])^m,x]
 

Output:

(2^(1 + m/2)*(e*Cos[c + d*x])^(2 - m)*Hypergeometric2F1[1 - m/2, -1/2*m, 2 
 - m/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1 - m/2)*(a*(1 + Sin[c 
+ d*x]))^m)/(d*e*(-2 + m))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.27, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {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.

\(\displaystyle \int (a \sin (c+d x)+a)^m (e \cos (c+d x))^{1-m} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (c+d x)+a)^m (e \cos (c+d x))^{1-m}dx\)

\(\Big \downarrow \) 3168

\(\displaystyle \frac {a^2 (a-a \sin (c+d x))^{\frac {m-2}{2}} (a \sin (c+d x)+a)^{\frac {m-2}{2}} (e \cos (c+d x))^{2-m} \int (a-a \sin (c+d x))^{-m/2} (\sin (c+d x) a+a)^{m/2}d\sin (c+d x)}{d e}\)

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {a 2^{1-\frac {m}{2}} (1-\sin (c+d x))^{m/2} (a-a \sin (c+d x))^{\frac {m-2}{2}-\frac {m}{2}} (a \sin (c+d x)+a)^{\frac {m-2}{2}+\frac {m+2}{2}} (e \cos (c+d x))^{2-m} \operatorname {Hypergeometric2F1}\left (\frac {m}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {1}{2} (\sin (c+d x)+1)\right )}{d e (m+2)}\)

Input:

Int[(e*Cos[c + d*x])^(1 - m)*(a + a*Sin[c + d*x])^m,x]
 

Output:

(2^(1 - m/2)*a*(e*Cos[c + d*x])^(2 - m)*Hypergeometric2F1[m/2, (2 + m)/2, 
(4 + m)/2, (1 + Sin[c + d*x])/2]*(1 - Sin[c + d*x])^(m/2)*(a - a*Sin[c + d 
*x])^((-2 + m)/2 - m/2)*(a + a*Sin[c + d*x])^((-2 + m)/2 + (2 + m)/2))/(d* 
e*(2 + 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]
 

Maple [F]

\[\int \left (e \cos \left (d x +c \right )\right )^{1-m} \left (a +a \sin \left (d x +c \right )\right )^{m}d x\]

Input:

int((e*cos(d*x+c))^(1-m)*(a+a*sin(d*x+c))^m,x)
 

Output:

int((e*cos(d*x+c))^(1-m)*(a+a*sin(d*x+c))^m,x)
 

Fricas [F]

\[ \int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{-m + 1} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \] Input:

integrate((e*cos(d*x+c))^(1-m)*(a+a*sin(d*x+c))^m,x, algorithm="fricas")
 

Output:

integral((e*cos(d*x + c))^(-m + 1)*(a*sin(d*x + c) + a)^m, x)
 

Sympy [F]

\[ \int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \left (e \cos {\left (c + d x \right )}\right )^{1 - m}\, dx \] Input:

integrate((e*cos(d*x+c))**(1-m)*(a+a*sin(d*x+c))**m,x)
 

Output:

Integral((a*(sin(c + d*x) + 1))**m*(e*cos(c + d*x))**(1 - m), x)
 

Maxima [F]

\[ \int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{-m + 1} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \] Input:

integrate((e*cos(d*x+c))^(1-m)*(a+a*sin(d*x+c))^m,x, algorithm="maxima")
 

Output:

integrate((e*cos(d*x + c))^(-m + 1)*(a*sin(d*x + c) + a)^m, x)
 

Giac [F]

\[ \int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{-m + 1} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \] Input:

integrate((e*cos(d*x+c))^(1-m)*(a+a*sin(d*x+c))^m,x, algorithm="giac")
 

Output:

integrate((e*cos(d*x + c))^(-m + 1)*(a*sin(d*x + c) + a)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{1-m}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m \,d x \] Input:

int((e*cos(c + d*x))^(1 - m)*(a + a*sin(c + d*x))^m,x)
 

Output:

int((e*cos(c + d*x))^(1 - m)*(a + a*sin(c + d*x))^m, x)
 

Reduce [F]

\[ \int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx=\frac {\left (\int \frac {\left (\sin \left (d x +c \right ) a +a \right )^{m} \cos \left (d x +c \right )}{\cos \left (d x +c \right )^{m}}d x \right ) e}{e^{m}} \] Input:

int((e*cos(d*x+c))^(1-m)*(a+a*sin(d*x+c))^m,x)
 

Output:

(int(((sin(c + d*x)*a + a)**m*cos(c + d*x))/cos(c + d*x)**m,x)*e)/e**m